violin pedagogy and the physics of the bowed string
TRANSCRIPT
Violin Pedagogy and the Physics of the Bowed String
by
Alexander Rhodes McLeod
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Musical Arts
Faculty of MusicUniversity of Toronto
© Copyright by Alexander Rhodes McLeod 2014
Violin Pedagogy and the Physics of the Bowed String
Alexander Rhodes McLeod
Doctor of Musical Arts
Faculty of MusicUniversity of Toronto
Abstract
The paper describes the mechanics of violin tone production using non-specialist language, in
order to present a scientific understanding of tone production accessible to a broad readership. As
well as offering an objective understanding of tone production, this model provides a powerful
tool for analyzing the technique of string playing.
The interaction between the bow and the string is quite complex. Literature reviewed for this
study reveals that scientific investigations have provided important insights into the mechanics of
string playing, offering explanations for factors which both contribute to and limit the range of
tone colours and dynamics that stringed instruments can produce. Also examined in the
literature review are significant works of twentieth century violin pedagogy exploring tone
production on the violin, based on the practical experience of generations of teachers and
performers.
Hermann von Helmholtz described the stick-slip cycle which drives the string in 1863, which
replaced earlier ideas about the vibration of violin strings. Later, scientists such as John
Schelleng and Lothar Cremer were able to demonstrate how the mechanics of the bow-string
interaction can create different tone colours. Recent research by Anders Askenfelt, Knut
Guettler, and Erwin Schoonderwaldt have continued to refine earlier research in this area.
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The writings of Lucien Capet, Leopold Auer, Carl Flesch, Paul Rolland, Kató Havas, Ivan
Galamian, and Simon Fischer are examined and analyzed. Each author describes a different
approach to tone production on the violin, representing a different understanding of the
underlying mechanism. Analyzing these writings within the context of a scientific understanding
of tone production makes it possible to compare these approaches more consistently, and to
synthesize different concepts drawn from the diverse sources evaluated.
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Acknowledgements
I would like to sincerely thank Dr. Cameron Walter, Dr. Robin Elliott, and Prof. Annalee
Patipatanakoon, each of whom offered many hours of patient help and instruction. The main
chapter of this paper could never have come together without the help of Dr. Stephen Morris,
who patiently walked me through many aspects of the physics which eluded or confused me, or
Jason Leung, who challenged and helped me to further expand my research.
To Steven Dann, my most recent viola teacher, go many thanks for the inspiration to continue
developing as a violist, and for the patience with my divided attention, and to Eric Nowlin, for
the opportunity to put some of my ideas into practice in the teaching studio and the guidance
which helped me to do so. I would also like to thank Katherine Rapoport, without whom I would
never have become a musician, and whose inspired approach to technique set me on the path that
ended in this paper, and Johannes Lüthy, who saw me through the most difficult time in my life,
both musically and personally, with great kindness and patience. I hope that my own teaching
will follow the wonderful examples set for me by each of these four great teachers.
My quartet-mates, Sarah Steeves, Linnea Thacker, and Alexa Wilks, who have made playing
such a pleasure these last four years, will always have my gratitude for helping me through many
setbacks, listening to me rant and rave about the difficulties of writing, and lightening many days
with laughter and great music.
I would also like to thank my parents, Norman and Elaine McLeod, who have seen me through
many difficult times and have always supported me both emotionally and financially, and my
grandparents, who also offered much financial support and created a family in which music,
physics, and teaching seemed like natural companions.
Finally, I would like to profusely thank my loving and patient wife Victoria Leigh who
represented an unflagging and invaluable source of support and inspiration during the writing of
this dissertation, and my children, Ryleigh and Oliver, who literally give me a reason to get out
of bed every morning.
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Table of Contents
Introduction................................................................................................................................1
Chapter 1: Literature Review.....................................................................................................4
1 Helmholtz and the Founding of Modern Acoustics...........................................................5
2 Capet and Auer: Ending the 19th Century.........................................................................8
Flesch and Trendelenburg: Launching the 20th Century....................................................10
Pedagogy in the 1960s: Havas, Rolland and Galamian......................................................12
Schelleng, Cremer and the New Science of Violin Physics................................................14
Experimental Developments After Cremer.........................................................................20
Contemporary Pedagogy: Simon Fischer's Basics..............................................................22
Revisiting McIntyre and Schelleng.....................................................................................23
Physics in the Classroom: Cheri Collins.............................................................................25
Chapter 2: Physics for Violinists.............................................................................................26
Note, Tone, and Overtone: Understanding Sound Waves..................................................28
Underlying Forces: Tension, Friction, and Bow Force.......................................................32
1 Tension and Restoring Force...........................................................................................32
2 Static and Kinetic Friction...............................................................................................36
3 Bow Force and Bow Speed..............................................................................................38
The String in Action: Helmholtz Motion............................................................................40
1 Initiating the Stick-Slip Cycle..........................................................................................40
2 The Slip Phase..................................................................................................................41
3 The Stick Phase................................................................................................................43
The Ideal Limits of Helmholtz Motion...............................................................................44
The Rounded Corner: Tone Colour Explained...................................................................47
Friction, Contact Point and Timbre.....................................................................................49
The Bow Hair......................................................................................................................50
Other Factors Affecting the Tone.......................................................................................53
1 String Length and Timbre................................................................................................53
2 The Fingers of the Left Hand...........................................................................................54
The Bow Stick.....................................................................................................................54
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Putting it all Together..........................................................................................................57
Chapter 3: Analysis of Twentieth-Century Pedagogical Works..............................................61
Capet...................................................................................................................................63
Auer.....................................................................................................................................66
Flesch..................................................................................................................................68
Havas...................................................................................................................................71
Rolland................................................................................................................................75
Galamian.............................................................................................................................80
Fischer.................................................................................................................................84
Conclusion...............................................................................................................................89
Ideas for Future Research....................................................................................................90
Bibliography............................................................................................................................93
Appendix: Exercises and Examples from Pedagogical Sources..............................................97
I Capet.................................................................................................................................97
II Auer.................................................................................................................................99
III Flesch.............................................................................................................................99
IV Havas............................................................................................................................100
V Rolland..........................................................................................................................102
VI Galamian......................................................................................................................102
VII Fischer........................................................................................................................104
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List of Tables
Table 1: Bow Force, Contact Point, and Bow Speed...............................................................58
Table 2: Secondary Factors......................................................................................................59
Table 3: Effects of changing tone-producing factors...............................................................60
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List of Figures
Figure 1. The Schelleng Diagram..................................................................................................15
Figure 2. A Sine Wave...................................................................................................................29
Figure 3. Strength of Ideal Harmonic Series..................................................................................30
Figure 4. Tension in the Resting String.........................................................................................33
Figure 5. The Stretched String.......................................................................................................33
Figure 6. Restoring Forces in the Stretched String........................................................................33
Figure 7. Component Forces 1.......................................................................................................34
Figure 8. Component Forces 2.......................................................................................................34
Figure 9. Adding the Component Forces.......................................................................................34
Figure 10. Angles in the String......................................................................................................35
Figure 11. Friction and the Normal Force......................................................................................37
Figure 12: Static and Kinetic Friction............................................................................................37
Figure 13. Restoring Force and the Stretched String.....................................................................40
Figure 14. The Corner Slips...........................................................................................................42
Figure 15. Motion in the String......................................................................................................42
Figure 16. The Slip Phase and the Bridge......................................................................................43
Figure 17. The Stick Phase............................................................................................................43
Figure 18. The Corner Reflects Off the Nut..................................................................................44
Figure 19. Double-Slip Motion......................................................................................................45
Figure 20. Simplified Schelleng Diagram......................................................................................45
Figure 21: Idealized Envelope of String Vibration........................................................................46
Figure 22. The Sawtoothed Wave..................................................................................................48
Figure 23. A Rounded Corner........................................................................................................48
Figure 24. The Bow Reinforcing the Helmholtz Corner...............................................................48
Figure 25. A Miniature Corner......................................................................................................51
Figure 26. Corner Rounding by the Bow.......................................................................................52
Figure 27. Rolland's “Zones”.........................................................................................................77
Figure 28. Fischer's “Soundpoints”................................................................................................85
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Glossary
Bow-bridge distance. The distance between the bridge and the bow. See Contact point.
Bow force. The total vertical force exerted by the bow on the string. The bow force affects thenormal force, and thus the force of friction between the bow and the string. The bow forceincludes the bow pressure, as well as the weight of the bow which presses on the string. Thisis distinct from the term “bow pressure”, which some teachers use to describe the forceexerted by the player on the bow itself.
Bow pressure. Describes the force exerted by the player on the bow. When the string touchesclose to the tip of the bow, the player must exert more bow pressure than at the frog, tomaintain the same bow force.
Bow Speed. The speed at which the bow moves along the surface of the string, perpendicular tothe string's length. Most scientific sources will refer to the “bow velocity”, which includesthe direction of the bowing motion but is otherwise constant.
Contact point. Refers to the place that the bow hair touches the string. An exact point of contactwould be difficult to define, since real bows contact the string with a ribbon of hair thatvaries in width based on how tilted the bow is. Some sources refer instead to the “bow-bridgedistance”, which is another way of describing the contact point. Studies which have measuredthe bow-bridge distance have used various methods to do so, such as embedding a wire in thehair of the bow and measuring the position of the bow through the distance between the wireand the bridge.
Damping. Scientists discuss the violin in terms of the vibrational energy that is transmittedthrough the string, the bridge, and into the body of the instrument. Anything which absorbsenergy from this system can be thought of as having a “damping” effect, including thefingers of the left hand, or the energy which is lost in the string itself.
Harmonic. The frequency content of a sound is said to be harmonic when the partial frequenciesform a mathematically predictable series, each of which is a whole number multiple of asingle “fundamental” frequency. Such a set of frequencies can be referred to as a “harmonicseries”, and any partial frequency which is part of such a series can be referred to as aharmonic partial of its fundamental.
Overtone. Any partial frequency other than the fundamental. Many scientists prefer the term“partial”, because the term “overtone” was originally used to refer only to upper partials, butbecause so many musicians use “overtone” to refer to any partial frequency, this paper willuse overtone and partial interchangeably.
Restoring force. The force which pulls the string back towards equilibrium, when it is stretched.
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Spectral Analysis, Spectral Content. “Spectral analysis” is a way of determining the frequencyand amplitude of each partial component in a complex sound wave. The “spectral content” ofa sound refers to the sum total of all of its frequency components, as determined by spectralanalysis. Real sounds, even real musical sounds, often contain non-harmonic partialcomponents, which would still be considered part of the “spectrum” of that sound.
Technique. The physical methods used by a musician to play their instrument. Technique in thissense is a form of embodied knowledge. A player's understanding of technique informs thisonly indirectly, through practice and performance.
Tone. Musicians use the word tone in a variety of ways. In this paper it is to refer to the soundproduced by a player or an instrument. A player can be said to have a good tone if theyreliably produce pleasing sounds, or a bad tone if they reliably produce displeasing sounds.For the most part, “tone” and “tone production” can be understood as synonymous with“sound” and “sound production”.
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IntroductionThe physics of violin tone production are much better understood today than they were one hundred
years ago. Building on the work of Hermann von Helmholtz, who effectively founded modern
acoustics in the nineteenth century, physicists have developed a working model of the behaviour of the
bowed string that explains the variety of dynamics, tone colours, and effects it is possible to produce on
a stringed instrument. Approaches to violin technique have also changed over time, arguably improving
and being refined with each generation of players, and teachers have developed their own language to
describe these same phenomena. Playing the violin is, after all, a very direct way to explore the
behaviour of the bowed string and to observe its effect on the resulting sound. Using very different
methods, and describing their results in very different languages, the two disciplines have investigated
some of the same questions over the years:
• How does the action of the bow produce sound?
• Under what conditions will the bow and string produce a stable tone?
• What are the limits on violin timbre and loudness, and what determines these limits?
• What are the physical factors which affect tone quality and timbre?
• To what extent can these factors be controlled by the player?
Violinists and physicists have answered many of these questions to their satisfaction, each in their own
way. The violinist seeks to understand how best to manipulate the instrument, in order to achieve
certain sonic results, while the physicist wants to understand how and why the instrument does what it
does. The violinist is like the race-car driver who wants to understand how best to drive the car, and the
physicist the mechanic who wants to understand how the engine works. Few violinists have the
scientific background necessary to understand the work done by physicists, and few scientists play the
violin well enough to explain the implications of their work to string players, so they have worked in
parallel and only occasionally communicated. Although a large and sophisticated body of scientific
work on stringed instruments has existed for decades, its impact on violin teaching has been minimal,
partly because most string players lack the scientific background which would make this work
accessible to them. This paper will attempt to bridge that gap, to make some of the fruits of scientific
investigation available to teachers and practitioners of violin performance.
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In the twentieth century, violin pedagogy and violin physics remained largely separate endeavours.
Few violinists were aware of the scientific literature on the violin and few if any scientists were
involved with the pedagogical community. The first chapter will review the most significant written
works of the physics of tone production and of violin pedagogy, and outline the parallel development
of the two fields of research.
The second part of this paper will explore the physics of tone production, and develop a simplified
model using terms that any violinist can understand. This model will allow string players to more
accurately and efficiently understand the interaction between the string and the bow, and to better
understand the basis of their technique. The advantages of the scientific method include its ruthless
efficiency and its rigour. Whereas performers and teachers have developed a variety of tools to acquire
control over tone production, and a language to explain those tools to each other, physicists have
worked to develop a single model which describes the workings of the bowed string. This model is by
no means complete, as far as physicists are concerned, but it does offer explanations for the phenomena
which most concern players. Unfortunately, the mathematical language used by physicists to describe
this model is well beyond the understanding of most musicians. By translating this work into common
language, this section of the paper will aim to make the benefits of scientific research available to
players, teachers, and students of violin performance.
Violinists themselves, of course, have developed their own sophisticated understanding of tone
production through intrepid physical experimentation. For the violinist, the advantages of practical
investigation are obvious. The tonal possibilities of the instrument are revealed through the long and
arduous practice which is already necessary to master the violin, and through the struggle to explore
and understand countless varied and unique pieces of music. Over the course of the twentieth century,
ideas about how to produce a good tone on the instrument, and how to teach people to do so, have
developed significantly. From the work of Lucien Capet, widely considered the father of modern bow
technique, to the work of Simon Fischer, one of today's most widely celebrated pedagogues, the third
section of the paper will explore the twentieth century's most significant written works on violin
technique. Each work describes a different approach to tone production, and a different understanding
of the mechanism which underlies it. The authors of these works developed their ideas the same way
that modern players do, inheriting some of them from their teachers and peers, discovering some
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through exercises and musical problems, hearing others in masterclasses and coachings, and inventing
the rest themselves. Understanding the roots of various technical ideas can help players to better
understand their own technique and the logic which underlies it, but this type of analysis is only at its
best when grounded in objective fact. The teachers whose works are represented here all display some
understanding of physics, but few make explicit reference to scientific texts, and some misstate or
distort the information they do present. By analyzing each teacher's approach to tone production in the
light of modern physics, this section will help players to better understand their own inherited ideas
about technique, and to objectively evaluate technical ideas and practices.
By better understanding the underlying physics, players and teachers will not only be able to more
accurately and efficiently describe the workings of the instrument, but they will also be better able to
understand and evaluate violin technique. Technique is, after all, a set of solutions to various problems
on an instrument, and the physics provide a language that makes the problems clearer and easier to
understand. Understanding the problem makes it easier to understand the solutions to these problems,
to relate different technical ideas to one another, to explain them more clearly, and to effectively
combine ideas from different sources. Having a model to describe the interaction between the bow and
the string also makes it possible to build more consistent and comprehensive models for technique, and
to design new exercises and pedagogical methods which exploit the underlying mechanisms more
efficiently. This paper aims to provide teachers of stringed instruments with the means to better
understand and communicate about tone production, and thus help them to teach it more effectively and
efficiently.
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Chapter 1: Literature ReviewThe scientific understanding of tone production developed in fits and starts, as technological and
mathematical advances made new theoretical models and experiments possible. For the most part, the
scientific literature has the advantage of being unified by the academic community. Each scientist
whose work is represented here was aware of the work of their predecessors, and built on their
understanding to further the body of knowledge available to all scientists.
The major works of string pedagogy, on the other hand, are somewhat difficult to integrate into an
academic context. They are not academic texts, but are written for an audience who are both highly
specialized, because they are all violinists, and very general, because they come from any number of
different technical, musical, and educational backgrounds. It is, for example, very difficult to tell how
deeply aware the authors are of each other's work. Few of the authors make explicit reference to one
another, and when they do it is often in passing. Citations are few, and with some notable exceptions
the authors seem determined to develop their own ideas, without reference to any authority other than
their own. This makes it very difficult to trace ideas through the literature, and to describe the
development of violin technique as a unified field. Instead, the reader must be content to observe the
differences and commonalities between these approaches; to treat them as representative examples of
violin teaching, rather than a comprehensive summation of the whole world of violin pedagogy.
The chronological summary which follows outlines the development of pedagogical and scientific
ideas over the course of the twentieth century. In spite of the lack of communication between physicists
and violin teachers, their work followed parallel paths. Helmholtz provided a foundational explanation
for the basic motion of the string, but he could not explain why stringed instruments can produce
different timbres. Violinists gradually learned how to exploit the range of tonal possibilities which the
instrument can produce, without being able to explain why it produces them. Later scientists discovered
the physical basis for these different timbral effects, explored the limits of violin tone, and measured
the effects of various factors involved in tone production. Now it remains for violinists to integrate this
new model into the language of violin pedagogy and the practice of violin technique.
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1 Helmholtz and the Founding of Modern AcousticsHermann von Helmholtz’s Die Lehre von den Tonempfindungen als physiologische Grundlage für die
Theorie der Musik (On the Sensations of Tone as a Physiological Basis for the Theory of Music, 1863)1
forms the foundation for the modern understanding of acoustics and the physics of sound. The work
documents Helmholtz's experiments and discoveries in a wide range of areas, including investigations
into the physics of musical instruments, acoustics and the perception of sound, and the nature and
behaviour of sound waves. The most important of these, for the purposes of this paper, are Helmholtz's
work on the harmonic series and the motion of the bowed string.
Helmholtz's discovery of the harmonic series revolutionized the modern understanding of sound and
tone. Through a series of experiments, Helmholtz determined that most sounds can be described as a
set of regular waves, each of which made the air vibrate at a different rate. By building devices which
were designed to vibrate at one, precise frequency, now known as “Helmholtz resonators”, Helmholtz
was able to isolate individual waves from a more complex sound. These “partial” components of the
sound each have a different frequency, meaning they vibrate at a different rate. Many natural sounds
contain a chaotic mix of frequencies, and their pitch content is considered to be “inharmonic”.
Helmholtz found, however, that most musical instruments, especially string and wind instruments,
produce sounds in which the individual frequencies are mathematically related to one another. The
perceived pitch of these musical notes, Helmholtz discovered, is that of the lowest frequency present,
which he referred to as the “fundamental frequency”. The remaining pitches, referred to as “partial
frequencies” or “partials”, form a mathematically predictable series, each of which is a whole number
multiple of this fundamental. The note A which is played in first position on the cello G string, for
example, has a fundamental frequency of 110 cycles per second, and partial components with
frequencies of 220, 330, 440, 550 and so on. The pitch content of this note is said to be “harmonic”,
because the partial frequencies relate to one another in this way, and the set of partials which relate to
the fundamental frequency in this way is referred to as its “harmonic series”. Any partial which is a
member of this series can also be referred to as one of the note's harmonics. A second important aspect
of these harmonic sound waves is that they are periodic, which is to say that the waves repeat over a
1Citations throughout will use the original date of publication, to clarify issues of chronology, except when referring to editorial content from a more recent edition. The page numbers, on the other hand, refer to the most recent edition of each work available, as listed in the bibliography.
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given period of time. By definition any sound wave which is harmonic is also periodic, and any sound
wave which is not periodic is inharmonic. Sounds which contain an inharmonic set of partial
frequencies, like those of many percussion instruments, are often difficult or impossible to perceive as
having a pitch, because the component frequencies do not repeat periodically or form a harmonic
series. Sounds which are periodic are more likely to be perceived as having a distinct pitch.
Helmholtz deduced that for a human listener, the quality of a sound is determined by the relative
strength of these partial frequencies (Helmholtz 1863, 118-119). Noises and natural sounds are
distinguished by the relative strength of the various frequencies present, and the timbre of musical tone
is determined by the relative strengths of its harmonics. By comparing the sounds of various
instruments playing the same note, Helmholtz found that string instruments produce a complete series
of harmonically related frequencies, each of which is a whole number multiple of the fundamental.
Some wind instruments, on the other hand, produce only odd multiples of the fundamental. In both
cases as the harmonics rise in frequency, they fall in amplitude. The mathematical understanding of
vibrating strings at the time Helmholtz was writing was based on an ideal string, which is infinitely thin
and flexible, and which does not lose any energy to internal forces or to its interaction with the
environment. Helmholtz calculated that in such a string the harmonic series would be infinite, and there
would be an predictable relationship between the frequency of the harmonic and its amplitude, relative
to the fundamental, such that “the amplitude and the intensity of the second partial is one-fourth of that
of the prime tone, that of the third partial a ninth, that of a fourth a sixteenth, and so on” (Helmholtz
1863, 83-84). Real strings are not like this. They suffer energy loss in a number of ways that Helmholtz
did not account for. The harmonic content of real violin tones is therefore much more varied and
limited than Helmholtz predicted.
The harmonic series has implications for harmony and intonation, as well as for tone quality. Many
musical cultures have discovered that a vibrating string can produce the series of intervals we now
think of as the harmonic series, when its length is divided into whole number ratios. Other instruments
replicate the series more or less spontaneously because of their own construction. Reproducing it in this
way, musicians have effectively been experimenting with the harmonic series since antiquity. Many of
the world's systems of melody and harmony are thus, either directly or indirectly, based on the natural
harmonic series. The perception of intonation, and that of dissonance, is also thought to be related to
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the harmonic series. When a listener hears two pitches which are close in frequency, but not identical,
they can hear an audible series of “beats” in the sound. Helmholtz argued that perceptions of
consonance and dissonance come from these beats, which are caused by the upper partials of certain
intervals, and that this was the basis through which intervals can be said to be either in tune, or out of
tune (Helmholtz 1863, 181-182). Subsequent work has shown that these beats are caused by the
physiology of the ear. An interval is perceived to be more in tune when more of its frequencies are
shared by the two notes, and less in tune when the two notes contain many different frequencies.
Consonant intervals, similarly, share many partial frequencies, whereas dissonant intervals share fewer.
Helmholtz's work on the harmonic series provided a basis for new ways to understand harmony and
dissonance which many composers, notably Paul Hindemith, have since explored. Orchestral wind
players and players of string quartets in particular, use the harmonic series as a vital tool for
understanding and solving problems of intonation.
Helmholtz was also the first to observe the true behaviour of the bowed string. By marking the string
with a few grains of starch and observing their behaviour through a special microscope, Helmholtz
observed that “during the greater part of each vibration the string . . . clings to the bow, and is carried
on by it; then it suddenly detaches itself and rebounds, whereupon it is seized by other points in the
bow and again carried forward” (Helmholtz 1863, 83). This form of “stick-slip” motion is still referred
to as “Helmholtz motion”. Helmholtz also confirmed that the string does not move in smooth curves, as
had previously been thought, but in “a system of rectilinear zigzag lines” (Helmholtz 1863, 83n), and
worked out the mathematical basis for this behaviour. His equation shows that a corner is formed by
the bow, and then released to travel along the string. This motion is too fast to be visible to the naked
eye. A television refreshes its picture 24 times a second, which is fast enough to create the illusion of
smooth motion. The lowest note on the double bass has a frequency of about 41 cycles per second,
which is more than fast enough to create this same illusion. Since the strings move so fast, the eye is
fooled into seeing a curved motion rather than seeing the jerky cycle described by Helmholtz. Rather
than seeing the corner itself, a human observer sees the path travelled by the corner, which takes the
form of a smooth curve.
Helmholtz's groundbreaking work forms the basis for most of the research into the physics of sound
which was carried out in the twentieth century. Physicists and musicians alike have used many of his
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theories as a basis for further enquiry The basic, stick-slip behaviour of the string proved to be one of
the keys to understanding the motion of the vibrating string, and musicians still use the harmonic series
to understand harmony, dissonance, and intonation.
For violin players, Helmholtz's model of the motion of the bowed string provides a useful, if
incomplete, tool for understanding the motion of the bowed string. It provides an elementary
explanation for the relationship between the bow speed, the point of contact between the bow and the
string, and the volume of the resulting sound. According to Helmholtz's model, the volume of the tone
will be proportional to the bow speed, and inversely proportional to the distance between the bridge
and the bow (Cremer 1981, 77; Askenfelt 1988, 2). The faster the bow is moved, and the closer to the
bridge it is placed, the louder the tone will be, and vice versa. Helmholtz was unable to explain the
variety of tone colours which a violin can produce, because his model is based on an ideal, infinitely
flexible string. He observed that differences in timbre were the result of differences in the spectral
content of the notes, but he was unable to explain why different regimes of contact point, bow speed,
and bow force would yield different timbres. Later researchers, notably John C. Schelleng and Lothar
Cremer, were able to extend his model to explain these differences in timbre, and to explore the role of
bow force and contact point in determining the volume and timbre of notes, but in the mean time
players were left to their own devices.
2 Capet and Auer: Ending the 19th CenturyLucien Capet is one of the most important figures in the development of modern bow technique. Capet
is still remembered as a soloist and as a quartet player, for a style “characterized by faithfulness to the
composer's score, purity of tone, and finesse rather than force of expression” (New Grove Dictionary of
Music and Musicians 2nd Ed., s.v. “Capet, Lucien”). Capet's book, La Technique Supérieure de
l'Archet (Superior Bow Technique, 1916), is relatively neglected in the academic literature, partly
because it was only recently translated into English, and partly because his flowery language
occasionally makes the text difficult to follow. “Despite the grand scale of his ideas”, notes his first
English translator, Margaret Schmidt, “Capet was not a meticulous writer” (Capet 1993, 3). Capet
viewed music as much as a spiritual pursuit as an artistic and professional pursuit, and his work is
peppered with poetic digressions and meditations on the beauty of music. In regards to bow technique,
however, Capet's accomplishments were undoubtedly significant.
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In spite of its rhapsodic tone, Superior Bow Technique is focused on the eminently practical problem of
controlling the bow. Capet is especially concerned with establishing a sensitive bow hand, in which the
fingers are able to act individually to control the bow, and with regulating the bow speed. “The speed
of the bow on the string”, he argued, “should be determined by the tempo of the composition: it seems
natural that in an Adagio, the bow would not make rapid motions, while in an Allegro the bow would
not move too slowly” (Capet 1916, 24). Capet was unequivocally opposed to using bow speed as an
expressive tool, advocating instead that expressive nuances and accents alike should be achieved with
bow pressure. The contact point is never addressed explicitly in the text. Its importance is only hinted
at through Capet's discussion of the horizontal pressures exerted by the fingers.
One of the most famous aspects of Capet's teaching is the roulé stroke, sometimes called “bow vibrato”
which consists of rocking the bow back and forth so that it tilts variously toward and away from the
bridge. Debate still rages about what exactly is meant by Capet's description of the roulé, and how
Capet meant for the technique to be used. Capet posited that the roulé could improve the tone in a
similar way to left hand vibrato, without moving the left hand at all, but it is also one of the indirect
ways in which he addresses the contact point.
Much of the lasting value of Superior Bow Technique comes from the exercises contained within it.
Similar exercises are still used by teachers today, especially for students learning to compensate for the
uneven weight of the bow and to move the bow from one string to another. Capet's place in the
pedagogical firmament is also assured by the success of Ivan Galamian, who attributed much of his
own approach to bow technique to Capet's influence.
Leopold Auer's Violin Playing as I Teach It (1920), is almost as much a memoir as it is a pedagogical
treatise. As a teacher, Auer accepted almost exclusively advanced students, who had already formed
the basic substance of their technique. Violin Playing as I Teach It contains many anecdotes about
Auer's life, and about the famous violinists he met and heard in concert, but it does not for the most
part deal with specifics. Throughout the work, Auer displays a mild distaste for the codification of
knowledge. In spite of the fact that Auer's distinctive bow hold became quite famous, for example, he
writes that the bow hold is “a purely individual matter, based on physical and mental laws which it is
impossible to analyze or explain mathematically” (Auer 1920, 12). When it comes to tone production,
9
Auer shows more interest in discussing factors which can inhibit or limit tone than in developing a
positive model for how to best produce it. He is particularly cautious about the dangers of excess bow
pressure, which can choke the sound. Violin Playing as I Teach It cannot be considered a full record of
Auer's technique, but it reveals some tantalizing hints about his approach to tone production, and
contains several ideas which continue to influence contemporary violin teaching. Like Capet, Auer
alludes to the variety of colours which the violin can produce, but does not include instructions for how
to exploit them. Auer's caution against excessive use of bow force, however, put him more in line with
later teachers like Flesch, who manipulate the contact point and bow speed to control the tone.
Flesch and Trendelenburg: Launching the 20th CenturyCarl Flesch's Die Kunst des Violin-Spiels (The Art of Violin Playing) is a much more ambitious work
than either Auer's or Capet's. First published in 1923, The Art covers all aspects of technique and
musicianship in great detail, from purchasing an instrument to playing advanced concertos, and
includes a comprehensive discussion of tone production. Flesch's exploration of violin tone is much
more thorough than that of Auer or Capet, and his is the first major work of pedagogy to explore the
relationship between the timbre of a note and the contact point, bow force, and bow speed used to
produce it. The Art includes a description of different tone colours which can be produced at different
contact points, an explanation of the trade-offs between contact point, bow speed, and bow force, and a
discussion of how these factors affect the perceived volume of the tone. The novelty of Flesch's
approach lies in his focus on the contact point. Like Capet, Flesch considers the bow speed as basically
pre-determined by the composer, who indirectly dictates bow distribution through the tempo and the
notated bowings. Like Auer, Flesch considers excessive bow force to be undesirable, and instructs the
player to use no more than is necessary to achieve a solid tone. With the bow speed restricted, and the
bow force contingent on the other factors, the contact point becomes the primary focus of the player's
attention, both for determining the tone colour and for solving problems of bow distribution. Flesch
also explores the tonal and technical ramifications of playing on different strings and in different
positions, and includes a discussion of how to balance these factors against each other in practice. Die
Kunst des Violin-Spiels represents a huge advance in the pedagogy of tone production. It is both more
complete and more accurate than its predecessors in its description of tone production, and remains a
useful resource for modern players.
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Flesch considered tone production so important that he produced a second book on the subject.
Exhaustive as The Art is, it covers a wide range of subjects, and the discussion of tone production is
distributed among sections devoted to other issues. Published in 1931, some eight years after Die
Kunst, Problems of Tone Production in Violin Playing presents much of the material from the older
work in a leaner, more focused text. Like many parts of the earlier work, Problems of Tone Production
focuses on how a player's existing technique can be modified to address specific difficulties or issues.
Some of Flesch's ideas about tone colour are left out of this more compact work, notably his discussion
about how the choice of which position to play a passage in can affect the tone.
Flesch is also one of the only pedagogues discussed in this paper to refer directly to the work of a
scientist. In Problems of Tone Production, Flesch credits Wilhelm Trendelenburg with the discovery of
a relationship between contact point, bow pressure, and tone colour, and the claim that “the importance
of bow pressure for production of tonal volume is, in most cases, considerably over-estimated” (Flesch
1934, 8). This may indicate that Flesch was aware of Trendelenburg's work for a long time, or that they
knew one another, since Trendelenburg's major work, Die Natürlichen Grundlagen der Kunst des
Streichinstrumentspiels, was published in 1925, two years after Flesch's own The Art of Violin Playing.
Trendelenburg seems to have been the first scientist to investigate the role of bow pressure, contact
point, and bow speed in determining the tone colour. Helmholtz based his mathematical models on an
ideal string, which is perfectly flexible and yields an infinite harmonic series. If this were the case, the
timbre of a note would be independent of the contact point because all notes would include all
harmonic partials. Instead, the volume would be determined by the combination of bow-bridge distance
and bow speed, and the timbre would be strictly related to volume. It would be possible, for example,
to produce the exact same piano dynamic over the fingerboard and very close to the bridge, and they
would have identical timbres. Experience shows, however, that this is not the case. Just as Flesch
describes, the timbre and volume of a note vary wildly depending on the mix of contact point, bow
speed and bow force. Trendelenburg observed that:
The strength of the tone does not correspond to the displacement of the string. A basicexperiment is, for example, to bow a cello string with low bowing pressure andrelatively high bowing speed: it is easy in this way to achieve a displacement so greatthat the string nearly hits the adjacent ones. The tone is dull and not very loud. Now, ifthe bow is moved more slowly and with somewhat more pressure, the tone becomesbrighter, and somewhat louder, and so it carries to greater distances. In this case,however, the displacement is considerably smaller than before (Trendelenburg, W.,
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1925. Die Natürlichen Grundlagen der Kunst des Streichinstrumentspiels. [The NaturalBasis of the Art of String-Instrument Playing]. Berlin. Quoted in Cremer 1981, 78).
The question of what is meant by the “strength of the tone” is one that plagues performers and teachers
to this day, because it depends on complexities of both the tone generating mechanism, the violin, and
the human perception of sound. Although Flesch and Trendelenburg were able to identify many of the
effects of contact point, bow force, and bow speed on the resulting sound through practical
experimentation, scientists only began to discover the physical basis for this in the early 1970s.
Pedagogy in the 1960s: Havas, Rolland and GalamianBeginning with A New Approach to Violin Playing (1961), Kató Havas produced a series of works
which aim to develop a healthier approach to violin technique. She argued that older approaches like
Flesch's pay little attention to the player's comfort, which can lead to injuries. Havas also pointed out
that players will find it more difficult to control their tone when they are less comfortable physically. A
New Approach lays out this approach to the physiology of violin performance, focusing on minimizing
tension and simplifying the mechanism through which sound is produced. By developing a simpler,
more efficient physiological mechanism, Havas hoped to free players of distraction and allow them to
concentrate on musical ideas.
Havas was part of a reaction to traditional approaches to pedagogy and musicality which also produced
the period performance movement. Her most famous admirer was Yehudi Menuhin, who famously
struggled to find a comfortable way of playing in the middle portion of his performance career. Havas'
approach to technique focuses on the left hand, especially on the movements of the proximal knuckles,
which she calls the “base knuckles”. Once they have established a balanced and efficient physiological
approach to playing, Havas argues that the motions of the rest of the body should be subconsciously
related to those of the base knuckles. This allows the player to focus their conscious attention on a
single part of the body rather than dividing their focus between different parts of the playing
mechanism. Some of Havas's ideas are quite radical, particularly this claim that tone production is best
understood in terms of the base knuckles of the left hand, and her claim that vibrato can be achieved by
the base knuckles alone, without the participation of the left arm. The position she advocates for the left
hand is also unusual, with the violin sitting directly on the flesh between the thumb and index finger,
rather than held between the thumb and the base of the finger. A New Approach must be understood as
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a response to older approaches, as much as a positive program for technical practice. A second work,
The Twelve Lesson Course in a New Approach to Violin Playing (1964) attempts to remedy this, laying
out a series of exercises designed to develop the technique described in the earlier work. Although The
Twelve Lesson Course clarifies some of Havas's ideas, it reflects the same overall approach to tone and
technique.
Paul Rolland is most famous for his work on the Illinois String Project, which produced his most
famous work, The Teaching of Action in String Playing (1974), but the best expression of his approach
to technique and tone production is an earlier, shorter work, Basic Principles of Violin Playing (1960).
Of the two, The Teaching of Action in String Playing is much more substantial and academic in tone,
but it focuses exclusively on pedagogical methods and exercises, especially on movement training. The
earlier, briefer work, on the other hand, contains a compact model of tone production which clearly
describes relationships between string length and thickness, bow force, bow speed and contact point.
Rolland addresses many factors which can affect the tone, and argues that each element must be
adjusted to account for all of the others. According to Rolland, changing either the bow speed, the bow
force, or the contact point, by even the tiniest amount, must entail changing at least one other factor.
Similarly, he argues that the thickness of the string and the effective string length both affect the bow
speed, the bow force and the contact point, such that a player who follows all of his instructions
literally will need to change their bow speed, contact point and bow force for every note they ever play.
Rolland provides no guidance as to how the player should prioritize one factor over another, and
suggests no hierarchy or decision tree for determining in what order they should be modified under
different circumstances This flaw aside, Basic Principles of Violin Playing represents an extremely
compact, and quite comprehensive overview of basic technique.
Ivan Galamian's Principles of Violin Playing and Teaching, like Flesch's work, represents a
comprehensive manual of violin technique and performance pedagogy. Galamian's teaching lives on in
the continuing work of his students, who came to dominate the world of North American violin playing
and teaching, and the work of their students in turn. Principles of Violin Playing and Teaching covers
almost all aspects of its subject matter, although Galamian is notably silent on questions of posture and
the use of a shoulder rest. He also advocates an open-minded approach to style and interpretation, and a
flexible approach to teaching, modified to suit the needs and personality of each student.
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Galamian's approach to tone colour is very closely related to Rolland's, although he clarifies the
relationships between contact point, bow speed, and pressure, and string length and thickness.
Galamian introduces the term “sounding point”, which he defines as the ideal contact point for a given
bow speed and pressure. He describes the properties of the string, the length which vibrates freely and
thickness, as directly affecting the sounding point, and only indirectly affecting the pressure and bow
speed. Like Flesch, Galamian presents the pressure as playing a purely enabling role in the sound;
excessive pressure can inhibit the tone, so the pressure must be adjusted to suit the contact point and
bow speed, but for Galamian the bow pressure plays no positive role in determining the timbre or the
volume. In a notable departure from earlier teachers, he encourages the active use of bow speed as an
expressive tool. Galamian also includes exercises which are designed to demonstrate how the bow
speed and pressure can affect the appropriate sounding point. Galamian's oddest claim is that the tone
quality will be superior if the bow is very slightly slanted relative to the bridge, rather than exactly
parallel. No physical explanation is offered in Principles for why this might be the case, although one
later study noted that players use it as a strategy to change contact point (Askenfelt 1988, 13). As of yet
no experiments have been done to verify or disprove it.
Schelleng, Cremer and the New Science of Violin PhysicsIn the 1960s and 1970s, scientists began to discover the physical basis for many of the tonal
phenomena identified by Flesch and Trendelenburg, and explored by Havas, Rolland and Galamian.
The work of these scientists is summarized and synthesized in Lothar Cremer's 1981 work Physik der
Geige (The Physics of the Violin), but some of the notable papers which are still easily available will
be described here. Helmholtz's model had explained the motion of the string purely in terms of bow-
bridge distance and bow speed. Friction obviously played a role, in the alternation between the “stick”
and the “slip” relationships between the bow and the string, but neither Helmholtz nor his immediate
inheritors were able to explain the limits of this motion or to develop a theory of timbre. An early
article by the electrical engineer and amateur cellist John C. Schelleng, “The Violin as a Circuit”,
provided a new paradigm for analyzing stringed instruments (Hutchins 1980, 1074). Schelleng's
subsequent publications, first in the Journal of the Acoustical Society of America and then in a well-
known article in Scientific American, introduced many of the key ideas explored by the physicists of
this era. These articles remain some of the best known on the subject of stringed instrument physics.
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Schelleng's second important paper, “The Bowed String and the Player”, was published in the Journal
of the Acoustical Society of America in 1973. The article covers a wide range of topics, but its most
lasting contribution is almost certainly a diagram, which illustrates the maximum and minimum bow
forces that delimit Helmholtz motion. Schelleng predicts that the minimum bow pressure should be
inversely proportional to the bow-bridge distance, which is to say that as the bow moves away from the
bridge, the amount of bow force required to set the string in motion falls in proportion to the distance
(Schelleng 1973a, 28). Below this minimum, the motion of the string will shift into higher modes of
Helmholtz motion, with multiple slips per cycle. This will produce a wispy, insubstantial, and unstable
tone. The maximum bow force falls similarly, relative to the contact point, but at a lesser rate, such that
the range of forces which can produce Helmholtz motion expands as the bow moves towards the
fingerboard. Above the maximum, the bow fails to release at the end of each cycle, entering a regime
Schelleng referred to as “raucous”. The tone becomes increasingly irregular and noisy because the
release of the string is unpredictable and violent. Schelleng reproduced this diagram (see Figure 1) in a
well-known article in Scientific American, “The Physics of the Bowed String” (1973b), and it has been
used in many subsequent papers which attempt to explain Helmholtz motion to a lay audience.
Figure 1. The Schelleng Diagramfrom Schelleng 1973b
As the distance between the bridge and the bow increases, the upper and lower limits forbow force decrease, but the range of forces which will create a stable tone increase.Different bow speeds would yield different limits of bow force at each part of the string, andwould transition into sul ponticello and sul tasto at different places, but according toSchelleng they should produce the same general shape of graph.
15
Schelleng notes that the limits also depend on bow speed, and the three limits are interrelated, such that
one could just as easily refer to the minimum bow force for a given bow speed and contact point, the
maximum bow speed for a given contact point and bow force, or the contact point furthest from the
bridge for a given bow speed and bow force (Schelleng 1973a, 32). The graph above reflects the
relationship between bow-bridge distance and contact point, for a given bow speed. If the bow speed
were higher, more bow force would be required and permitted at every contact point, so the lower and
upper limits of the tone producing region would both rise. Bow speed thus forms a third limit of
Helmholtz motion, even though it is not represented in the graph. Schelleng predicts that the upper and
lower limits of bow force would rise in direct proportion to the bow speed, just as they fall in direct
proportion to the bow-bridge distance. Recent work by Schoonderwaldt, Guettler, and Askenfelt
(2008)2 has shown that in a real string, the lower limit of bow force which will create Helmholtz
motion is so dramatically affected by other factors that in practice it does does not behave quite as
simply as Schelleng predicted.
One notable consequence of Schelleng's work is that it contradicts Rolland's assertions about the
relationship between bow speed, bow force, and contact point (which is another way of describing the
bow-bridge distance). Rolland and Galamian both insist that these factors must be perfectly adjusted in
order to produce good tone. Luckily, Helmholtz motion is more robust than they suggest. Schelleng
points out that “the extreme limits of bow force are wider than most people would probably expect. In
the rapid changes of condition involved in virtuoso performance each note cannot be optimized”
(Schelleng 1973a, 28). The stick-slip system which drives the string is, to some degree at least, self-
correcting and self-sustaining.
Schelleng's article also addresses a second key point about the vibrating string, which is that different
regimes of bow force, bow speed, and contact point produce different harmonic content in the resulting
tone. Schelleng attributes this to the stiffness of real strings, which absorb higher partials. Helmholtz’s
predictions about the relative strength of harmonics, Schelleng notes, holds true for only the first few in
a real string (Schelleng 1973a, 30). He notes that when the real shapes of vibrating strings are recorded,
they differ significantly from Helmholtz's predictions3 (Schelleng 1973a, 28). Schelleng also notes that
2See “Revisiting McIntyre and Schelleng” below.
3See “The Rounded Corner: Tone Colour Explained”, beginning p. 47.
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tones produced closer to the bridge have a more brilliant and powerful tone, and that far from the
bridge they have fewer partial frequencies (Schelleng 1973a, 32), but he does not expand on this, or
develop a general theory of timbre.
Schelleng also identifies a phenomenon which might explain Kató Havas's concept of vibrato. Earlier
theories such as Helmholtz's predicted that the waveform of a vibrating string would be regular and
consistent, but Schelleng's measurements with modern instruments show that “the frequency and
amplitude of vibrations within short intervals . . . are not constants as ideally supposed, but that there is
sometimes a small variation which might be large enough to impart a warmth of tone colour somewhat
like that of the vibrato” (Schelleng 1973a, 36). The minute, natural variations in the tone that Schelleng
describes in this paper are nowhere near the amplitude of vibrato oscillations, but they do add to the
complexity of the sound that is produced. This natural richness and complexity could be what Havas
refers to as a natural vibrato, although it is by no means clear.
Schelleng's papers remain some of the best known on the physics of the violin. This is largely because
of the quality, originality, and scope of his work, and partly because he was a great communicator. As
an amateur cellist, he was well equipped to convey his ideas to other musicians, and passionate about
the work. The “Schelleng diagram” is still often used to explain Helmholtz motion, and has been
revisited in recent work by Schoondewaldt, Guettler, and Askenfelt (2008). Schelleng also had a key
role in the founding of the Catgut Acoustical Society, which would produce much valuable work on the
physics and acoustics of stringed instruments (Hutchins 1980, 1075). As if that wasn't enough, his
article in Scientific American, “The Physics of the Bowed String” made this work available to a wider
audience than that reached by any of his contemporaries.
In 1977, in a paper submitted to the Catgut Acoustical Society Newsletter (“New Results on the Bowed
String”), McIntyre, Schumacher, and Woodhouse identified two important phenomena that occur at
high bow forces, one which makes the tone rough and noisy and one which flattens the pitch. Both of
these undesirable effects, the authors argue, can be explained by the width of the bow hair.
Previous mathematical models of the bowed string had assumed an ideal bow, which was infinitely thin
and touched the string at a single point, but “New Results on the Bowed String” demonstrates that the
width of the hair can make the tone rough and noisy when high bow forces are present. The waveforms
17
produced by real strings, stimulated by real bows, contain irregular spikes, which decrease in frequency
and increase in intensity as the bow force rises. McIntyre, Schumacher, and Woodhouse show that
these spikes can be explained by the “differential” slipping of the bow hair during the stick phase of the
cycle: “A simple calculation shows that within the tolerance range of normal bowing, one or other [sic]
part of the short section of string in contact with the bow hair (or double-stick4) must slip momentarily
during the nominal 'sticking' part of the Helmholtz cycle” (1977, 30). The width of the bow hair, in
effect, creates and then releases a series of miniature corners, which produce small spikes in the sound.
When the bow force is low, these spikes are very frequent and very small, and thus difficult to
perceive, but as the bow force rises it releases the corners less frequently, so they become larger and
more noticeable. Well before the sound crosses into Schelleng's “raucous” regime, these spikes make it
undesirably noisy and rough.
A second important effect of the bow hair's real and finite width is that it can flatten the pitch. Pitch
flattening, the article argues, is caused by slight lags in the release of the string from the bow hair.
These lags, in turn, are caused by the interaction between the width of the bow hair and the rounded
corners which occur in real strings. When a rounded corner is released from the bow hair, the velocity
of the hair itself rises rapidly. Ideal models predict that it will rise instantly from zero to a large
number, but in reality “each time a rounded corner passes the bow, it has a 'bite' taken from it as the
velocity jumps the relevant gap” (McIntyre, Schumacher, and Woodhouse 1977, 28). These “bites”
delay the release of the corner, prolong the stick phase of the Helmholtz cycle and thus decrease its
frequency, flattening the pitch of the resulting note.
Lothar Cremer's work, Physik der Geige (The Physics of the Violin, 1981), attempts to summarize and
synthesize the work of his contemporaries, and to construct a complete model of the physics of the
violin. Cremer had worked together with a colleague, Hans Lazarus, on some of the same problems
addressed that Schelleng addressed in “The Bowed String and the Player”. Cremer does acknowledge
Schelleng's work on the subject, which was widely known, but he credits Lazarus with the discovery
that “the bowing pressure necessary to produce Helmholtz motion decreases when the bow is farther
from the bridge, and that the minimum bowing pressure increases with bowing speed” (Cremer 1981,
4The special device used in these experiments to mimic a ribbon of bow hair.
18
74). Cremer also offers a more detailed explanation for the differences in timbre which are achievable
on the violin than Schelleng, and incorporates them into his mathematical model.
According to a simple Helmholtz model, the interaction between the bow and the string will produce a
sharp corner regardless of where it is bowed. The sharpness of this corner reflects the ideal, infinite
harmonic series that Helmholtz predicted would be produced by a vibrating string, but a real string is
not a perfect medium for the transmission of energy, nor is it perfectly flexible. Cremer and Lazarus
were able to show, through a series of experiments, that the sharp corners predicted by Helmholtz
become, in a real string, “rounded corners of finite length and constant radius” (Cremer 1981, 79).
During each cycle of vibration, the string has a damping effect, absorbing some of the energy generated
by the bow. Higher frequencies are, for various reasons, more vulnerable to factors which dampen the
sound, which has the effect of rounding out the shape of the corner as it travels through the string. Even
though the interaction between the string and the bow might theoretically generate a perfect harmonic
series, therefore, some of the partials will be lost or weakened as the energy travels through the string.
Schelleng also observed that Helmholtz's predictions only held true for the first few harmonics, and
that the real waveforms observed in a modern oscilloscope are rounded, but Cremer and Lazarus were
able to create a robust model to explain this behaviour, with greater explanatory power and
mathematical utility.
One of the consequences of this new model is that the bow force becomes much more important to
understanding the tone. During each cycle of the bow-string interaction, the friction between the bow
and the string reinforces the shape of the corners, sharpening the corners: “As soon as the rounded
corner attempts to move past the bow, sticking friction works to hold the string onto the bow longer,
and so to straighten out the curved part of the string” (Cremer 1981, 79). Bow force, therefore, helps to
determine the strength of the partial frequencies of the resulting notes, and thus their timbre. Greater
bow force, within certain limits, will increase the force of friction between the bow and the string, and
produce a sound with stronger partials. Lesser amounts of bow force will have a smaller straightening
effect, and therefore produce weaker partials. The limits themselves, as illustrated by Schelleng, vary
according to the contact point. Playing close to the bridge requires a greater amount of bow force,
which is one reason that players have long associated this area of the string with a richer tone.
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Cremer shows how several other factors affect the vibration of the string and influence the resulting
sound, including the twisting motion of the string, the stiffness of the string, friction between the string
and the surrounding air, and the absorption of energy by the fingers of the left hand, the nut, or the
bridge. In addition to complicating the mathematical model he constructs, each of these also contributes
to the damping effect of the string, and thus affects the shape of the waves which travel through it and
volume of the resulting tone. The mathematics of Cremer's model are well beyond the grasp of most
musicians, but the fundamental interactions it describes can be understood fairly easily.
Experimental Developments After CremerIn the period since Cremer's work, the field of violin physics has been dominated by experimental,
rather than theoretical, exploration. Early work by McIntyre, Schumacher, and Woodhouse, some of
which is referred to in Cremer's Physik der Geige, was followed by work by Anders Askenfelt, Knut
Guettler, and Erwin Schoonderwaldt at Sweden's Royal Institute of Technology (KTH). The key
development proved to be sensors which could effectively measure the bowing parameters used by live
violinists. This allowed Guettler and Askenfelt to analyze the playing of skilled musicians, and
compare their use of the bow to the theoretical models developed by earlier scientists. Other examples,
like Askenfelt's “Measurement of the Bowing Parameters in Violin Playing” (1988), reflect new ways
to analyze and describe violin technique.
In “Measurement of the Bowing Parameters in Violin Playing”, Askenfelt describes an attempt to
measure the effects of different bowing parameter regimes on a real violin. By developing a method
which would measure the bow force, bow-bridge distance, and bow speed, and track the part of the
bow used by a real player, Askenfelt aimed to confirm that theoretical concepts developed by physicists
were really exploited by expert violinists. Two professional players were asked to carry out various
tasks, such as playing at several different dynamic levels, changing from one dynamic to another, and
playing in different positions. Generally, the two players were found to use the contact point and bow
force to change the volume more readily than the bow speed. When playing an octave scale on one
string, the players were observed to move their contact point closer to the bridge, as advocated by
Flesch, Galamian and Trendelenburg. “In principle”, Askenfelt notes, “the player has a wide choice of
combinations of bow-bridge distance, bow velocity, and bow force for producing a given dynamic level
and tone quality”, and their selection is affected by such factors as the length of the note, the bowing
20
pattern which is called for, and the surrounding dynamics (Askenfelt 1988, 27). Askenfelt argues that
although the results were collected from only two players, “there is . . . little reason to expect that the
basic scheme in these two professionals' choice of bowing parameters is not representative of the
recognized principles of violin playing today”5 (28). Many musicians would assert, on the contrary, that
different schools of playing call for quite different combinations of bowing parameters from one
another, even for identical passages. It seems likely that a wider sample size, selected from players with
a greater range of musical backgrounds, would yield a more diverse set of results.
In “Some Aspects of Bow Resonances - Conditions for Spectral Influence on the Bowed String”
(1995), Guettler and Askenfelt examine the effects of the vibrations within the bow itself on the
vibration of the string, in the hope that they might discover how the quality of a bow can influence the
sound of a violin. Noting that their own earlier experiments, as well as those by Schumacher and
Cremer, had observed oscillations in the bow hair, and that these oscillations can cause fluctuations in
the force of friction between the bow and the string, Guettler and Askenfelt designed an experiment to
determine whether these fluctuations can have a meaningful effect on the sound. By running several
simulations using a computer model, they discovered that it was indeed possible for the vibrations of
the bow to affect the vibrations of the string. They note, however, that “in experiments with real bows
and strings, there is currently no clear evidence of that [sic] the bow resonances are excited sufficiently
strongly, as to have a significant influence on the output system” (Guettler and Askenfelt 1995a, 115).
In other words, although the effect is measurable, it is not large enough to have a perceptible effect on
the sound. Theoretically, this work might suggest some explanation for differences in bow quality, but
it doesn't appear to have significant ramifications for the technique of tone production.
“What is a Proper Start of a Bowed String?” (Guettler and Askenfelt 1995b) analyzes the perception of
notes with different attacks, some recorded by real players and some simulated by computer. Many
players believe that it is possible to create a perfect attack, in which the string is caught by the bow and
released exactly once, creating an ideal martelé accent. Guettler and Askenfelt note, on the contrary,
that each bow stroke begins with a chaotic “onset period” in which the bow either delays the release of
the string, similar to the “raucous” regime described by Schelleng, or releases it more than once per
cycle, as in the higher modes of Helmholtz motion. Guettler refers to these as “delayed triggering” and
5The emphasis is Askenfelt's.
21
“multiple flyback” attacks, respectively. Sample recordings of each type of attack were compared,
along with samples of “perfect” attacks, which transition directly into perfect Helmholtz motion. By
polling twenty string students, Guettler and Askenfelt aimed to determine the length of an acceptable
attack of each type. The attacks with delayed triggering, which correspond to a pressure accent, or a
martelé style attack, were found to be acceptable when their durations were less than fifty milliseconds.
Those with multiple flybacks, which correspond to softer attacks, were deemed acceptable when their
durations were less than ninety milliseconds. Listeners proved unable to differentiate between
simulated, “perfect”, attacks, and the most precise examples produced by real players.
“On the Kinematics of Spiccato and Ricochet Bowing” (Guettler and Askenfelt 1998) takes a different
tack than earlier work by the two authors, developing a model which can be used to simulate spiccato
and ricochet bowings. Spiccato combines a transverse motion, the back and forth of the up and down
bows, with a rotational motion, caused by the player raising and lowering their hand as the bow
bounces on the string. Since the tip and the frog move in different directions, the bow is effectively
rotating around an axis, with the center of rotation somewhere in the neighbourhood of the player's
thumb (Guettler and Askenfelt 1998, 10). Guettler and Askenfelt were able to simulate an ideal
spiccato by combining two equations which represent the transverse and rotational motions. Adjusting
the various parameters of the two equations allowed them to analyze the factors which produce clear
attacks and even repetitions of the notes. Real players, they note, can control the timing of the stroke
and the rotational motion, and can also effectively alter the stiffness of the bow by tilting it (13)6. “On
the Kinematics of Spiccato and Ricochet Bowing” develops a model to describe spiccato and ricochet
bowing which could be used as a basis for analyzing a real player's technique. Measuring specific
parameters in their playing and comparing these to the synthesized ideal could help players who are
having difficulty to develop a more successful spiccato stroke.
Contemporary Pedagogy: Simon Fischer's BasicsSimon Fischer's Basics: 300 Exercises and Practice Routines for the Violin (1997) is not an expository
work like the others considered in this paper. Few teachers since Galamian have attempted to present a
comprehensive manual of violin technique and artistry, and none have produced works of the same
lasting significance as Galamian and Flesch. Fischer takes a different approach than these earlier
6Normally violin and viola players tilt the bow stick away from the bridge.
22
authors, presenting a series of individual exercises with only brief explanations for each. Although he
does not describe a theory of technique as such, his exercises do reflect a systematic approach to the
violin. Fischer's ideas about tone production build on the tradition established by Capet, Flesch, and
Galamian. He includes an exercise which is a variation of Capet's roulé stroke, and expands on
Galamian's exercises for changing the contact point, speed, and pressure. Fischer's exercises
demonstrate the effects of bow speed, bow force, contact point, string length, and string thickness on
sound, and show the player how to improve their control over the bow speed, bow force and contact
point, and how to compensate for changes in string length and thickness. Like Rolland, Fischer
acknowledges that bow force can have a direct effect on timbre, although he does not explore the effect
in depth.
The physical explanations offered in Basics are somewhat problematic. They seem to reflect the
science to some degree, but also to simplify it to the point of distortion. It is by no means clear whether
this is due to problems with Fischer's own understanding, or to the manner in which he chooses to
express himself. He describes an ideal tone as being maximally resonant, for example, and equates this
resonance with the size of the envelope of the string's motion. In a way, this is correct, as the envelope
of the string's motion correlates with the strength of the fundamental frequency, but in another way it is
incorrect, as the human perception of volume does not depend on this alone (Cremer 1981, 78). A
second example would be his description of the beginning of a note, which implies that in a proper
attack the Helmholtz corner is formed exactly once and then released, as it is in a pizzicato. Both of
these descriptions are incorrect, but potentially helpful to the student. Fischer's exercises demonstrate
how each aspect of sound production with the bow can be mastered, and how they can be balanced
against each other to produce good tone, but he does not deal with tone colour in depth, and his
explanations for the underlying physics leave much to be desired.
Revisiting McIntyre and SchellengIn “Effect of the Width of the Bow Hair on the Violin String Spectrum” (Schoonderwaldt, Guettler, and
Askenfelt 2003), the authors follow up on earlier work by McIntyre, Woodhouse, and Schumacher on
the role of the bow hair in determining the tone. Using a real violin bow and a monochord, the same
special instrument Schelleng used to mimic a single violin string, nine combinations of bow force, bow
speed and bow-hair width were tested. Additional tests were performed with the bow tilted at an angle
23
of 45º, to mimic a common practice among string players which effectively narrows the ribbon of hair
touching the string. The experiments show that the bow hair can have a significant role in inhibiting the
strength of upper partials. The artificially narrow bow hair used in the experiment boosted the strength
of harmonics starting at the eleventh partial tone, compared to the normal width of the bow, and tilting
the bow had a similar effect. The authors conclude that tilting the bow hair is an effective strategy to
create richer tone.
“An Empirical Investigation of Bow-Force Limits in the Schelleng Diagram” (Schoonderwaldt,
Guettler, and Askenfelt 2008) describes the results of three experiments meant to test the results of
Schelleng's research into the limits of Helmholtz motion. Using an automatic bowing machine, the
authors were able to measure the response of real violin strings, played by a real bow at various
regimes of bow force, bow speed, and bow-bridge distance. “In the classical Schelleng diagram”, the
authors note, “the upper and lower bow-force limits form straight lines with slopes of -1 and -2
respectively, demarcating a triangular-shaped playable region with Helmholtz motion”
(Schoonderwaldt, Guettler, and Askenfelt 2008, 604-605). Graphing the results of their own
experiments, the authors show that Schelleng's original diagram predicts the shape of the playable
region quite accurately, but that his predictions about the precise limits of the region turn out to be less
accurate.
The experiment showed, first of all, that minimum bow force does not vary according to the bow
velocity as predicted by Schelleng. The upper bow-force limit, as expected, rose in proportion to the
bow speed, but the lower limit of Helmholtz motion turned out to be essentially constant over the range
of bow speeds tested (Schoonderwaldt, Guettler, and Askenfelt 2008, 613). Second, it turned out that
damping had a much more dramatic effect on the lower limit than Schelleng had predicted, meaning
that the real limits on bow force are effectively much higher than he predicted. The authors note that
players may find the exact lower limit to be less critical than the upper limit, because it is usually used
at such quiet dynamics that “a certain amount of multiple slipping in the waveform is not easily
perceptible, particularly not in orchestral playing” (620). The upper limit, on the other hand, is more
obvious, both because the sound is louder and because “pitch flattening and noise content are helpful
by giving continuous indications of how far away the disastrous switch-over into raucous motion
actually is” (620).
24
The main implication of the study is that the lower limit of Helmholtz motion is defined by factors
which Schelleng did not consider. Rather than the clean transition into multiple-slip motion predicted
by Schelleng's work, the lower limit consists of region in which multiple factors interfere with the
stick-slip cycle, including ripples in the string such as those identified by McIntyre, Schumacher, and
Woodhouse (1977) and corner rounding as described by Cremer (1981). Although Schelleng's diagram
remains a good representation of the Helmholtz region, Schoonderwaldt, Guettler, and Askenfelt
rightly conclude that Schelleng's diagram should be expanded and updated to reflect these additional
factors.
Physics in the Classroom: Cheri CollinsIn “Connecting Science and the Musical Arts in Teaching Tone Quality: Integrating Helmholtz Motion
and Master Violin Teachers’ Pedagogies” (2009), Cheri Collins argues that the physics of tone
production can be used as the basis for more effective teaching. “When Helmholtz Motion and the
Schelleng Diagram are used together”, Collins points out, “They illustrate the basic characteristics of
good tone-production on the violin” (Collins 2009, 35). Even for a teacher with limited understanding
of violin technique, she goes on to argue, this provides a basis for understanding violin technique and
for guiding the teacher's use of language. The physical model presented in the paper is very
rudimentary, and Collins limits her discussion of tone production to establishing the limits of
Helmholtz motion and teaching a basic, stable tone. The paper also proposes a set of exercises, based
on Helmholtz motion and the Schelleng diagram, to help students in a strings class improve their tone
(Collins 2009, 113-140). Although the paper does not address more advanced issues of tone quality, or
the underlying physics of the rounded corner, it makes a compelling case that understanding the limits
of Helmholtz motion can help beginner students understand the basic problem of how to generate an
acceptable sound.
25
Chapter 2: Physics for ViolinistsMost musicians have only a vague idea of how the violin generates sound. This makes it harder for
them to understand how their actions affect the resulting tone, and harder to analyze their own
technique. The instrument itself remains an object of obsessive study, as scientists and luthiers struggle
to understand the mysterious qualities that make an instrument by Stradivarius or Amati so exceptional.
Even the best instruments, however, are limited by the technique of their players, and the violinist's art
focuses on the sound that their instrument produces, which is created by the motion of the strings. The
vibrations which produce sound in the body of a stringed instrument are generated by the strings, which
are bowed by the player7. Once they arrive at the bridge, the body of the instrument takes over and the
player's work is done. The behaviour of the bowed string then, is the key to understanding the
technique of tone production on the violin, viola, cello, and bass.
Scientists are still struggling to figure out exactly how the shape and construction of a given violin
affect the sound, but the basics are relatively simple. The vibrations generated in the string travel
through the bridge to the top of the instrument, and through the sound post to the bottom of the
instrument. These two vibrating plates move back and forth, agitating the air which is trapped inside
the instrument, which generates sound. The body of the instrument acts like a filter, absorbing some
frequencies and amplifying others at different rates, which transforms the resulting sound. Each
individual violin has its own complex profile for which frequencies it amplifies and which it absorbs.
Importantly, the behaviour of a given stringed instrument is mathematically linear, which is to say that
it will react consistently and predictably to a given input. If violins produced simple sound waves, this
would simply mean that some notes were less loud than others (for example an A might be louder than
a C#). But real violins produce very rich spectral content, and the filtering effect of the violin body
transforms all of the partial frequencies produced by the instrument, the violin body's influence on the
resulting tone is rather complicated8. The effect is also, generally, rather subtle. Expert players can, of
course, distinguish between the tone produced by different instruments, but recent work has shown that
7This paper will almost exclusively address bowed string playing. For more information on the physics of plucked strings, see Beament 1997.
8For more on this subject, see Jim Woodhouse's excellent lecture available at http://youtu.be/wf_FfC9Uq3U. His discussion
of the filtering effects of the violin body begins 39 minutes and 19 seconds into the lecture.
26
player's preferences are very difficult to define, let alone quantify. The effect of the instrument body on
sound will generally be ignored in this paper, because players cannot directly influence it, certainly not
during performance. It should be taken for granted, however, that whatever the player does to transform
the sound through their interactions with the vibrating string is further transformed, for better or for
worse, by the instrument body.
One reason that the vibrating string has been of such interest to physicists and mathematicians is that
even the motion of a freely vibrating string, excited by a simpler mechanism than the bow-string
interaction, can be nonlinear, which can be difficult to accurately represent mathematically. A complete
model of the behaviour of the bowed string, accounting for modern mathematical and physical
discoveries, would be tremendously complicated. The mathematician Nicholas Tufillaro describes the
freely vibrating string as “one of the simplest spatially distributed nonlinear systems imaginable”
(Tufillaro 1989, 409), but the mathematics of even the simplest nonlinear system are well outside the
grasp of the average musician. Thankfully, the nonlinear effects which are so interesting to
mathematicians and physicists are also extremely small and difficult to observe, even in an experiment
designed for that purpose (Tufillaro et al 1995, 164), and their impact on the sound of a violin is
negligible. Several other aspects of the bowed string also have observable effects on its motion which
do not significantly effect the sound. The string twists, for example, when the bow rubs against it. This
twisting has a small, observable, and measurable effect on the interaction between the bow and string,
but not an audible one. This section of the paper will describe a simplified model of the bowed string,
which will explain the basic interactions between the bow which affect the sound, and describe to what
extent they can be controlled by the player. The aim of this section is to make some of the benefits of
the discoveries made by physicists available to performers.
The basic physical phenomenon which explains the motion of the vibrating violin string is now often
referred to as Helmholtz motion. It was Helmholtz who discovered that the string alternates between
sticking to the bow and slipping off of it, and that this causes the string to move in a more complicated
manner than the traditional picture of a freely oscillating string. The stick-slip motion of the string is
caused by the effects of tension in the string and friction between the string and the bow, which can be
manipulated by the player. Several factors affect the waves which travel through the string, and thus the
resulting sound, but once they arrive at the bridge they move beyond the player's ability to control.
27
Aside from choosing another instrument or bow, manipulating the vibration of the string represents the
player's only way of affecting the sound. The effects of any given action by a player are mediated by
the behaviour of the string, and in order to understand the relationship between technique and tone, it is
necessary to understand Helmholtz motion and the vibration of the bowed string.
Note, Tone, and Overtone: Understanding Sound WavesOne problem confronts us immediately, which is that musicians and physicists use very different
language to describe sounds. Many musicians, for example, use the term “overtone” to describe partial
frequencies, a term which scientists use only rarely to refer to upper partials9. Musicians also use the
word “tone” in a variety of ways which would be confusing in a scientific context, sometimes referring
to timbre, sometimes to a specific musical sound or note, and sometimes to refer to the sound quality
produced by a certain player (for example, “Christian Tetzlaff plays with a beautiful tone”). Another
problematic term is “dynamic”, which players often think of as describing loudness. In practice a
musical dynamic is a much more complicated concept than loudness. It is defined by the timbre and the
musical context as well as the absolute loudness of the sound. Avoiding such problematic musical
terms, this part of the paper will focus on three terms which are relatively easy to define and explain:
pitch, timbre, and loudness. These terms are the building blocks out of which complex musical
concepts like a “dynamic” are built on, and each describes an aspects of the human perception of sound
which are related to concrete physical quantities which are present in sound waves. The pitch, timbre,
and loudness of a musical note are determined by the nature of the sound wave which produces it, and
physicists describe these sound waves in terms of their frequency and their amplitude.
The frequency of a wave refers to the number of times it completes a cycle of vibration in a given unit
of time, usually expressed in cycles per second, which are called “Hertz” (Hz). If Figure 2 is thought of
as taking place over one second, the wave it represents would have a frequency of two Hz. In the case
of a sound wave the amplitude describes the range of displacement of air molecules, which is to say the
distance within which air molecules are moving back and forth. This determines the sound pressure
level, literally the air pressure which is generated by the motion of the air molecules, which is one of
the factors affecting perceived loudness. In a vibrating string, the amplitude refers to the range of
motion of the string itself. If Figure 2 was thought of as having an amplitude of one centimetre, that
9See the Translator's note, Helmholtz 1954 pp. 24-25, for further discussion of the issues underlying this terminology.
28
would mean the distance between the highest and the lowest points on the wave represented one
centimetre. In the case of the sound wave, this would mean that the air molecules excited by the wave
moved back and forth over a range one centimetre wide, and in the case of the string it would mean that
the particular point represented was moving back and forth over a distance of one centimetre.
Figure 2. A Sine Wave
All sounds can be broken down into a series of component waves, each of which is “sinusoidal”, which
is to say that its shape is similar to that of a “sine” wave (see Figure 2). Sinusoidal waves are periodic,
meaning they repeat themselves at regular intervals, and any sinusoidal wave can be represented
mathematically by its frequency and its amplitude. Most natural sounds are very complex, but any
periodic wave can be described as the sum of a number of sinusoidal component waves, each of which
will have a frequency that is a whole number multiple of the fundamental frequency and an amplitude
that reflects its strength Scientists refer to the complete set of these component waves as the “spectral
content” of a sound. “Spectral analysis” is a way of breaking down a complex sound wave into its
component waves, determining their frequency and amplitude. By analyzing the spectral content of
different sounds, through such criteria as the presence, absence, and relative strength of various partial
frequencies, it becomes possible to mathematically compare different sounds with one another.
Even in simple tones, the perception of loudness is complicated. Like the behaviour of the bowed
string, the human ear is non-linear, and the perception of loudness is difficult to express
mathematically. In addition to the sound pressure level, perceived loudness is also affected by the
frequency of the sound wave. The human ear is particularly sensitive to frequencies in certain ranges,
particularly within the range produced by human speech, and relatively less sensitive to high and low
frequency sounds. This means that the relative strength of each partial frequency will be perceived
differently by the ear than it is by a machine. Generally, the amplitude of a tone must be increased by
roughly a factor of three for it to be perceived as twice as loud (Beament 1997, 5), but this is only a
rough approximation.
29
One of the distinctions between notes produced by a violin and sounds produced in nature is that the
frequencies produced by a violin form a harmonic series. Violins also produce various anomalous
noises and irregularities, but when it is played appropriately the spectral content of violin sound is
dominated by the harmonic series. The open A string of a violin, for example, is usually tuned to 440
Hz. The ear perceives this tone as having the same pitch as a tuning fork, even though the violin also
contains frequencies of 880 Hz, 1320 Hz, and so on. In fact, it is possible to train the ear to perceive
some of the individual partials of a note, as Helmholtz demonstrated, or to trick the ear into perceiving
a certain pitch by producing members of its harmonic series, even though the fundamental frequency is
entirely absent (Pierce 1999, 15). Under normal circumstances, however, the ear will perceive the
fundamental frequency of a given tone as its pitch.
The timbre of a note is determined by its spectral content. Timbre, unlike pitch, depends not just on
whether certain frequencies are present, but how strong they are in relation to each other and to the
fundamental. A note played on the violin typically contains over twenty harmonic partials of varying
strengths, which the listener perceives as a single tone (Beament 1997, 11). In normal violin tone, the
fundamental is always of higher amplitude than the partials. Each harmonic has a maximum strength,
relative to the fundamental, which is determined by the overall structure of the instrument. The
following graph (Figure 3) shows the upper limit of possible amplitudes for each harmonic partial,
derived from a computer model of an idealized string.
Figure 3. Strength of Ideal Harmonic Series
Beament 1997, 21
Each subsequent harmonic is quieter as the pitch rises, and although in theory the series extends to
infinity, in practice the string and the instrument absorb the higher partials. Very broadly speaking,
30
tones will be perceived as more rich and full when the harmonic content is stronger, particularly in the
middle and lower frequency range, and more thin when harmonic content is weaker.
The perceived loudness of a musical note is also related to its spectral content. In simple tones the
loudness is determined by the amplitude, and by the relative sensitivity of the ear to tones of that
frequency, but in a musical tone the partial frequencies all contribute to the perception of loudness. The
harmonics are naturally much weaker than the fundamental, even without taking into account
absorption by the string and the violin, but the spectral content of a note can have a significant effect on
perceived loudness (Cremer 1981, 78). It is possible to produce two tones on a violin which are
perceived as equally loud even though one has a much stronger fundamental than the other, or two
notes which are not perceived as equally loud, even though their fundamentals have the same
amplitude.
The perception of pitch, timbre, and loudness depend as much on the listening apparatus as they do the
underlying physics. Perception of pitch is relatively straightforward, at least for the purposes of this
paper. Except in the cases of certain effects, like over-pressure, the pitch of a violin tone is normally
determined by the frequency of the fundamental, which depends on the length of the string which is
free to vibrate. Timbre is determined by the spectral content of the sound, i.e. the relative strength of
the partial frequencies. Loudness, like timbre, depends on the amplitude of the partials as well as that of
the fundamental. Both loudness and timbre are also affected by the complexity of the hearing
mechanism, which perceives certain frequency ranges more strongly than others. Although this paper
will explain the general relationship between the actions of the bow and the harmonic content of the
resulting tone, it is difficult to explain the perception of timbre and loudness in anything other than
these general terms. Increased harmonic content, particularly in the lower ranges, makes a tone sound
fuller and more robust, while decreased harmonic content makes a tone sound more pure and delicate.
The violin is theoretically capable of producing a complete harmonic series, and normal playing indeed
produces a very rich spectrum of sound. The actions of the player will either preserve or dampen these
partial tones, which will determine the timbre of the resulting note.
31
Underlying Forces: Tension, Friction, and Bow Force
1 Tension and Restoring ForceThe interaction between the bow and the string is ruled by three fundamental forces: the tension in the
string itself, the force the player exerts on the bow and the force of friction between the hairs of the
bow and the string. Each of these forces can be either controlled or manipulated by the player, and each
must be understood to some degree in order to construct a model of how they interact. The simplest
force, which can be considered as an underlying constant, is the force of tension in the string.
The strings of the violin are stretched between two relatively rigid ends, the bridge and the nut. The
player can adjust the tension of the string, by turning the pegs or by using fine tuners, but during
normal playing conditions the tension will remain constant. When the string is at rest, the force of
tension pulls all points equally in two directions. The string pulls on the nut at one end and the bridge at
the other, and each of these pulls with equal force in the opposite direction. Even though there is
tension present in the string, it is said to be in a state of equilibrium, because all the forces present are
balanced against each other. A real violin string is not quite as simple as this. It is stretched by the
bridge, which holds the string up above the body of the instrument, so that the bridge will absorb
energy from the string and transmit it into the body of the violin where the sound is generated. This
vertical stretching means that the forces experienced by the string asymmetrical, because the vertical
forces experienced by the string are greater than the horizontal forces. In order to fully describe the
behaviour of the bowed violin string, it would be necessary to account for the three-dimensional shape
of the string itself, as well as for this asymmetrical tension. Real playing involves some twisting of the
string (Beament 1997, 29; Cremer 1981, 112-126), as well as some lateral stretching, which causes
fluctuations in the length of the string (Tufillaro 1989, 408). The asymmetrical tension experienced by
the string further complicates the mathematics, but these factors are far less significant to the sound
than the vibration caused by the bow. This chapter will address only the horizontal motion of the bow
and string, and ignore the asymmetrical tension on the string, the effects of twisting, and of lateral
stretching. For the purposes of this paper, the ends of the string will be treated as fixed, and only the
section of the string between the nut and the bridge will be addressed.
32
When the string is at rest, each part of the string experiences two forces which pull in opposite
directions (see Figure 4). The string, in turn, pulls on the nut and the bridge. Since these forces are
equal, and evenly distributed, the string stays still. It is said to be in a state of “equilibrium”.
Figure 4. Tension in the Resting String
When the string is pulled out of this equilibrium, for example when a player pulls the string with their
finger in order to play a pizzicato note, the string is further stretched and displaced (see Figure 5).
Figure 5. The Stretched String
Figures 5 and 6 depicts a string which is stretched in just such a way. When the string is displaced from
equilibrium, internal spring-like forces act to restore it to equilibrium. The total tension in the string
increases, with the new “restoring” forces pulling towards equilibrium, resisted by the fixed ends of the
bridge and the nut. The easiest place to understand this is the exact point where the string is divided.
This point is pulled toward the bridge and the nut with equal force (see Figure 6).
Figure 6. Restoring Forces in the Stretched String
33
A B
Bridge Nut
Finger
Force Towardsthe Nut
Force towardthe bridge
Bridgepulls onstring
Nut pullson stringString pulls on bridge
String pulls on nut
Another way to think about this new situation is to divide these forces into two components: those that
pull towards either end of the string and those that pull the string back towards its resting position.
Figure 7 depicts the force pulling towards the bridge, divided into two components. The component
which pulls towards resting position is depicted as vertical, pulling toward the bottom of the page, and
the remaining component, parallel to the length of the string, is depicted as horizontal, in this case
pulling to the left.
Figure 7. Component Forces 1
Figure 8 depicts the force towards the nut as similarly divided into two components, one which pulls
vertically, back towards the string's resting position, and another pulling along the length of the string,
this time to the left.
Figure 8. Component Forces 2
When these components are added together, the forces which pull towards the nut and bridge cancel
each other out, and the components which pull towards equilibrium add together (see Figure 9).
Figure 9. Adding the Component Forces
Technically all of the new forces which arise when the string is stretched can be considered as
“restoring” forces, because they work to restore the string to its equilibrium position. But the forces
pulling along the length of the string cancel each other out, and for the purposes of this paper the
“restoring force” should be understood to refer solely to those components that are parallel to the
bridge, which work to pull the string back towards its resting position. The forces pulling along the
34
+ = 0 ; + =
length of the string (horizontal in the above diagrams) will always be equal to one another and opposite
in direction, but the transverse restoring force (vertical in the diagrams) increases as the string is pulled
away from its resting position, and decreases as it returns to it.
The player cannot directly affect the force of tension during normal playing. The tension of the strings
is adjusted during tuning, but it remains basically constant as long as the instrument remains in tune. In
fact, the tension remains the same even when the string is changed by stopping it with a finger. The
restoring force, however, is affected by changing the position of the bow, or by stopping the string with
a finger, because it depends on the angle formed by the string. When the string is pulled away from
equilibrium by a given distance, closer to either the bridge or the nut, it will form a larger angle than if
it is pulled the same distance closer to the middle of the string (see Figure 10). Similarly, changing the
length of the string which vibrates, by stopping it with the left hand, will change the angles which are
created by a given contact point.
Figure 10. Angles in the String
When the forces are divided into their transverse and lateral components to calculate the transverse
restoring force, as discussed above, a greater portion of the total force will translate into restoring force
when the angle is large than if the angles were more shallow. This is why it is easier to pull the string
by a given distance in the middle than it is at either end, but no easier to stretch the string lengthwise at
any particular part of its length.
The different angles which are created by the string, and which determine the restoring force, are the
reason that the bow speed and bow force must be altered as the contact point changes. The closer the
bow is to the bridge, the more restoring force is generated for a given displacement (which, as will be
shown, is determined by the bow speed), and the more force must be used to displace it the same
35
Large Angle Smaller Angle
distance. When the string is stopped by a finger, its length is effectively shortened, which means that
the same displacement of the string would yield a larger angle between the bow and the finger. As will
be explained below, the more restoring force is generated, the more bow force will be required to move
the string, which is why it is necessary to press harder when playing closer to the bridge.
2 Static and Kinetic FrictionThe second force which must be understood in order to construct a model of the bowed string is
friction. When the bow is moved across the surface of the string, it is friction that makes the string stick
to the bow, acting against tension and pulling it out of equilibrium. The motion of the bowed string
involves alternating between two kinds of friction, “sticking”, or static friction, and “slipping” or
kinetic friction.
The classic model of static and kinetic friction describes the interactions between an object and the
surface it rests on, like a textbook sitting on a table. The force of static friction is determined by two
things:
1. The “normal” force. This force reflects the total forces which push the objects together.In the example below, the normal force is equal to the force that gravity exerts on thetextbook, which is determined by its mass.
2. The coefficient of friction10. This is a number which is determined by how smooth orrough the objects are, and although it can be affected by a number of things, it is usuallyapproximated as a constant for a given object and surface.
The normal force and the coefficient of friction determine the force of friction. When attempting to
push a textbook along the surface of a table, the force of friction acts in opposition to the force applied
to the textbook (see Figure 11).
10A “Coefficient” is a mathematical term that refers to a known or constant multiplier, for example in the equation y=14x, x
has a coefficient of 14.
36
Figure 11. Friction and the Normal Force
As more force is applied to the textbook, the force of friction which resists it increases, until a threshold
is reached and the object begins to move. At this point, the force of friction drops.
Kinetic friction, that is the friction between a sliding object and the surface that supports it, is less than
static friction. The relationship can be represented by the graph in Figure 12.
Figure 12: Static and Kinetic Friction
In the graph, the force applied to a static object increases as the graph moves to the right. As the
amount of force applied to the object increases, so too does the force of friction, until a threshold is
reached and the object begins to move. At this point, the force of friction drops. This is why, once an
object is sliding, it is relatively easier to keep it moving than it is to get it to move in the first place.
Once the textbook begins to slide along the table, it is only necessary to apply a force equal to that of
kinetic friction in order to keep it going. Any additional force will increase the speed of the object, and
37
Force of Push
Force of Friction
Static Friction
Kinetic Friction
Threshold
Table
Normal Force (of the textbook pushing down)
Force of HandTextbookForce of Friction
Normal Force (of the table pushing up)
if the force applied to the object falls below the threshold of the resisting kinetic friction, it will stick
again, and the system will revert to the dynamics of static friction.
Sliding friction is not quite as simple as the straight line indicated on the graph. It depends to some
degree on the relative velocity of the sliding object and the surface, and increases as more force is
applied, but the dynamics of this interaction depend on advanced math, and has only a small effect on
the limits of stick-slip vibration. In the case of a bow sliding along a violin string, additional
complexities are introduced by the behaviour of the rosin (Rossing 2010, 282-283, Smith and
Woodhouse 2000).
It seems sensible that the area where the object and the surface touch would affect the force of friction,
but it turns out not to be the case. When less of the book touches the table, the surface area is reduced,
but each part of the area experiences a larger normal force. There is now more book sitting on top of
each square centimetre of table than there was previously. When more of the object touches the surface,
for example when the book is lying flat on its cover, the normal force is divided over a larger area, and
the mass above each part of increased surface area is less. Unless an additional force is applied, the
mass of the object changes, or the nature of the material is altered, the normal force will remain
constant, and the effects of friction will stay the same.
3 Bow Force and Bow SpeedThe third force which must be understood in order to construct a model of the bowed string is that
which the player exerts on the bow. In the simplest case, when the bow is perfectly vertical relative to
string, this force can be thought of as having two components: one which is vertical relative to the
string (pushing down towards the ground or up towards the ceiling) and one which is horizontal
(parallel to the bridge). These are best understood in terms of their effect on the bow, rather than in
terms of the mechanism through which the player affects the bow itself. The vertical component of this
force is part of the normal force. If the player exerts a net upward force on the bow, this will have the
same effect on the system as making the bow lighter, and if the player exerts a net downward force, it
will have the same effect as making the bow heavier. A bow which is simply balanced on the string
will experience a total force of friction proportional to its weight, while a bow which is either lifted or
pressed into the string will experience a force of friction proportional to its weight plus or minus the
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force exerted by the player. The net force of friction between the bow and the string, which is
determined by the coefficient of friction and the normal force is often referred to as the “bow force”.
This is distinct from “bow pressure”, a term often used by teachers, which refers to the work done by
the player.
The horizontal component of the force exerted by the player is what sets the system into motion. The
more horizontal force is applied to the string, the further it will stretch. In the model of static friction
presented above, the horizontal force of the bow is equivalent to the hand pushing on the textbook. The
horizontal force is resisted by the force of friction. When the player pushes or pulls on the bow, the
force of friction between the string and the bow will resist this motion. If the string were to remain
stationary, like the table, the player would eventually exceed the force of static friction and the bow
would slide along the stationary string. The string, however, does not remain stationary; it stretches.
The bow continues to move, because the string is stretching, and the force of friction keeps the bow and
string together. As the string stretches more and more, the restoring force increases, pulling in the
opposite direction of the friction. Eventually, the restoring force will exceed the force of static friction
which holds the string and the bow together. At this moment, the string will unhitch from the bow, the
force of friction will drop to that of kinetic friction, and the string will begin to slide across the surface
of the bow hair.
The influence of the horizontal force on the system is usually discussed in terms of the bow speed. The
more force is applied to the bow, the faster the bow moves, and the more energy is delivered into the
string over a given period of time. When the other parameters are properly adjusted, the horizontal
motion of the bow determines the displacement of the string during each cycle of motion, which affects
the amplitude of the resulting vibrations and thus the loudness of the tone. In order to understand the
cyclic vibrations which take place in the string, the bow speed is usually considered as a constant.
When the bow speed is changed, the system adjusts instantly to the new regime, and the displacement
of the string (and thus the loudness) changes accordingly.
The tension of the string, the friction between the string and the bow, and the force exerted by the
player interact to determine the behaviour of the bowed string. These forces are the principle means
through which a violinist is able to manipulate the sound. The player can affect the restoring force by
39
stopping the string, restricting the length of the string which vibrates, or by changing the contact point.
By changing the amount of vertical force they exert on the string, the player can alter the normal force,
and thus the force of friction. Finally, by changing the bow speed, the player can affect the horizontal
forces which act on the string. Understanding the interaction between these forces and the violin string,
which together create Helmholtz motion, makes it possible to understand the relationship between the
actions of the player and the resulting sound.
The String in Action: Helmholtz Motion
1 Initiating the Stick-Slip CycleHelmholtz discovered that when the three forces which govern the motion of the string are balanced
against each other, they can create a dynamic, physical system which ultimately creates sound. The
system is set in motion by the horizontal force which the player exerts on the string. The force of
friction causes the string to stick to the hairs of the bow, and the string stretches. As the string moves
away from equilibrium, the restoring force increases (see Figure 13), until it exceeds that of the static
friction between the string and the hairs of the bow. When the restoring force exceeds the force of
friction, the string slips away from the bow and flies back towards its resting position.
Figure 13. Restoring Force and the Stretched String
As it does, the restoring force falls, but the string picks up speed and thus momentum. Eventually, the
combination of the restoring force and the momentum of the moving string fall below that of even
kinetic friction, and the string sticks to the bow once more. When the friction, tension, and bow speed
fall within certain limits, this system will create a cyclic motion which efficiently transfers energy from
the bow, through the string, into the bridge and thus the body of the instrument.
40
Bridge
Bow
Nut
StringRestoring Force
Force of Static Friction
This initial phase of the motion of the string is very difficult to control. When the string releases from
the bow, the string itself begins to vibrate. The stick-slip cycle between the bow and the string causes
vibrations within the string, which then affect the stick-slip cycle itself, and eventually the two
oscillating cycles stabilize into a single system. Guettler and Askenfelt observe that “The onset of a
note on a bowed instrument shows a transition period during which the periodic “steady state” motion
develops” (Guettler and Askenfelt 1995b, 97). This onset period, during which the string establishes
the cyclic motion of a stable violin tone, represents the part of a note that players think of as the
“attack”. Guettler and Askenfelt were also able to demonstrate that capable players can control these
onset periods more effectively than listeners can detect them (Guettler and Askenfelt 1995b, 97).
Players exploit the differences between onset periods which are more characterized by sticking to
create martelé attacks and other accents, and those which are characterized by more slipping to create
notes with soft beginnings.
Helmholtz motion occurs when the cycle of sticking and slipping friction, which occurs between the
bow and the string, becomes coordinated with the vibrations of the string. For this to happen, the
chaotic onset period must stabilize into a simpler system, in which both the oscillations of the string
and the stick-slip cycle become more regular. Cremer notes that “the bow always tries to make the
interval of sticking as long as possible; when there are two possibilities, the bow always forces the one
with the longer interval of sticking friction” (Cremer 1981, 43). This helps the system to stabilize into a
predictable cycle. When the Helmholtz cycle is established, the friction between the bow and the string,
the tension in the string and the horizontal force provided by the player must all fall within certain
ranges. The friction between the bow and the string must be sufficient to hold them together for most of
the cycle, and the motion of the string must be sufficiently strong to cause them to release at the correct
moment.
2 The Slip PhaseA stable slip-stick cycle can be thought of as beginning when restoring forces within the string are
sufficient to pull the string away from the surface of the hair. When the string is displaced by the bow
the string takes on a triangular shape (see Figures 5, 6, and 13). A corner is formed at the sight of the
bow. When the restoring forces pull the string away from the bow, the string begins to slide along the
surface of the bow hair, the force of friction between the bow and the string falls to that of kinetic
41
friction, and the corner which is formed at the site of the bow hair begins to move through the string
(see Figure 14).
Figure 14. The Corner Slips
The forces moving through the string excite it to change shape, such that corner formed at the sight of
the bow seems to move through the string. The corner is often referred to as a “kink” because, as will
be shown below, it does not take on the shape of a sharp angle in real strings, but for the moment
“corner” will be clearer. On one side of the corner, the string is moving out of equilibrium, and on the
other side, towards it (see Figure 15).
Figure 15. Motion in the String
When the corner arrives at the bridge, it loses some energy to the bridge, and an even smaller amount
to the remaining length of string and the tailpiece on the other side of the bridge. Slightly weakened, it
reflects off of the bridge (see Figure 16)11.
11http://plus.maths.org/content/why-violin-so-hard-play includes an excellent explanation of the stick-slip cycle, and a very
good animation that depicts a complete cycle.
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Nut
The string slips
The restoring force
Corner
Motion of the string
Motion of the string
Figure 16. The Slip Phase and the Bridge
The energy that is transferred from the string into the bridge produces sound on the violin, and the way
that the bridge moves determines the nature of the resulting sound. Each violin behaves differently,
given a certain input at the bridge, because of its unique construction. Since the motions of the
instrument itself are extremely small in amplitude, and the body of the violin is relatively rigid, the
motion of a given violin is relatively easy to predict, given a certain input at the bridge. Cremer goes so
far as to say that the best way to think of this is that “only the alternating forces at the input of the
bridge 'make music'” (Cremer 1981, 60). By manipulating the behaviour of the string, the player
changes the forces which arrive at the bridge, and thus the sound produced by the instrument.
3 The Stick PhaseWhen the corner returns to the bow from the bridge, this time it works in the same direction as the
friction between the bow hair and the string, so the string re-attaches to the bow hair, restoring the force
of static friction.
Figure 17. The Stick Phase
The corner continues to move along the string towards the nut (see Figure 17), where it reflects back
into the string and travels along the opposite side (see Figure 18).
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The bridge vibrates
Some energy is lostSome energy is reflected
Nut
The string sticks
Corner
RestoringForce
Figure 18. The Corner Reflects Off the Nut
Finally, the corner returns to the site of the bow hair. When it does, the forces which generate the
corner add themselves to the restoring force and the force of friction. During Helmholtz motion, this
combination suffices to pull the string away from the bow hair, causing a new slip and renewing the
Helmholtz cycle.
The three forces which govern Helmholtz motion do not have to be exactly balanced in order to
maintain it. If they were, any change in the player's actions would interrupt the cycle of motion, and
making a good tone on the violin would be even more difficult than it is. What is required is that in
each cycle, when the restoring force and the forces which generate the corner are added together, they
are greater than the force of static friction between the bow and the string. As long as these conditions
are met, the system will stabilize itself, and transmit energy from the bow to the bridge.
The Ideal Limits of Helmholtz MotionEven in an ideal model, with a perfectly flexible string and a bow that is infinitely thin, Helmholtz
motion would have limits. The friction between the bow and the string, the restoring force, and the
horizontal force applied to the string by the bow must all be sufficient to set the system into motion, but
not so great that they destabilize it. The restoring force changes based on the distance between the bow
and the bridge, and the length of the string which vibrates, because these affect the angle which is
formed by the string. The interaction between the changing forces of static and sliding friction and the
restoring force produce stick-slip motion. If the friction is too little, or the restoring force is too great,
the bow will slip again before the cycle is completed, producing two corners (see Figure 19). This is
often referred to as “double-slip” motion.
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Nut
Corner
Figure 19. Double-Slip Motion
Alternatively, the friction may be too great, or the restoring force too little, and the the bow will fail to
release at the end of each cycle. This means the length of the cycles becomes uneven, and the sound
becomes choked and rough.
The restoring force thus determines the range of other forces within which Helmholtz motion will be
possible. It must be enough to unhitch the bow from the string in each cycle, when the corner returns to
the bow, but not so much that it creates a double-slip. The restoring force, in turn, is affected by the
length and thickness of the string, and the distance between the bow and the bridge. Schelleng's famous
diagram (see Figure 20) can be viewed as an illustration of the effect of bow-bridge distance on the
restoring force, and of the adjustments that must be made to the bow force to compensate.
Figure 20. Simplified Schelleng DiagramWoodhouse and Galuzzo 2004
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Nut
Energy
Energy
The lower left region represents situations in which the force of friction is not sufficient to create a
stick-slip cycle, and results in double-slip (and potential triple-slip or quadruple-slip) motion, what
Schelleng referred to as the higher modes of Helmholtz motion. The upper-right area represents
situations where the force of friction is too great, and a noisy, scratchy sound results. For a given
contact point, Schelleng demonstrates that there will be a range of bow force within which Helmholtz
motion can take place. Although in principle the range of bow speeds and forces within which it is
possible to create Helmholtz motion increases as the bow approaches the fingerboard, this is limited in
practice because the string becomes increasingly flexible. When playing this far from the bridge, even a
fairly low bow force will displace the string sufficiently so that it touches its neighbours. The speed
which is necessary to produce sound also rises, making it increasingly hard to control the bow.
In this idealized situation, with a perfectly flexible string which is stimulated at exactly one point of
infinitesimal size, the violin will produce an infinite harmonic series, as predicted by Helmholtz. If this
were true of a real violin, there would be essentially no variation in timbre, except that caused by the
filtering of the instrument's body, only in loudness. The loudness, in this case, would be basically
determined by the bow speed and the bow bridge distance. As long as the bow force fell within the
limits outlined above, it would have no effect on the resulting sound. The volume would be
proportional to the bow speed, and inversely proportional to the bow-bridge distance, such that
doubling the speed of the bow would double the amplitude of the resulting sound wave. This would
make it possible to create identical tones by adjusting the bow speed and contact point, as illustrated by
Figure 21.
Figure 21: Idealized Envelope of String VibrationBeament 1997, 19
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According to Helmholtz's theories, each of the four illustrated contact points would yield identical
sounds, as long as the bow speed was adjusted accordingly.
For several reasons, this turns out not to be the case. For one thing, the bow is not infinitely thin.
Neither is the string perfectly flexible, and it is certainly not a perfect transmitter of energy. Each of
these has a dampening effect on the sound, and a noticeable effect on the timbre. These inefficiencies
in the system are the reason that the violin can create such a stunning variety of tone colours.
Understanding how they affect the tone makes it possible to describe the relationship between the
player's actions and the resulting timbre, and to provide a more realistic account of the limits of
Helmholtz motion and violin tone.
The Rounded Corner: Tone Colour ExplainedHelmholtz predicted that the violin string could produce a complete harmonic series. Theoretically this
series can extend to infinity, but only in an infinitely flexible string. Real strings are somewhat stiff,
and they are made of materials which absorb energy. They vibrate through the air, which resists their
passage. Neither the bridge nor the nut are perfectly rigid, and some energy is lost at each end of the
string. When the string is stopped by one of the fingers of the left hand, that finger will also absorb
even more energy, because it is softer than the nut. All of these factors remove energy from the system,
and dampen the energy which travels through the string towards the bridge.
If these each absorbed energy indiscriminately, they would only affect the volume of the notes. The
ratio between the amplitude of the fundamental and that of the partials would be preserved, because
each would be equally dampened, and the corners which move through the string would still be sharp.
But the frequencies are not equally vulnerable to these damping factors; higher frequency waves are
more vulnerable to damping than lower frequency waves. This means that upper partials are more
affected by dampening than their fundamentals, which is why damping affects the timbre of notes on
the violin, as well as the volume. The more energy is absorbed by the string, the duller the tone will be,
because the upper partials are weakened.
The shape of the corner formed by the bow correlates with its spectral content. The richer the harmonic
content of the sound, the stronger are the various frequencies moving through the string, which makes
the point of the corner sharper. The infinite harmonic series predicted by Helmholtz produces the
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distinctive pattern of a sawtoothed wave at the bridge, as a series of perfectly sharp corners transmit the
full harmonic series into it (see Figure 22).
Figure 22. The Sawtoothed WaveBeament 1997, 19
The damping properties of the string, and those of the nut or finger, act to smooth out this corner as
they absorb spectral content. In a real string, the corner is rounded because of damping, not sharp as
predicted by Helmholtz (see Figure 23).
Figure 23. A Rounded Corner
During each cycle of Helmholtz motion, the bow works to counteract this damping, and reinforce the
shape of the corner. Cremer notes that “as soon as the rounded corner attempts to move past the bow,
sticking friction works to hold the string onto the bow longer, and so to straighten out the curved part of
the string” (Cremer 1981, 79). This restores some of the lost energy, and acts to strengthen the spectral
content (see Figure 24). Just as the upper partials are more vulnerable to damping, they are more
strongly affected by the corner-sharpening effects of friction. The more friction is present, the less the
harmonics will be weakened by damping, and the brighter the tone will sound.
Figure 24. The Bow Reinforcing the Helmholtz Corner
It is the interplay between factors which dampen the harmonics and factors which reinforce them, that
causes the violin to produce such a great range of tone colours and dynamics. Some of these factors are
48
Bow reinforces corner
beyond the player's control, such as the different damping properties of the four strings, or the energy
which is lost at the bridge. Others can be manipulated more or less freely to affect the tone, like the use
of friction to reinforce the upper partials. Understanding how these factors affect the shape of the
corner is the key to understanding their effect on the tone.
Friction, Contact Point and TimbreDifferent regimes of bow force, bow-bridge distance and bow speed produce different timbres, not just
different volumes, because they affect the shape of the vibrating string in different ways. Bow speed
increases the amount of energy which is transferred into the string during each cycle of vibration, but it
affects all partial frequencies equally. Its main effect on the tone is therefore to increase or decrease
perceived loudness. Bow force, on the other hand, increases friction and affects the tone quality.
Damping acts to dull the Helmholtz corner, which also dulls the sound by weakening the upper partials.
Friction acts to sharpen the Helmholtz corner, which strengthens the upper partials and makes for a
brighter tone.
To some degree, this means that simply by applying more or less force to the bow, increasing and
decreasing the Normal force, it is possible to change the tone. As long as the bow force remains inside
the limits which allow Helmholtz motion, changing the bow force will thus change the timbre even if
the other factors remain constant. It turns out that excessive bow force has other undesirable effects
even within these limits, which will be discussed later, but changing the bow force is still an important
tool in the violinist's arsenal for changing the timbre.
More important still is the effect of different contact points on the tone. Greater friction means greater
spectral content, particularly in higher frequency ranges, but when the friction is too great it causes
sticking and interrupts the Helmholtz cycle. Ultimately, the amount of friction which the system will
allow is determined by the angles which create the restoring force. When the contact point is close to
the bridge, a large restoring force is generated for a relatively small displacement, which is why a large
amount of bow force must be used for even a relatively slow bow speed. Far from the bridge, it takes a
relatively fast bow and little bow force to generate tone. Indirectly, the brightness of the tone is thus
affected by the contact point. Further from the bridge, the tone can be dull because it is possible to
produce tone with relatively little friction. Within certain limits, bow force can be used to sharpen the
49
corner and make the tone brighter, but those limits are lower the further the contact point gets from the
bridge. Close to the bridge the tone is bright, because a high amount of friction is required to produce
Helmholtz motion.
Changing the amount of friction between the bow and the string directly, by changing the bow force, or
indirectly, by changing the contact point (and adjusting the bow force and bow speed), is the most
important tool a player has for controlling the timbre. The choice of contact point determines the range
of bow forces and bow speeds which will produce Helmholtz motion, and thus the range of timbres that
can be produced. This determines the basic shape of the corner which will travel through the string, and
has a dramatic effect on the resulting sound. Importantly, the player is relatively free to choose the
contact point for themselves, as long as they adjust the bow speed and bow force accordingly. Some
musical situations may restrict the choice of contact point and bow speed indirectly, because of the
dynamics and the bowings called for by the composer, but even then the player almost always has some
latitude. In general, the interplay between contact point, bow speed and bow force makes it possible to
produce a range of tones which are all perceived as being the same dynamic, and the player is able to
adjust the tone to suit their own taste.
The Bow HairMany of the mathematical models which describe the bowed string essentially ignore the bow itself.
They assume that the bow contacts the string at a single point, and thus its effect on the string is
relatively simple. In these idealized models, a perfectly sharp corner arrives at an infinitely thin bow
exactly once per cycle, and the combined forces unhitch the corner from the string in a single instant.
Real bows, of course, use a ribbon of horse hair to stimulate the string, which is not infinitesimally thin,
and real strings produce waves with rounded corners. When a real bow rubs against the string, the bow
exerts its friction across the whole width of the ribbon. This turns out to have three important effects on
violin sound: it makes the sound rougher, it flattens the pitch and it dampens some partials. Roughness
and pitch flattening are both exaggerated when the bow force is high, and can be somewhat mitigated
by tilting the bow.
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As the bow moves across the surface of the string, it exerts force across the whole width which touches
the string. Throughout the Helmholtz cycle, extra corners form, much smaller than the main Helmholtz
corner, across the width of the bow (see Figure 25).
Figure 25. A Miniature Corner
Figure 25 Illustrates the formation of one of these tiny corners. The angles ofthe string is vastly exaggerated, to make the corner visible. The mainHelmholtz corner is far off to the right, in this diagram, as the string iscurrently in a stick phase. The dotted line represents the straight line that anideal model would predict the string to form.
The effect of these tiny corners was observed in early experiments using oscilloscopes, and is
sometimes referred to as partial slipping. When the force of friction is low, these corners release very
often, and their effect on the tone is hardly noticeable. When friction is high, however, the bow holds
on to the hairs for longer, and they grow larger. This is what makes the sound rough, or noisy, at high
bow forces. This means that the real upper limit of Helmholtz motion is musical, rather than physical.
Well within the limits of Helmholtz motion, the width of the bow causes undesirable noise, and forces
the player to choose between volume and tone quality.
Pitch flattening is also caused by the width of the bow hair. In a real bow, the bow force is distributed
across the width of the hair. Since real Helmholtz corners are rounded, the force contained in them is
also distributed more evenly than would be the case with a sharp corner. An ideal model predicts that
the corner will unhitch instantly, but since the corner in a real string is rounded and the bow force is
distributed across the width of the hair, the real interaction between them is more complicated than this
and the corner doesn't always release perfectly. At high bow forces, the release of the string can be
slightly delayed. This prolongs the stick phase of each cycle, making the whole cycle longer and thus
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Bow Hair
Tiny Corner
flattening the pitch. The phenomenon is easy to observe when an open G string is played with great
vigour, but in stopped notes it is relatively easy to compensate for by adjusting the left hand.
Finally, the width of the bow contributes to corner rounding. The total amount of friction remains the
same regardless of the width of the bow, but it is spread across the width of the hair. The wider the bow
hair, the more evenly its friction will be spread out and the more it will dampen the tone. Comparing
two extreme cases will make this obvious:
Figure 26. Corner Rounding by the Bow
Figure 26(a) shows how an ideal bow would rub across the string at a singlepoint, and figure 26(b) how the distributed friction of a real bow rounds outthe corner. The wider the ribbon of bow hair, and the more evenly force isdistributed, the more it will dampen the partials.
All of these effects can be mitigated by tilting the bow. Tilting the bow does not change the total
amount of friction between the bow and the string, but it does change the way this force is distributed
across the ribbon of bow hair. Experiments by Schoonderwaldt, Guettler and Askenfelt (2003) found
that tilting the bow towards the fingerboard by 45º reduced the damping of some partial frequencies
significantly, which would also make the tone brighter. At high volumes, a tilted bow can reduce
undesirable noise and pitch-flattening for the same reasons. When the friction is concentrated on one
side of the ribbon of hair, as it is when a player tilts their bow toward the fingerboard, the friction on
the other side is reduced, which mitigates the effects of the width of the hair. Schoonderwaldt, Guettler
and Askenfelt also experimented with altering the width of the bow hair directly, and discovered that a
thinner ribbon of hair dampened the sound less.
The width of the bow hair has its advantages too. When executing a martelé attack, for example, it is
necessary to briefly exert a great amount of bow force, to make a sharp accent at the beginning of the
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(a) (b)
note. Not only is it easier to control the bow, when it is vertical, but the even distribution of bow hair
makes it easier to create the noisy attack, which is in this case deliberate. Because of the way it is
curved, the bow is also at its bounciest when it is vertical, and the hair is flat. This can be desirable, in
some bouncing strokes, or undesirable, when using a very light fast bow.
Much work remains to be done before the effects of the bow hair on tone production are fully
understood. The experiments performed by Schoonderwaldt et al (2003, 2008) represent an excellent
follow-up to the work of Schelleng, McIntyre and Cremer, but some details of the interaction remain
mysterious. More work on the effects of tilting the bow, for example, would be of great interest to
players, who use this technique constantly. The effects which are described above explain the most
noticeable effects of the bow hair on the sound, and have to some extent been understood by musicians
for a long time, and variously compensated for or avoided. Integrating them into a functional model of
the bowed string makes it easier to understand how they interact with other factors affecting the tone,
and thus easier to consider their role in the tone.
Other Factors Affecting the Tone
1 String Length and TimbreStopping the string in different places, as string players do with the fingers of their left hand, affects the
angles which are created in the string. This changes the dynamics of the restoring force, and thus the
bow-force limits for a given contact point. When the string is stopped high on the A string, for
example, the length of the string which vibrates is greatly reduced. This has a similar effect to moving
the contact point away from the bridge, because the shorter string will produce different angles when
displaced by the same amount. Changing the length of the string which vibrates, however, changes
nothing else about that string. The tension remains the same, which means that even when the string is
displaced by the same distance (i.e. the bow speed remains constant), the angles created in the string
change and more restoring force is generated from the same displacement. The player must either
increase the bow force, to compensate for this, reduce the bow speed, which will also reduce the
loudness, or adjust more than one bowing parameter to maintain a similar and acceptable tone. The
relationship between the contact point and speed will be different, therefore, depending on where the
left hand stops the string, and the higher the string is stopped, the narrower the range of bow forces and
speeds which will produce Helmholtz motion. Since the strings are all different thicknesses, and often
53
made of different materials, it is hard to separate the effects of string length from the other changes in
the tone and pitch, but they contribute significantly to the differences between notes played in low and
high positions.
The string length is, of course, less available to the player as a tool than the contact point, bow force or
bow speed, because it affects the pitch. Rather than freely choosing from a continuous range of options,
the player must select from either two, three, or at the absolute most four finger positions which will
produce the appropriate pitch. Since it is difficult to change positions, the choice of which position to
play in is often motivated by other technical considerations, not by the sound. Nonetheless, players
often exploit the tonal differences produced by different positions and strings, and adjust the bow force,
contact point, and bow speed to compensate.
2 The Fingers of the Left HandOften, the behaviour of the left hand is determined by other technical concerns, or covered by the use
of vibrato, but it can also be used to subtly affect the tone. This is because the fingers of the left hand
absorb energy from the string. The surface of a finger is softer than the nut, which stops an open string.
The softer the finger is, the more energy it will absorb from the string. Pressing harder on the fingertips
stiffens the flesh, and makes them better reflectors of energy. This means that to a small degree, the
player can affect the tone by pressing harder or softer with the fingers of their left hand and making
them, respectively, less or more absorptive. Damping by the fingers has a very small effect on the
sound, but it is noticeable under certain circumstances, and it is relatively easy for the player to control.
The Bow StickThe stick of the bow is incredibly important to players. It mediates between the bow hand and the
instrument, absorbing and storing some force and transmitting the rest through the hair and into the
string. This spring-like characteristic of the bow is vital to string players, because it helps to
compensate for the natural deficiencies of the human body. Most of the experiments described in the
first section of the paper use a stiff, rosined stick to stimulate the string, not a real bow, and little has
been discovered about the effect of the bow stick itself on the string. Its effect on the player, however,
is undoubtedly great.
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The weight of the bow stick varies throughout its length, such that it is effectively lighter at the tip, and
heavier at the frog. The mechanics of the bow arm exaggerate this effect, because the tip of the bow is
also the end of a lever, which has its fulcrum in the bow hand. The most obvious way to compensate
for this is to vary the pressure exerted by the bow arm, so that the arm presses down harder when the
string is nearer to the tip than the frog. This also stiffens the fingers of the bow hand. It is possible,
instead, to compensate for this with bow speed, but this can cause undesirable accents at bow changes.
The resiliency, or springiness of the bow also varies through its length, and is also affected by the
leverage of the arm. Bouncing strokes rely on the natural springiness of the bow stick, and
understanding how this changes along the length of the bow is necessary to optimize these strokes. This
is exaggerated because the tip of the bow moves through a wider arc when the bow swings up and
down than the middle or the frog do. Even in a regular, on-the-string stroke, the springiness of the bow
can make it harder to control at the tip when the bow force is low, because the bow bounces too much.
On the other hand, many players favour the tip when using more bow force, it is more forgiving than
the lower parts of the bow. Closer to the frog, the bow is less forgiving, and it is also controlled by
larger, clumsier muscles than when playing at the tip. This leads many players to avoid the lowest
portion of the bow altogether.
The tilt of the bow also affects its behaviour. When the bow is more vertical, it is bouncier and more
resilient, and when it is more tilted it is less bouncy and more forgiving. In lifted or bouncing strokes,
this means the player can adjust the tilt of the bow, as well as the length and height of their own stroke,
or the part of the bow they are using, to make it bounce higher or lower. In detaché and legato playing,
especially when the bow force is low, tilting will reduce undesirable bouncing. Tilting the bow also
complicates the mechanics of its relationship with the string.
The bow is at its mechanical simplest when it is held perpendicular to the string, and both the bow and
the string are held parallel to the ground. In this case, the bow exerts a downward force on the string, in
the same direction that gravity acts on the bow, and the string pushes directly upward. The influence of
the bow on the string, in this case, would be relatively simple. The bow would exert a net force
downward, equal to the force of gravity on the bow plus or minus the work of the player, and the string
would exert an equal force upward. In this case, the normal force would be exactly equal to this
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combined force, and the force of friction between the bow and the string would be directly proportional
to the work done by the player.
In real violin playing, however, the violin and the bow both are almost always angled, relative to the
ground and to each other, and both are in constant motion. In order to keep the bow tilted at a
consistent angle, and prevent it from falling over, the player must compensate for the force used to tilt
it by pressing with the thumb. The player must exert force in two different directions, and balance the
work done by their arm and thumb against each other. Tilting the bow thus increases the amount of
tension in the bow hold, and makes the tone-producing mechanism harder to understand and to
manage.
It is very difficult for the player to separate out these mechanical and physiological effects of tilting the
bow, which affect their interaction with it, from the effect that the bow itself has on the tone. This has
led to much confusion about the effect of tilting the bow on the tone. Rolland, for example, claims that
playing with more bow hair produces a more robust tone (Rolland 1960, 38). In fact, more bow hair
dampens the tone, as discussed above. Rolland's assertion reflects the physiological effect of tilting the
bow, which relaxes the hand and makes it easier to exert more bow force, not the physical effect, which
dampens the tone and increases surface noise. Auer, on the other hand, instructs the player to avoid
using the full ribbon of hair at the frog (Auer 1920, 53), and Galamian argues that tilting at the frog
helps to maintain a straight bow stroke, to keep the wrist flexible, and to compensate for the increased
weight of the bow (Galamian 1962, 54). This too has more to do with physiology than physics. The
physical effects of the bow hair's width on the tone, after all, do not vary over the length of the bow.
Tilting the bow, however, stiffens the hand and makes it harder to exert excessive downward force by
accident. Consciously changing the tilt of the bow will also draw the player's attention to the bow when
playing at the frog, causing them to adjust the bow force automatically.
Understanding the mechanical influence of the bow stick makes it easier to analyze different strategies
for controlling it. Being able to separate out the physical effects of the bow on the string from the
effects of the bow stick on the player, makes it possible to think more clearly about bow technique, and
to separate physiological problems (how best to control the bow) from physical ones (how to
understand the bow's interaction with the string).
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Putting it all TogetherThe model described in this chapter offers string players a better understanding of tone production on
the violin. Understanding how the bow and string interact makes it easier to solve technical problems
because it makes those problems easier to analyze, and their solutions easier to find. Understanding
how different sounds relate to one another can also make it easier to understand what the player hears
when they listen to their own sound, or to that of other players. This also makes it easier to
communicate ideas about tone quality and timbre in rehearsals and lessons. It also makes it possible to
think about technical problems more flexibly, and to find solutions more easily. A player who finds
their sound too rough or ugly, because they are pressing too hard, has at least four options:
1) Reduce the amount of bow force they are using
2) Increase the bow speed
3) Move their contact point closer to the bridge
4) Tilt the bow.
Each of these will have a different result, and understanding the physics of the bowed string will allow
the player to understand what those results will be, and why they occur. Reducing the amount of bow
force will take some of the roughness out of the sound, but will also weaken some of the partials.
Increasing the bow speed will increase the volume of the tone and may also reduce the roughness.
Moving their contact point closer to the bridge will reduce roughness, strengthen the spectral content,
and increase the loudness. Finally, depending on the circumstances, tilting the bow may reduce the
roughness and strengthen some of the higher partials. Any of these choices might solve the problem,
but knowing why and how they do so allows the player to better understand their options, and to
evaluate the success of each potential solution.
The main factors which affect the sound are the bow speed, the contact point and the bow force. The
contact point affects the underlying mechanics, by changing the restoring force which is generated by a
given displacement of the string. The bow speed affects the displacement of the string during each
cycle, as long as the bow force is adjusted appropriately, and affects the volume directly. The bow
force affects mainly the shape of the corner which travels through the string, and thus the spectral
content of the tone. It is the bow force that generates friction between the bow and the string, so it must
be adjusted according to the restoring force. Table 1 summarizes these interactions.
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Contact Point Bow Speed Bow Force
ContactPoint
Affects the restoring forceand the angles formed bythe string. Determines therange of bow forces andbow speeds that willproduce Helmholtzmotion, and thus therange of timbres andvolumes which arepossible.
The farther from thebridge the contactpoint, the more bowspeed is required todisplace the string thesame amount. Closer tothe bridge, a smallerrange of bow speeds ispermitted, and the bowspeeds will be slower.
The closer to the bridge,the more bow force isrequired to maintainHelmholtz motion, andthe narrower the range offorces which willgenerate it. Farther fromthe bridge, less bow forceis required, and a greaterrange of forces arepermitted.
Bow Speed Each bow speed is onlypossible at a certain rangeof contact points, evenwhen the bow force isadjusted suitably. Fartherfrom the bridge the rangeis greater.
Increases or decreasesthe displacement of thestring during eachcycle of motion.Affects the volume.
When the bow speed issufficiently high, the bowforce must also beincreased to maintainHelmholtz motion.
Bow Force Each bow force will onlyproduce normalHelmholtz motion at acertain range of contactpoints.
Determines the rangeof bow speeds whichwill produceHelmholtz motion at agiven contact point.
Sharpens the Helmholtzcorner. Determines theforce of friction betweenthe bow and the string(and thus the normalforce). Affects the timbre,and thus the perceivedvolume. High bow forcecan cause pitch flatteningand a rough tone qualityeven within the limits ofHelmholtz motion.
Table 1: Bow Force, Contact Point, and Bow Speed
Several other factors can affect the tone, and each can also be understood in terms of its effect on the
angles created in the string and the Helmholtz corner. The width of the bow hair absorbs some
vibrations from the string, rounding out the corner more than a perfectly thin bow would. It also causes
noise effects and pitch-flattening, both of which increase dramatically as the bow force rises. The
length of the string which is free to vibrate affects the angles formed by the bow, and the amount of
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bow force which is required at a given contact point. This means that the contact point must be moved
closer to the bridge, the higher up the fingerboard the string is stopped, and that the new contact point
will allow a different range of bow forces and speeds than were permitted at the previous contact point.
The fingers of the left hand can also have a greater or lesser damping effect on the string, smoothing
out the shape of the corner more when the fingertip is soft and less when it is harder. Table 2
summarizes these effects.
Width of the bow hair Dampens the overtones. More width = rounder corner.
Causes noise and pitch flattening.
Length of string vibrating Affects the angles formed by the string, and the effect of differentcontact points.
Affects the range of bow forces allowable at a given contact point.
Fingers of the left hand Absorb some partials, rounding the corner. More harmonic contentis absorbed when the contact with the finger is softer, less when thecontact is harder.
Table 2: Secondary Factors
Even easier to understand is how a given tone can be transformed by changing one or more of the tone-
producing factors. Adding bow force makes the corner sharper, and reducing the bow force makes it
smoother. Moving the bow closer to the bridge increases the angle formed by the string and makes the
corner sharper, and moving it away does the opposite. Table 3 summarizes how changes in five of the
six factors will affect a given tone, if each of the others is held constant (String length is left out
because it cannot be freely altered).
Contact point moves closer to thebridge.
Changes the angles created in the string, and the restoringforce. As long as bow force and bow speed are stillwithin limits, tone becomes louder and brighter.
Contact point moves farther from thebridge.
Changes the angles created in the string, and the restoringforce. As long as bow force and bow speed are stillwithin limits, tone becomes quieter and less bright.
Bow speed increases. String is displaced farther during each cycle of vibration.As long as the other factors allow, the volume increases.
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Bow speed decreases. String is displaced less during each cycle of vibration. Aslong as the other factors allow, the volume decreases.
Bow force increases. Increases the force of friction between the bow and thestring. As long as everything still falls within acceptablelimits, the corner is sharpened and the tone becomesbrighter. When bow force is high, it can delay the releaseof the corner, flattening the pitch, and increase thesurface noise created by the bow hair.
Bow force decreases. Decreases the force of friction. As long as everything stillfalls within the limits, the corner becomes more roundedand the tone less bright. Can also decrease surface noiseand pitch-flattening.
Bow tilt increased. Decreases the noise-producing and pitch-flatteningeffects of the bow hair. Bow becomes less bouncy.Interactions between bow and string become morecomplicated, because the angles are increased.
Bow tilt decreased (bow morevertical).
Increases noise-producing and pitch-flattening effects.Bow becomes more bouncy. Interactions between bowand string are simplified.
Fingers of left hand press harder. If the flesh of the finger pad is stiffened, the fingersabsorb less energy from the string and dampingdecreases. If the finger pads are already as stiff aspossible, has no effect.
Fingers of left hand press morelightly.
Increases damping, until the pressure no longer sufficesto stop the string.
Table 3: Effects of changing tone-producing factors
Players and teachers who understand these fundamentals will find it much easier to communicate
clearly in the studio and the rehearsal hall. A clear understanding of these phenomena allows players to
efficiently think through problems of tone production, and visualizing their effect on the corner makes
it easy to predict the results of their actions. Armed with an objective model of the mechanism of tone
production, players will find it much easier to evaluate the results of their actions and solve technical
problems, and teachers will find it easier to explain their ideas to students.
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Chapter 3: Analysis of Twentieth-Century Pedagogical WorksUnderstanding the physics of tone production makes it possible to analyze the ideas, exercises and
opinions that violinists encounter in the studio. String teaching is firmly rooted in its pedagogical past,
and all experienced players inherit a mix of concepts and practices from their teachers. Many of these
ideas and exercises have been passed on from teacher to student over generations, and have been
variously recycled, re-jigged, re-purposed, and re-focused over the years. The works examined here
represent, at best, an incomplete record of pedagogy in the twentieth century. Even as a record of their
own authors' practice, they are occasionally inadequate, but they do suggest a gradual development in
concepts of tone production and technique in the pedagogical community.
Over time, descriptions of tone production become more clear and more accurate. The two earliest
authors examined, Lucien Capet and Leopold Auer, clearly indicate that moving the contact point can
affect the tone, but they do not explain what effect it has, or how to adjust the bow force and speed as it
does so. Carl Flesch offers a more complete picture, discussing the relationship between contact point,
bow force, and bow speed, and exploring a range of different timbral possibilities. Kató Havas focuses
her discussion on the left hand, largely ignoring the bowing parameters, but Rolland and Galamian
describe the model even more neatly and completely than Flesch. Finally, Simon Fischer attempts to
update the model completely. His descriptions of the underlying physics are somewhat problematic, but
his exercises represent a thorough exploration of the mechanism of tone production which accounts for
nearly all of its important aspects.
As the relationship between bow force, contact point, and bow speed became better understood, it
became easier to describe and to teach. This manifests itself both in the clarity of the language the
authors use to describe the relationship, and in the increasing variety of expressive tools it makes
available. A notable example is the bow speed. Capet's most important contribution to the tradition of
violin pedagogy was his work on bow distribution. By learning to play with even bow speed, the player
must also learn to compensate for the weight of the bow, and in doing so they gain significantly better
control over their bow arm. Capet actively discouraged his readers from altering their bow speed for
expressive effects, because he thought that a constant bow speed was necessary to maintain tone
quality. Flesch similarly argues that the tempo and written bowings in a piece of music will limit the
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player's freedom so much that bow speed should largely be considered as fixed, and he makes the
contact point the center of his model of tone production. By the time Rolland and Galamian were
writing, Capet's techniques and Flesch's ideas had been thoroughly digested by the teaching
community. Establishing control over the bow speed was still a challenge every student had to face, but
it was no longer cutting edge technique, and conscious manipulation of the bow speed could be more
fully integrated into a mature technical practice.
Fischer integrates much of the work of these earlier pedagogues into his own exercises. Many of his
explanations echo the language and ideas of the works presented in this paper, and he openly
acknowledges his debt to the past:
Many of the exercises in this collection have been used widely for decades, and in somecases for centuries. Their exact origin is difficult to trace because they have been sowidely practised. Others have been used so much by certain teachers, though notnecessarily invented by them, that the exercises have become associated with them. . . .Some of the exercises are adaptations of traditional methods, while others are my own.But in a field as old and widespread as violin playing, new ideas usually turn out to havebeen thought of before (Fischer 1997, vii).
What the physical model of tone production offers, above all, is a chance to move past this situation. Of
course, teaching methods and exercises will always change and evolve, disappearing and reappearing
as the needs of students, teachers, and players change. The basic physical reality of tone production,
however, does not change over time. Science is ever refining its understanding, but many of the
fundamental mechanics of the system, at least as they affect the violinist, are clear, and this makes it
easier to objectively understand the problems that technique is designed to solve.
Aside from illustrating some of the changes that have taken place in teaching practice, this section aims
to show how an understanding of the underlying physics can make it easier to engage with written
works of violin pedagogy. Without an objective sense of how the violin works, it is difficult to engage
with these works usefully. With that understanding, it is easier to sift through these works and to
fruitfully analyze the ideas found within them. Havas, for example, wildly exaggerates the importance
of the left hand on physical tone production. This does not invalidate her other ideas, it does mean they
must be re-evaluated in the light of a more accurate understanding. More importantly it means they can
be re-evaluated, rather than simply accepted or rejected. Parsing the works in this way, separating out
physiological/technical concepts from physical/mechanical ones, and correcting for misconceptions,
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will make these works more useful to future players, because they can better engage with them
critically.
CapetThe central concern of Capet's great work, Superior Bow Technique, is overcoming the mechanical
complexities of the bow itself. Capet waxes philosophical about the variety of tone colours it is
possible to produce on the violin, but in fact the work is largely focused on maintaining consistent tone
quality under different circumstances. To this end, Superior Bow Technique lays out a series of
exercises which are designed to teach even bow division. With the bow speed thus regulated, Capet
argues that dynamic nuance can be created by the use of bow force, which can be controlled by the
fingers of the right hand. Training these fingers to function individually is the other great theme of the
work. Capet writes in great detail about the role of each finger, and how they can be used to regulate
the motion of the bow by exerting horizontal and vertical pressure. The contact point is discussed only
obliquely, in discussions on the effect of the horizontal pressure of the fingers. It clearly played a role
in Capet's technique, but the role is not well defined. Capet never discusses how the contact point
affects the sound, except to imply that changing the contact point is an important way to vary the tone.
The roulé stroke, which is often called the “bow vibrato”, is in fact primarily a way of sensitizing the
fingers of the bow hand, so that they can better control the contact point. Roulé involves rolling the
bow back and forth along the surface of the string, which has the effect of very slightly changing the
contact point, or at least the way that the bow force is distributed across the ribbon of bow hair which
touches the string. In practising roulé, players will experience, however slightly and subtly, the effects
of different contact points, and will come to understand the bow position subconsciously. Even though
he does not provide an explanatory model for the effects of the contact point, Capet's system thus
makes it available for the player to manipulate unconsciously
One of the fundamental aspects of Capet's technique is his preference for a constant bow speed. Pages
and pages of examples illustrate how the bow speed can be regulated by assigning rhythmic units to
different portions of the bow's length. Generally, in long runs or slurs, Capet asserts that “it is always
preferable to divide the bow in as many equal parts as there are notes in the same bow stroke. Dynamic
nuances should be obtained by different pressures of the fingers on the stick and not by irregular
divisions of the bow” (Capet 1916, 17). Capet's italics in this passage typify his recurring expressions
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of distaste for the deliberate use of bow speed to achieve expressive effects. Maintaining a constant
bow speed makes it easier to keep a consistent tone through long passages, especially those involving
slurs, and it clearly separates the large-scale decisions, like dynamic or tone colour, from the smaller-
scale nuances that interest Capet.
A second important aspect of Capet's technique is his emphasis on the individual fingers of the bow
hand. The lever which is formed by the contrary pressures of the index and pinky fingers, with the
thumb and middle finger acting as a fulcrum, makes it possible for the player to precisely control the
motion of the bow by exerting force in different directions. In order to control these competing
pressures, Capet explains, the player must develop a fine-grained sensitivity in the bow hand: “Our
fingers must be literally [sic] antennae to allow us to penetrate this mysterious world where few beings
occupy the circumference and of which the center is Beauty”12 (Capet 1916, 13). Capet identifies two
primary types of finger motion, which can be applied in various combinations for different effects
(Capet 1916, 12). Vertical motion can be used to apply or alleviate bow pressure, affecting the bow
force, or to regulate the balance of the bow, as in lifted and bouncing bow strokes. Horizontal pressure
can be used to change the contact point, when uneven bowing patterns make this necessary, and to
augment the vertical force in heavy bow strokes. Capet's discussion of the bow force is much more
detailed and clear than his discussion of the contact point, and uniquely among the authors discussed in
this chapter, he considers it to be the player's primary means of expression.
The role of the contact point in Superior Bow Technique is somewhat ill-defined. Capet seems to have
been unclear about the relationship between contact point and bow force, and his language often
conflates them. He talks about “possessing” and “penetrating” the string, in order to master its tonal
possibilities, but he never discusses the relationship between contact point and tone directly. His
clearest statement on the subject is that in order to produce a variety of tones, “it is necessary to add to
the vertical pressure – which is due to the resilience of the stick on the hair – a sort of horizontal
flexibility, which increases the sensitivity of this pressure” (Capet 1992, 28). “Horizontal flexibility”,
of course, refers to the ability to change the contact point, but acknowledging that the contact point has
an effect on the tone and the dynamics is as far as he goes in explaining its effect on sound. Capet does
assign the contact point a role in determining the dynamic (Capet 1916, 24), and in solving problems of
12The capitalization is Capet's.
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bow distribution, but this role is only secondary to the role of the bow force. “It sometimes happens”,
Capet notes, “that one finds oneself in the presence of bow divisions that are unequal and necessary”.
In these cases, “It falls upon the sensitivity of the fingers, in the horizontal movement to overcome this
inconvenience, preserving the same quality of sound on the two notes, as well as the same strength of
attack” (Capet 1916, 21). In other words, the contact point must be moved, so that the dynamic can be
maintained while the bow speed changes. Even without a general model of how the contact point
affects the tone, Capet understood that the contact point must be altered under some circumstances. His
most famous technique, the roulé stroke, represents an attempt to sensitize the player to the bow's
horizontal motion, and to teach them how to control it, even without an abstract model to explain how
and why the contact point should be altered.
The roulé stroke is well remembered by its unfortunate nickname, “bow vibrato”. Capet seems to have
intended it as a way to sensitize the player to motions of the bow, particularly the horizontal motions
which determine the contact point. Roulé involves rocking the bow back and forth across the string, so
that it alternates between leaning towards the bridge and away from it. “By means of this bow stroke”,
Capet claims, “one has at his disposal a kind of vibrato of the bow which is an extreme sensitivity to
the feeling of penetration” into the string (Capet 1916, 29). The “feeling of penetration” Capet
describes, seems to refer to the sensation in the hand as the bow is moved to different contact points.
Different amounts of vertical pressure must be used in different parts of the string, and the horizontal
force must be used to move the bow, so the sensitivity Capet describes can be understood as a way of
understanding the contact point as well as the bow force. The association of this stroke with vibrato has
led many to assume that the motion of the bow affects the sound directly, causing some kind of
oscillation equivalent to left hand vibrato. The back and forth motion of the bow used to practice the
roulé suggests this connection, but Capet states explicitly that “this movement of the penetration of the
hair on-the-strings should not be visible at the time of performance” (Capet 1916, 30). Instead, the tone
will be so improved by the sensitive use of vertical and horizontal pressure to guide the bow, that it will
sound more refined even without the help of the left hand.
With the role of the contact point so obscurely explained, and the bow speed kept constant, Superior
Bow Technique relies on the bow force to control most expressive effects. Accents, string changes, and
dynamic nuances are all assigned to the bow force, and the importance of adjusting the vertical
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pressure in double-stops and high positions are also minutely examined. All of this speaks to Capet's
confusion about the role of the contact point and the bow force. As careful as he is to distinguish
between the vertical and horizontal forces at the beginning of the work, his descriptions in the body of
the text often blend the two. Accents, for example, can be produced with vertical force alone, as Capet
suggests, but dynamic nuances require either changes in the bow speed, or bow force and contact point.
The exact mix of pressures which will produce this is left vague in Superior Bow Technique, because of
Capet's limited understanding of the underlying physics.
A modern player, armed with a robust model of the physics of tone production, can easily fill in the
blanks in Capet's model and put his exercises to good use. The roulé, for example, is clearly designed
to teach players how to adjust their bow force and contact point. Without a clear model of the
relationship between bow force and contact point, Capet's attempts to tone production get bogged down
in vague poetic descriptions, but a player who does understand this relationship can still use the roulé
exercises in Superior Bow Technique to build sensitivity in their bow hold. Even better, they can
expand on them and integrate them into an exploration of different contact points. Similarly, one need
not adopt Capet's bias against the expressive use of bow speed in order to use his bow division
exercises, or his string changing exercises.
AuerAuer's Violin Playing as I Teach It (1920) is a frustratingly inadequate summary of its author's
approach to technique. Auer shares many fascinating anecdotes from his own adventurous and globe-
trotting career, and many fascinating observations about the famous violinists he encountered in his
travels, but he shares much more about his teaching and aesthetic philosophies than he does about his
actual technique. Auer's main concern seems to be avoiding a harsh tone, either by playing too close to
the bridge or by using excessive bow force. To this end, he argues that the bow should be controlled by
the wrist of the bow arm, not the fingers or the arm itself.
Violin Playing as I Teach It, for the most part, provides only the most general advice on tone
production. The student should be taught to produce a good tone, using the full length of the bow. They
must learn to compensate for the uneven weight of the bow, and to play on all strings, in all positions.
Sufficient pressure must be used to set the string into motion, but no more should be used than is
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absolutely necessary, lest it inhibit the tone. The contact point should generally be kept between the
bridge and the fingerboard, “for it is within this compass that the tone is most full and sonorous” (Auer
1920, 21), but it can be moved over the fingerboard “when it is desirable to secure a very soft, sweet
tone” (Auer 1920, 21). Finally, occasional vibrato can be used to tastefully decorate the sound, but “the
tabasco of continuous vibrato” (Auer 1920, 23) must be avoided at all cost. These general descriptions
suggest that Auer's technique exploited many of the aspects which govern tone production effectively,
but do not explain how he did so.
Much of the advice contained in Auer's work concerns avoiding a harsh tone. “As soon as you play
near the bridge with any degree of strength”, he cautions, “the tone grows harsh” (Auer 1920, 21-22).
He also warns the student against the risks of excess bow force in general: “Above all, do not try to
bring out a big tone by pressing the bow on the strings. This is an art in itself, and can only be
developed by means of hard work and experience” (Auer 1920, 20). The task of regulating the bow
force is given not to the fingers of the bow hand, as Capet advocated, but to the wrist: “Do not press
down the bow with the arm: the whole body of sound should be produced by means of a light pressure
of the wrist, which may be increased, little by little, until it calls forth a full tone” (Auer 1920, 20). Like
Capet, Auer relies on experience rather than theory to teach the limits of this force: “The degree of
[force] to be applied to the stick is a question of experience, of observation from the instructive side,
and also of discipline” (Auer 1920, 21). Auer's desire to avoid rough sound reflects a sensitivity to the
noise effects described by McIntyre, Schumacher, and Woodhouse (1977), which several other
pedagogues shared. Building up the sound from the lowest possible bow force, and consciously
controlling the bow force with the wrist rather than the arm, both represent strategies to avoid these
rough sounds.
Of all the pedagogical works considered in this paper, Violin Playing as I Teach It yields the least to
analysis, because it contains the least information about its author's technique. Undoubtedly, the
teacher of so many famous violinists understood more about tone production than is contained within
its pages, but Auer seems to have been almost allergic to systematic thinking. Everything in Violin
Playing as I Teach It accords with the work of modern physicists and with common sense, but mostly
because it is so non-specific. Reading Auer's work makes the benefits of a scientific understanding of
tone production painfully obvious, because it so clearly lacks such a model.
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FleschThe Art of Violin Playing (1923) stands on its own as a monument of thoughtful, critical violin
pedagogy. Flesch's work has never had as large a following in North America as it does in Europe,
possibly because for years the first volume of the work was nearly impossible to find in English, but his
international influence is immense. Almost every aspect of tone production is examined in various
parts of The Art, but the discussion is not unified in a single section. Flesch's later work, Problems of
Tone Production (Flesch 1934), collects the core elements of his thinking on tone production into a
more compact form, but leaves out some of the elements which this paper is concerned with. The core
of Flesch's approach to tone production is his treatment of the contact point. Where Capet and Auer
drop hints about the rich variety of tones the violin can produce, Flesch provides a practical guide to
tone production which explains how these tones can be produced. Flesch's model covers nearly all the
aspects of tone production discussed in this paper, and his advice is largely consistent with a modern
physical understanding of the instrument.
Flesch saw the contact point as the key to understanding tone production. Long before physicists were
able to explain why changing the contact point would affect the timbre, Flesch was able to clearly
describe how it did so, and to describe its relationship with bow force and bow speed. In addition to the
tonal differences between the strings, he points out, the violin is able to vary the tone on each string
through various combinations of contact point, bow speed, and bow force. As a short-hand, Flesch
suggests dividing the range of possible contact points into five areas: On the fingerboard, near the
fingerboard, at the centre, near the bridge, and on the bridge (Flesch 1934, 8). Each of these will
typically, although not exclusively, be used for different types of tone. Flesch provides three examples
which illustrate how the different areas of the string can be used to produce different tones:
1. “Flute timbre”, which is produced near the fingerboard, with a very light and fast bow,typically for fast notes and quiet dynamics
2. “Clarinet timbre”, which is produced between the fingerboard and the bridge, withlonger, slower bows, for notes of moderate length and dynamic
3. “Oboe timbre”, which is produced close to the bridge, with very long and slow bow, andtypically used for very long notes and loud dynamics (Flesch 1923, 77)
Of course, it is equally possible to play a long, quiet note close to the bridge, with very slow bow (a
technique known as son filé, which Flesch discusses elsewhere), or to play loud dynamics far from the
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bridge. With the help of these three examples, Flesch argues that “It is possible to localize and establish
more precisely the vague ideas hitherto existing as to the so-called 'hard' or 'weak' parts of the string”
(Flesch 1934, 8), and develop a more consistent and unified theory of tone production.
Flesch was also well aware of the mutual interdependence between the contact point, bow speed, and
bow force. The bow force must be sufficient to set the string in motion, but not so great as to choke the
sound, and it must be varied according to the contact point and the bow speed. Flesch describes it as
contingent on all the other factors which determine the sound:
A healthy tone is . . . mainly dependent on the correct dosage of the weight to beapplied. This, in turn, is subject to the greatest variations, in particular the prescribeddynamics and shadings, the duration of the stroke, the properties of the part of the bowwhich is being used, the length of that part of the string which is vibrating, and finallythe number of strings which are involved (Flesch 1923, 62).
The bow speed, on the other hand, is often limited by the exigencies of bowing. Like Capet, Flesch
displays a preference for constant bow speed. Almost all of his instructions for altering the tone involve
only the contact point and bow force. In fact, Flesch only instructs players to alter their bow speed in
two exceptional cases: In the case of crescendos in a rapid tempo, where the bow speed is less
restricted because the tempo is fast, or in cases where the physics of the bow work against creating a
crescendo through pressure (Flesch 1923, 72). Generally, he considers the bow speed as a constant, or a
given, and makes the contact point responsible for changes in dynamic as well as tone colour.
Flesch also discusses the effects of playing on different strings, and in different positions, on the tone
and the contact point. In the same passage that he describes the flute, clarinet and oboe timbres, he
describes the characteristic tone of the four strings:
The E-string possesses the keen edge of the dramatic soprano, but also the lovelylightness of the coloratura soprano. The A-string, on the other hand approaches thetimbre of the mezzo-soprano. Does the D-string not remind one of a 'pastose' (mellow)alto voice, and can the G-string not rival a triumphant and enthusiastic tenor – withoutthe pitch limit of the high C? (Flesch 1923, 76).
The basic tone of the strings is altered by the use of the bow, but playing in different positions also
affects the tone. Many of the notes on the violin can be played in three or four different positions, but in
practice the position must be chosen to suit the whole passage, not just the individual notes. “One
should never hesitate in deciding upon a difficult change of position if the tonal picture can be brought
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to greater perfection through it” (Flesch 1934, 17), argues Flesch, but he is also aware that the choice
has practical ramifications for the right hand.
Playing in high positions, Flesch notes, requires that the contact point be moved closer to the bridge
(Flesch 1923, 63), and also requires that the bow force be changed. For some passages, a lower string
may yield a darker tone, but make the bow stroke impractical. For others, a lower position might make
for smoother left hand action, but an undesirably harsh tone. When moving between a low and a high
position, the contact point must be consciously adjusted to suit the new situation (Flesch 1934, 17).
Double-stops and chords, Flesch notes, present a special problem, because each string is stopped in a
different place and thus requires a different contact point (see Problems of Tone Production, 12-13). In
a section of The Art titled “Fingerings in Relation to Tone Colours” (Flesch 1923, 133-136), Flesch
presents a series of musical examples which illustrate how the player can balance these musical and
technical requirements. He also suggests a few guiding aesthetic principles:
• “For fast passages in forte, in principle bright-sounding strings should be used, alsolower positions rather than higher” (134)
• “In echo effects, the darker or brighter character of the different strings comes fully to the fore”,and can be used to differentiate the repeated notes (135)
• “Phrases in one colour should be played, as much as possible, on the same string” (135)
• “The appearance of a new tone color should take place suddenly; the effect of the newcolor should not be diluted by an unnecessary, premature use of the respective string”(136)
Flesch also recommends against using the middle strings in high positions, staying on one string for too
long, and using an open string to disguise a shift. Each of these concerns will affect the player's choice
of position, and each position will allow a different range of contact points, bow speeds, and bow
forces.
Flesch's discussion of tone production is remarkably complete. He even covers the effect of the
pressure of the fingers of the left hand, noting that “too much finger pressure causes glassy, brittle tone
quality” (Flesch 1934, 11). A clear understanding of the underlying physics makes it possible to
construct a nearly complete and basically accurate model of the underlying mechanics from his work,
but not without difficulty, because the ideas are scattered throughout his two works on the subject. A
physical understanding of the system also reveals some of the limits of Flesch's approach. For example,
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his desire for a constant bow speed severely delimits the tonal possibilities by effectively neutering one
of the three main factors the player can use to affect the tone. Also, although he outlines and describes
many of the tonal possibilities on the instrument, he provides little explanation of how the tones are
related to one another. A clear understanding of the relationship between the tone producing
mechanism, the harmonic content of the resulting sound, and the perceived timbre helps to unify these
ideas into a single system, which is easier to grasp than the series of disparate instructions Flesch
offers.
HavasOf all the pedagogues discussed in this paper, none would have benefited from a clear understanding of
the underlying physics more than Kató Havas. Havas raises many important points, and her work
contains much of value, but it also contains many misleading statements about tone production, and
cries out for a clear analysis. Havas's work is best understood as a critical response to older methods, an
attempt to develop new psychological and physical strategies which avoid pain and injury. Her primary
interest is in developing a healthy approach to playing the violin, reducing the amount of physical
effort and strain experienced by the player and simplifying the mechanics of performance. To this end,
she encourages the player to consider the balance of forces within their own body which are used to
generate motion. The simpler and more balanced the physiological mechanism which is used by the
player, the easier it will be for them to understand its use. Curiously, for a writer so concerned with
tone quality, Havas almost completely ignores the technique of the right hand. The left hand, she
argues, is the key to good tone production, because of its effect on the harmonic content of the sound.
Havas seems to take the technique of the right arm largely for granted, and her works almost
completely ignore its effect on the tone. This is not to say that she was unaware of it. Early in the first
chapter of A New Approach, she exhorts the reader that:
It is a fallacy to think that the purity and sweetness of tone, that quality which has thepower to move us so much, depends on depth of soul or feeling, talent, or goodnessknows what else, of the individual player. . . . No-as I discovered over and over again, awarm and beautiful tone has nothing to do with talent or individual personality. It is notthe outcome of hours and hours of practise and perseverance. It is merely putting theright pressure, on the right spot, at the right moment! (Havas 1961, 3-4).
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This would seem to refer to the bow force and the contact point, but Havas makes no other mention of
their effect on the tone. It could simply be that she assumed players would already be aware of it. A
New Approach (Havas 1961) assumes a working knowledge of older approaches to technique, and
directly refers to Flesch several times, so she may have believed its readers were already able to
manipulate the contact point in the manner Flesch describes. The Twelve Lesson Course (Havas 1964),
on the other hand, is devoted to beginners, and while it devotes considerable attention to playing on
open strings and establishing basic bow strokes, its discussion of tone quality is very elementary and
qualitative. Even though Havas seems to have understood the role of the bow in determining the tone,
her books largely ignore it, instead focusing on the role of the left hand in tone production.
Havas's emphasis on the left hand is partly motivated by her ideas about posture, and partly by her
understanding of partial tones. The timbre of a note, she writes, “depends on the fact that many waves
of sound that reach the ear are compound wave systems, built up of constituent waves, each of which is
capable of exciting a sensation of a simple tone if it be singled out and reinforced by a resonator”
(Havas 1961, 26). Havas understood that the relative strength of these constituent waves (partials)
determines the quality of the tone (Havas 1961, 28), but she believed that the left hand had a greater
role than the right in determining this. “The control of length of sound, its smoothness and volume, is
in the bowing arm”, she argues, but “The quality of the tone depends entirely on the left hand action”
(Havas 1964, 27). The fingers of the left hand, as has been established, can dampen the sound by
absorbing harmonic content from the string, but their effect is relatively minor compared to that of the
bowing parameters. Havas, however, treats them as a central concern, wildly exaggerating their
importance to the tone.
Ultimately, Havas argues that the actions of the whole body should be understood in terms of the left
hand, specifically in the knuckles at the base of the fingers. This represents a psychological strategy, as
well as a physiological one. “Our goal”, she explains, “is to focus all problems upon one musical point;
to collect them into one basket, so to speak. For if one concentrates only on one dominating factor,
everything else will not only follow by natural impulses but will allow the interpretation of music to
remain the uppermost point” (Havas 1961, 36). The actions of the right arm, she hoped, would become
completely subconscious, and the motions of the player's whole body could be directed by the base
knuckles of the left hand.
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The main focus of Havas's method is not the mechanics of tone production, but the physiology and
psychology of violin performance. By developing a simpler, healthier and more efficient method of
playing the violin, Havas hoped to make it easier to understand, and thus to control. “Most violinists”,
she argues, “have been concentrating on the development of those muscles which they could see in
action; hands, fingers, etc.” (Havas 1961, 14). This leads them to focus on developing those muscles,
instead of understanding the mechanism as a whole. It would be better, Havas argues, to think in terms
of balance, and to do as much work as possible with the larger muscles farther up the arm, and in the
shoulder and back.
Accounting for the balance of forces in the body allows the player to reduce the total amount of work
which is done to accomplish any motion, and creates a feeling of ease. “If from the very beginning we
create a continuous, natural see-saw-like balance in various parts of the body”, Havas posits, “the
necessary movements will spring from a self-propelled action without the need of forced or consciously
manufactured movements” (Havas 1961, 15). The key to this approach is understanding the joints.
Each joint is affected by the muscles on either side of it, and when the muscles are used in the most
efficient, natural way, the player can feel their joints moving more freely. Since “any dead weight from
the bow will immediately stop the natural vibration of the string, all artificial movement or force that
creates this pressure must be eliminated” (Havas 1961, 21). The more balanced the player's posture,
and the less their muscles work against each other, the more freely their body can move and the simpler
and more efficient its actions will be.
As this interplay of forces is better understood, it can be more easily and more efficiently controlled.
Her emphasis on the base joints does not exclude the participation of the bow arm from the action, it
merely argues that it is better to control its motion indirectly: “If the base joints are in absolute control,
the 'air-borne' right arm is free to respond and . . . all technique becomes a mere expression of tone
production” (Havas 1961, 30). Eventually, “The movement of the right arm will become so natural,
that eventually it can be completely forgotten” (Havas 1961, 25). The importance of the base knuckles
to the action of the left hand is obvious, but Havas also treats it as the focal point for understanding the
actions of the right arm. Only one technique in all of A New Approach, the martelé accent, is assigned
directly to the right hand. All other aspects of bow technique, including string crossings, legato playing
and bow strokes - even spiccato - are explained in relation to the left hand and the base knuckles. In
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each of these cases, it is explained that a focus on the base knuckles can solve problems of tone quality
and technique. Little is said about the actions of the bow arm, in each of these cases, except that it must
be coordinated with the left hand in order to function correctly.
Probably the most remarkable claim Havas makes, in regards to tone production, is that it is possible to
create a vibrato effect without oscillations in the left hand. If, instead of consciously moving the hand,
the player concentrates on relaxed and free motion of the base joints, “with the release of artificial,
forced pressure from the finger, the actual contact between fingertip and string is so delicate that the
free and regular vibration of the fundamental note and its harmonics is assured and there is a definite
sound of vibrato” (Havas 1961, 29-30). Havas argues that this is due to to the physical effects of the
lighter finger on the sound, but it could equally well refer to the natural fluctuations in the sound noted
by Schelleng, as discussed above, which were also investigated by McIntyre (1981). These smaller-
scale fluctuations in the sound would be more noticeable without vibrato than with it, and might
explain what Havas is referring to. It is also possible that Havas is describing what happens when the
tension in the left hand is reduced: players produce small vibrato oscillations subconsciously,
seemingly without exerting any effort.
Havas's work cannot be understood except as a reaction against earlier pedagogical approaches. This
reactive quality makes it difficult to assess some of her claims, because the pedagogical environment of
today is so different from the one she hoped to change. For a player first learning the instrument,
Havas's claim that the left hand was the key to good tone production might do great harm, because it
ignores the importance of the bow. For an experienced player, however, with a well-developed bow
technique, it might be helpful to redirect their attention to the left hand. Many of Havas's arguments
contain a curious blend of true and misleading statements. She states correctly, for example, that “the
quality of tone will depend on a natural and undisturbed vibration” (Havas 1961, 21) of the string, but
she incorrectly argues that the most important influence on these vibrations is the left hand. It is
possible that Havas assumed her audience would already be aware of the effects of the right arm on the
sound, but even if that were the case, her language is no less misleading for it.
An objective understanding of tone production makes it possible for a modern player to distinguish
between the physiological, psychological, and physical aspects of Havas's approach, and thus to engage
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with her ideas critically. It is not true that the left hand influences the tone more dramatically than the
right, but it is true that the left hand has a physical influence on the sound, and an individual player
might find that focusing on their left hand does improve their tone either because of its physical or its
psychological effects. It is also untrue that pressing more lightly with the left hand will, in and of itself,
create vibrato, but it might cause the player to adjust the actions of the right arm automatically, or it
might refocus their attention on the tone. Being able to analyze Havas's works objectively allows a
modern player to explore her ideas about technique, posture, and tone production without being misled
about the nature of tone production.
RollandRolland can be considered one of the most influential modern writers on string pedagogy, because of
the success of the book and video series, The Teaching of Action in String Playing, which is still well
regarded in the pedagogical community. The Teaching of Action in String Playing focuses on teaching
strategies and movement training, but largely ignores the question of what technique should be taught.
Rolland's earlier and lesser-known work, Basic Principles of Violin Playing, on the other hand, is a
remarkably succinct guide to basic violin technique. In it, Rolland lists nine factors which can affect the
tone:
1. Contact point2. Bow Speed3. Bow Force, which Rolland calls the Bow Pressure4. The “tension” of the stroke5. The amount of bow hair touching the string6. Articulation7. Vibrato8. The instrument itself, how well it is adjusted, set up, and accessorized9. The acoustics of the room (Rolland 1960, 32-33)
The last three of these are beyond the scope of this paper, and the fourth, “tension”, really just means
the proportion between the bow force and the bow speed. Rolland's discussion of the effects of the left
hand however, which he discusses as an aspect of articulation, and his description of the effects of the
bow hair, will bear some discussion.
Rolland describes the contact point, bow speed, and bow force individually, but he argues that they
must be adjusted to suit each other, and that each depends on the thickness and vibrating length of the
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string. His emphasis on the mutual interdependence of these factors leads him to exaggerate, arguing
that “the triumvirate of bow speed - pressure - contact point is such that a good tone will result only
with a specific constellation of all three factors. A good tone is the evidence of correct proportions, and
if we change one of these factors, the tone will necessarily be spoiled” (Rolland 1960, 37). Rolland also
describes the effect of playing on different strings and in different positions on each factor. “The lower
strings are less responsive near the bridge”, he points out, “therefore, the CP [contact point] generally
shifts toward the fingerboard when playing on them. In general, the higher the string, the closer the
contact point should be to the bridge” (Rolland 1960, 34). More bow speed is also recommended on the
upper strings (Rolland 1960, 35), and more pressure on the lower strings (Rolland 1960, 36). The
shorter the length of the string, Rolland observes, the closer the contact point should be to the bridge
(Rolland 1960, 34), the more bow speed will be required (Rolland 1960, 35), and “the less pressure will
be tolerated” (Rolland 1960, 37). This would mean that in order to produce an acceptable tone, the
contact point, bow force, and bow speed would need to be adjusted every time the player changed
notes.
In fact, the vibrating string is more forgiving than this. As Schelleng's diagram illustrates, in a real
string each factor can be adjusted within a certain range, without either of the other factors needing to
be adjusted. When one factor is changed, the limits for each of the other factors changes, but the tone
will not suffer unduly as long as they still fall within the new limits. Some of Rolland's statements
indicate an awareness that this might be the case. He argues, for example, that “increased pressure has
less effect upon dynamics than is generally believed. It is the quality (timbre) of the tone that changes
considerably with increased pressure” (Rolland 1960, 36). If the factors had to be perfectly adjusted to
one another, this wouldn't even make sense, because it would be impossible to separate the effect of
one factor from another. His statement that “the volume of the tone is much more dependent on bow
speed than on bow pressure” (Rolland 1960, 35) suffers from the same problem. It is meaningless to
say that the volume is more dependent on bow speed than pressure if both they are completely
interdependent on each other in the way that Rolland describes. It is easy to understand why a work
aimed at beginners might over-emphasize the inter-dependence of bowing factors. For the player, it is
far more important to establish the general relationship between the factors than it is to establish their
limits, and the consequences of adjusting too conscientiously are much less dire than those of not
adjusting enough.
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Rolland identifies three areas in each string, which he uses to illustrate the effect of the thickness of the
strings on the contact point, and of the contact point on the sound (see Figure 27). The angled lines
which delineate these “zones” reflect the effect of string thickness on the contact point.
Figure 27. Rolland's “Zones”Rolland 1960, 33
Zone I, he argues, is “for long, sustained strokes and loud playing”. In zone one, “the string is stiff and
will not respond to a light touch of the bow unless the bow speed is quite slow” (Rolland 1960, 33).
This means that loud sounds, and slow bows are favoured. Rolland also points out that in zone I, “the
high partials of the string are favored and the tone has a brilliante and intense, oboe-like quality”
(Rolland 1960, 34). Zone II is “for normal playing conditions”. It represents “a medium position
between bridge and fingerboard [in which] the bow can be moved fairly fast with medium to high
pressure (loud), or slowly, with low or medium pressure” (Rolland 1960, 33). This offers the player a
great deal of flexibility in what means they use to generate sound, and it “therefore is the logical spot
for most playing, especially on the elementary level” (Rolland 1960, 33). Rolland describes the tone
quality in zone II as “neither too brilliant nor too mellow, but [with] a full round quality, somewhat
similar to the clarinet tone. The full singing quality is the most desireable normal sound of the violin”
(Rolland 1960, 34). Finally, in zone III, “for Soft playing and for swiftly moving light bows” (Rolland
1960, 33-34), Rolland argues that the string “responds easily, and lends itself to light, effortless
playing” (Rolland 1960, 34), and “the tone has a flute-like, velvety quality (flautando); soft and
mellow” (Rolland 1960, 34). Each of these zones will require different bow speeds and bow pressures,
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and will typically lend itself to creating certain timbres, as Rolland describes, but there is no need for a
player to limit their tone palette to these typical combinations. It is just as possible to play piano in zone
I, close to the bridge, and loud in zone III, over the fingerboard as it is to achieve these more typical
colours. Rolland's choice of oboe, clarinet and flute, to describe the three zones, is either consciously or
unconsciously copied from Flesch (Flesch 1923, 77).
Rolland's concept of the “tension” of a bow stroke must be understood as primarily a physiological and
acoustic concept, rather than a physical one. By the tension of the stroke, Rolland means the balance of
bow force and bow speed: “The greater the bow pressure and the slower the stroke, the higher is the
tension” (Rolland 1960, 38). Physiologically, these kinds of bow strokes must be understood to be quite
literally “tense”, as they require more energy and muscle control to execute. Acoustically, the tension
of a note must be understood as referring to the relationship between a note's volume and the richness
of its harmonic content. The most tense timbres will be played with a slow bow and high bow force.
These notes will have stronger partials, because the bow force sharpens the Helmholtz corner, but
relative to a less tense tone with the same apparent loudness, the fundamental frequency will be weaker
because of the slow bow speed. The least tense notes will have weaker partials, because of the low bow
force, but relatively strong fundamentals, because of the high bow speed. Two notes which have the
same apparent volume, but different timbres, can thus be compared. One will be less tense and the
other more tense, by Rolland's definition. The more tense the tone, the stronger the upper partials will
be relative to strength of the fundamental. Tension can also be used to describe tones more generally, in
the teaching studio or in rehearsal, and to compare tones of different dynamics and timbres.
Rolland also makes an important distinction between the pressure exerted by the player and the force
experienced by the string. Since the bow is unevenly heavy, and its relationship with the arm is
controlled with leverage in the hand, it is necessary for the player to exert significantly more force at
the tip than at the frog. This human pressure is treated separately from the pressure exerted on the
string, usually called the bow force. Rolland treats it as a fundamental technical issue, which must be
addressed early on. Practically the first task he sets before the beginner is playing an even tone with a
straight bow and a consistent speed, using “positive pressure”, which enters the bow through the index
finger, and negative pressure, which enters through the little finger, to compensate for the uneven
weight of the bow (Rolland 1960, 8). Once the player has learned to do this, the question of human
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pressure is treated as settled. Rolland clearly considers it separately from the bow force, which
represents the net pressure that must be exerted on the string at various contact points and bow speeds.
Basic Principles accurately describes a general theory of the bow hair's effect on the sound, but draws a
few questionable conclusions about how to exploit it. “If the bow is too heavily haired”, Rolland notes,
“The strings will not vibrate freely; surface noises and a dull tone will result, due to the absorption of
the upper partials. If the hair is too thin, a clear (but often thin), glassy tone will result” (Rolland 1960,
38). This describes the general damping effects of the bow hair noted above. On the question of surface
noise, however, Rolland is less reliable. He advises adjusting the tilt of the hair to avoid surface noise
in soft playing, arguing that the effect of surface noise is more obvious when the volume is low, but
using the full width of the bow in loud playing, where he argues the surface noises will be covered by
the loud tone (Rolland 1960, 38). Changing the tilt in this way, he points out, will also affect the
springiness of the bow, making it less bouncy in quiet playing, when the lower bow force makes it
harder for the player to control the bounce. While this strategy is convenient, because it makes the bow
easier to control, it does not necessarily represent an optimal approach to tone production. Surface
noises are hardly noticeable when the bow force is low, as it normally is for softer dynamics. Surface
noises become dramatically louder as the bow force increases, which makes it desirable to tilt the bow
for loud tones. Tilting the bow is therefore most desirable when it is least convenient, which makes the
player's decision making process about bow tilt much more complicated than Rolland implies.
Rolland's discussion of the fingers of the left hand is tied up with his discussion of articulation. By
articulation, Rolland means to include both the onset and ending of each note, as defined by the bow,
and the actions of the left hand, which must be coordinated with it. The articulation can have a
significant effect on the listener's perception of a tone. A note with a very clear beginning, for example,
can pop out of a fluid texture that it would otherwise blend into. Similarly, any problem with the
coordination between the left and right hands can interfere with the flow of the music, and make a
passage sound muddy or messy. Rolland argues that the general effect of finger pressure on the tone,
that “strong pressure hardens the fingertips, and the tone quality [becomes] harder, even glassy”
(Rolland 1960, 39), is related to the effect of articulation. “A slow initial contact with the string”, he
argues, “should be avoided because it will blur the beginning of the sound” (Rolland 1960, 39). “A
decisive, clear finger impact”, on the other hand, “will produce a ringing initial sound” (Rolland 1960,
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39). Accordingly, the player should seek for a clear, strong, initial contact, and then relax during the
note. The effect of clear or muddy articulation has a dramatic effect on the perception of a listener, but
it should not be confused with the relatively minor effect of the finger pressure itself on the tone.
Rolland's description confuses the two for the sake of simplicity.
Basic Principles of Violin Playing is remarkably comprehensive and succinct, and can be used as an
excellent resource for beginners, but it suffers from a slight lack of clarity. Describing the Helmholtz
model might not have helped Rolland to communicate with his intended audience, but understanding it
surely would have helped him avoid a few factual errors.
GalamianPrinciples of Violin Playing and Teaching (1962) presents a cogent and comprehensive overview of
violin technique. Although less exhaustive than Flesch's Art, it is also much better organized, and the
ideas are more clearly expressed. Galamian seems to have been aware of the work of many earlier
pedagogues, particularly Capet (whom he studied with), Auer and Flesch, and although he does not
refer to their work directly, some aspects of his own model seem to respond to these earlier models.
Like Rolland, Galamian describes the bow force, contact point, and bow speed as directly related.
“These three factors are interdependent”, he argues, “inasmuch as a change in any one of them will
require a corresponding adaptation in at least one of the others” (Galamian 1962, 55). He is less
insistent than Rolland, however, on the precision with which these factors must be adjusted to one
another. Galamian departs most notably from earlier models in his use of bow speed. In other respects,
his work can be seen as developing and refining earlier ideas, rather than challenging or rejecting them.
The best illustration of Galamian's approach to tone production can be found in the exercises he
includes. These demonstrate the relationship between bow speed, bow force, and contact point, and
how different combinations affect the tone and dynamic. The first of these demonstrates the
relationship between bow speed and contact point. Galamian asks the player to gradually increase the
bow speed as the contact point moves away from the bridge, keeping the bow force constant. The
player is instructed to “listen for the same resonant sound throughout” (Galamian 1962, 60), implying
that they should try to keep the dynamic steady. The second exercise demonstrates the relationship
between contact point and bow force. The player is instructed to begin far from the bridge, using fast
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bow strokes, then gradually increase the bow force and notice that the contact point must be moved
towards the bridge (Galamian 1962, 60). This would also create a crescendo, and illustrate the effect of
different regimes on the volume of the tone. Galamian also includes exercises to show how the bow
speed, contact point, and bow force can be altered to compensate for uneven bowing patterns
(Galamian 1962, 60-61). Finally, he asks that the player experiment with different dynamics, first
performing each exercise within a single dynamic and then introducing crescendos, decrescendos, and
sudden changes from one dynamic to another (Galamian 1962, 61). These exercises represent a
practical exploration of the effects of each factor on the other, and will lead the player through every
conceivable combination of factors.
Galamian's explanation of the three factors suggests a different set of priorities from earlier authors.
Although his explanations and exercises clearly indicate that each factor is equally dependent on the
others, some aspects of his writing suggest that the bow force and contact point should be less actively
exploited than the bow speed. Rather than discussing the contact point as an abstract location, for
example, Galamian prefers to use the term “sounding point”, which he defines as “that particular place,
in relationship to the bridge where the bow has to contact the string in order to get the best tonal
results” (Galamian 1962, 56). The sounding point is, therefore, the ideal contact point for a given bow
speed and bow force. This can be seen as a rejection of Flesch's approach, which treats the contact
point as the most important factor in determining the tone, and encourages the player to think about it
actively. Galamian's consistent use of the term implies that the contact point should be determined by
the other factors, a more passive conception than Flesch's.
Galamian notes that the length and thickness of the string affect the position of the sounding point. “It
is not necessary to go into the physical basis for this”, he claims, “all that needs to be understood is the
fact that on thinner strings the sounding point is closer to the bridge than on thicker strings and that in
higher positions the sounding point is also closer to the bridge than in lower positions” (Galamian, 58).
Like Flesch and Capet, Galamian mentions that this sometimes leads to complications in double-stops
and chords. He puts less emphasis on the effect of the thickness and length of strings on timbre than
earlier authors, perhaps considering this obvious.
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Galamian's description of the bow force implies that it should mainly be controlled subconsciously. At
first, he argues, the player must learn to control the uneven weight of the bow. Because of this uneven
distribution of weight, and the mechanics of the leverage created in the hand and arm, Galamian points
out that “an equal pressure or weight applied throughout the bow results in an unequal pressure on the
strings” (Galamian 1962, 57). The player's first task is thus to learn how to compensate for this, to play
with an even tone and a constant bow speed. Although he does not mention Capet specifically,
Galamian's explanation of the bow speed sheds light on Capet's approach, and that of others who call
for constant bow speed. “For a tone or tone-sequence that requires the same dynamics throughout”,
Galamian points out, “the simplest and therefore the best way to bow is with equalized speed; this will
also, under the circumstances, entail equal pressure and identical sounding point” (Galamian, 55-56).
Galamian's approach to bow technique can be seen as an extension of Capet's method. He suggests
much the same approach as Capet to uneven bowings, and includes some of the same exercises
regarding bow division. Galamian's whole approach to scales, as outlined in Contemporary Violin
Technique (1966), is designed to use even bow division to improve coordination between the right and
left hands, which echoes Capet's ideas. For Galamian, however, playing with constant bow speed is
only a preliminary requirement, along with learning to compensate for the weight of the bow.
Ultimately, he argues, bow force should be controlled automatically, through what he called the System
of Springs.
The System of Springs is Galamian's way of describing the interactions between the bow and the
player's right hand, arm, and shoulder. “The right proportion of muscular pressure and of arm and hand
pressure and hand weight”, Galamian argues, “cannot possibly be achieved by calculation” (Galamian
1962, 58). Instead, the flexible system of joints and muscles should respond automatically to the
physical parameters present, and to the player's intuition of what sound they would like to make. Once
the player has developed a good understanding of the workings of this system, and come to understand
how to compensate for the uneven weight of the bow, “the musical imagination, desiring certain
sounds, and the ear, listening attentively for positive results, will automatically bring forth the
necessary co-ordination of all elements involved” (Galamian 1962, 58). Galamian, like Auer and
Havas, argues that “what counts in tone production is not just the amount of pressure used but, if one
may so term it, the quality of the pressure . . . . It must not, under any circumstances, take effect as a
dead weight, inelastic and inarticulate, that would crush the vibrations of the string or, at best, produce
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a tone of inferior quality” (Galamian 1962, 57). When it is working properly, Galamian argued, the
System of Springs should be able to manage the bow force on its own, without the player's conscious
awareness.
The bow speed, on the other hand, was Galamian's main tool for creating nuance and expression. Its
most basic importance is its effect on the dynamic: “Greater speed of bow stroke per time unit means
greater energy transmitted to the violin. If pressure (the other energy producing factor) remains
constant, a change in speed will produce a change in dynamics” (Galamian 1962, 55). Therefore, “a
well-controlled and logical division of the bow is of the greatest importance” (Galamian 1962, 56), in
order to avoid unintended accents. But this does not mean that the bow speed should remain fixed.
Instead, it should be consciously manipulated as an expressive tool:
Where the dynamic is not meant to be even – be it for out-right accents, crescendos, thesubtle nuance that is necessary for good phrasing, the little inflection that gives life to asingle note, or whatever – a variation of speed in the stroke will be called for, mostly incombination with a corresponding increase in pressure. . . . Similarly, in phrasing, theclimax note should have more bow, while the closing note, in tapering off as it ought to,will have less. Very delicate shadings can be obtained. These subtle manipulations ofthe bow speed realize them in the smoothest way possible (Galamian 1962, 56-57).
This puts Galamian's model of tone production in direct opposition to Capet's, almost reversing the
roles of the bow speed and pressure.
In addition to his discussion of the normal bowing parameters, which generally accords with the
physical model outlined in chapter 3, Galamian makes the unusual claim that the tone quality can be
improved if the bow is slightly slanted relative to the bridge.
Until now it has been assumed that the bow is perfectly parallel to the bridge at alltimes. However, it is a fact that in drawing a singing tone at not too great a speed, themost resonant sound will be produced when the bow is at an extremely slight angle withthe bridge - in such a fashion that the point of the bow is always a little more toward thefingerboard and the frog slightly closer to the player's body (Galamian 1962, 61).
It is not clear what physical effect this would have on the interaction between the bow and the string, or
why it would have a positive effect on tone quality. Galamian insists that “the facts as stated here [that
the slanted stroke improves the sound] can be confirmed by anybody whose ear is sensitive to shades of
resonance and color” (Galamian 1962, 61), but nothing in the physics of the bow and string indicates
that there would be any mechanical benefit to the slanted stroke. It seems more likely that the Slightly
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Slanted Stroke, as Galamian calls it, would affect the player's relationship with the bow. In practice, it
is nearly impossible for a player to isolate the effect that such a slant would have on the instrument
from the effect it has on them. Slanting the bow slightly is not only more comfortable, because it
follows the body's natural movements, but it could make the player more likely to change the contact
point than if their bow was completely straight. Like Capet's roulé stroke, it represents at least in part a
strategy to encourage the player to move their contact point subconsciously.
Aside from the issue of the slightly slanted stroke, Galamian's model of tone production is remarkably
straight-forward, and follows the physics closely. His description of the different tone-producing
factors is more interesting in what it emphasizes than what it includes, and in this sense it represents a
significant development from earlier models. Even though he does not identify them, much of
Galamian's work can be seen as responding to or building on the work of earlier pedagogues. His
discussion of bow speed, for example, includes many ideas inherited from Capet, and encourages the
player to learn to play with constant bow speed and even bow distribution. These ideas, however, are
then integrated into a more complete picture of bow technique, one which leads Galamian to the
opposite conclusion as Capet in regards to the use of bow speed for expressive effect. Similarly,
Galamian builds on Flesch's model in his discussion of the interdependence of bowing parameters, but
de-emphasizes the conscious manipulation of the contact point. Galamian's practical approach to tone
production provides the tools that any player needs to explore and exploit the different tonal
possibilities of the instrument, but he shies away from discussing the physical basis for the system. As
useful as his instructions might be for the player, they provide a very approximate conceptual
framework for tone production.
FischerFischer's work stands out among the pedagogical texts examined in this paper, because it is primarily a
book of exercises and not an expository work. Fischer's ideas about technique are expressed in the
explanations for these exercises, not collected into a single argument. In a way, this exemplifies the
learning experience of most violin students, who acquire their ideas about technique and tone through
individual exercises and etudes. Some aspects of Fischer's writing suggest that he might be familiar
with some of the scientific literature on violin tone, but like earlier teachers his understanding seems to
be incomplete. The core concept in Fischer's approach to tone production is a type of sound he
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describes as “resonant”, which seems to refer to a strong tone which is free of surface noise. Many of
the exercises he includes involve adjusting the basic bowing parameters to produce different dynamics
and tone colours which are, by Fischer's definition, maximally resonant. Fischer also describes the
stick-slip cycle, the formation of the Helmholtz corner, and the raucous and double-slip regimes, but
not the cycle of motion of the string itself.
Like Galamian, Fischer uses the term soundpoint, but he does not use it in the same specialized way
that Galamian did. Fischer identifies five soundpoints, which he uses as the basis for many of the
exercises in Basics (see Figure 28). “Normally”, he notes, “The bow plays on different soundpoints
from note to note, and from phrase to phrase” (Fischer 1997, 41). Some allowance must also be made
for the lower strings, which “Are too thick and hard to respond easily when the bow is very close to the
bridge” (Fischer 1997, 41), and for playing in high positions, which changes the location of the five
soundpoints.
Figure 28. Fischer's “Soundpoints”Fischer 1997, 41.
Copyright © 1997 by C. F. Peters Ltd. Used by Permission. All Rights Reserved.
Fischer uses these soundpoints as a way to structure his exercises. Guiding the player through different
combinations of bow speed and bow force, in a variety of dynamics, Basics outlines the relationships
between the basic bowing parameters through practical exploration.
A second guiding principle in Fischer's method is his concept of resonance. This is a subjective term,
and Fischer seems to use it in this purely colloquial sense. “A very short, resonant note, such as third
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finger D on the A string, rings on after the end of the note”, he notes; “The same note played longer
should ring during the note, as well as after the note stops” (Fischer 1997, 37). Fischer links this
resonance with the ring of a pizzicato note, and his way of establishing a resonant tone is to begin with
very short bow strokes, which sound similar. These strokes, he argues, begin with a “Click”, which he
links with the formation of the Helmholtz corner and the start of the stick-slip cycle. What Fischer
seems to mean by a resonant tone is one in which a regular stick-slip cycle is established, using a
minimum of bow force. “Every shade of tone colour” he explains, “is available through the different
degrees of speed, pressure, and distance from the bridge. Proportions of more pressure to less speed are
used for more closed, darker or denser tone colours. But for the most open, freely speaking, resonant
sound, tone production is based on speed of bow, not pressure” (Fischer 1997, 48). Favouring speed
over bow force avoids the noise effects caused by the bow hair, and encourages the player to use a
relatively light, pure tone.
Fischer uses a description of the stick-slip cycle to explain the limits of acceptable tone. Even though
he does not directly refer to Helmholtz or Schelleng, this suggests some familiarity with the underlying
scientific concepts, but Fischer's explanations also suggest his understanding of the science to be
somewhat limited. He makes no attempt to describe the motion of the vibrating string, or to link the
stick-slip cycle to his discussion of timbre and dynamics, and the model he describes is somewhat
idealized:
The friction of the bow pulls and pushes the string from side to side. A magnified, slow-motion film of a bow stroke would show that during the down-bow the hair 'catches' thestring and pulls it to the right. The further the bow pulls or 'bends' the string, the morethe tension of the string increases until the tension is such that the string suddenly snapsback. The hair instantly catches the string again, and the 'catch, pull, snap-back, catch'repeats an infinite number of times (Fischer 1997, 36).
This describes the stick-slip cycle admirably, when it is in motion, but Fischer uses it as the basis for
his understanding of how notes begin. In the exercise he uses to illustrate this, he argues that “The hair
catches, pulls and releases the string once at a time, which produces a 'click' sound. This 'click' is the
sound at the very beginning of strokes such as collé, martelé, or sharply accented strokes” (Fischer
1997, 36). This certainly describes what the player will hear, but the beginning of a real stick-slip cycle
in a violin string is a much messier affair than this (See Guettler and Askenfelt 1995b), and the stick-
slip cycle does not establish itself in quite this idealized way.
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Fischer's description of the limits of violin tone also suggest familiarity with some version of
Schelleng's description of the bowed string. “If the bow is drawn too quickly or lightly”, he posits,
“there is a whistling sound because the hair skids over the surface of the string without catching it. Too
much pressure produces a torn, scraped sound because the string cannot move freely from side to side
underneath the hair” (Fischer 1997, 36). This describes the same multiple-slip and raucous regimes as
Schelleng's original work.
These passages appear to represent the limits of Fischer's understanding of stick-slip motion. He does
not describe the full motion of the vibrating string, or connect the Helmholtz corner, described above,
with the timbre of the resulting tone. Instead, exercises 56, 57, and 58 ask the player to establish a
ringing tone by gradually extending the length of notes that begin with strong accents. A resonant tone,
and thus by implication Helmholtz motion, is thus described purely in contrast to the 'click'.
Fischer equates the resonance of the sound with the amplitude of the string's motion, and he argues that
it is inhibited by bow force. “Enough pressure has to be used to engage the string”, he points out, “but
too much pressure constricts the side-to-side movements of the string and chokes the tone. For a pure
tone, pressure is always only as much as necessary for the bow speed” (Fischer 1997, 42). The
exercises, as outlined in the appendix to this paper, demonstrate how each factor can be used to modify
the sound, and how the factors can be adjusted to create this ideal tone. The player is repeatedly
instructed to “find exactly the right speed and pressure to make the string vibrate as widely as possible”
(Fischer 1997, 41) at different soundpoints. Flesch describes a desirable tone in similar language, and
argues that tones produced in this way will penetrate better in the concert hall (Flesch 1923, 69).
Although some of the exercises in Basics, such as number 69, will inevitably lead the player to use
combinations that do not satisfy Fischer's ideal, generally this idea of maximum amplitude is used as
the standard for having successfully adjusted the bow speed and bow force. If Fischer's descriptions are
taken literally, this would describe a set of tones with the weakest possible partials, relative to the
strength of their fundamentals. As noted by Schoonderwaldt, Guettler, and Askenfelt (2008), the
precise lower limits of Helmholtz motion are very difficult to define, but Fischer's “pure” sound will be
that which is closest to the lower limit of bow force, without risking the tone quality. This avoids the
noisy effects of the bow hair, which is one reason the sound is more pure, but it discourages the player
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from exploring rougher and more aggressive tone qualities, which may be desirable in some musical
contexts.
One other notable feature of Fischer's work is his discussion on the tilt of the bow. Like many of the
other authors, he notes that “more hair is used for the strongest, thickest, and deepest tone, less hair for
playing more p or dolce”, and that “lifted strokes respond differently with different amounts of hair”
(Fischer 1997, 38). Unlike other writers, Fischer claims that the tilt of the bow will affect the tone
directly. Exercise 60 is a variation of the roulé stroke. It asks the player to alternate between flat and
tilted bow hair. Fischer asserts that “the tone will be richer and thicker during the full hair” than during
the tilted part of the stroke. In practice, this could end up being true, simply because the player is likely
to unconsciously alter the bow force as they adjust the tilt of the bow. If only the tilt of the bow is
adjusted, it will have a very small effect on the sound, and the effect it will have is mostly tone-
dampening. Fischer's exercise, however, begs the question of whether the subconscious alteration of
bow force by the player might have been part of the “vibrato” effect of the original roulé.
As a theoretical explanation for tone production on the violin, Fischer's text leaves much to be desired.
As a practical resource, it is invaluable. The collection of exercises contained in the section on tone
production represent a rigorous exploration of the bowing parameters, and an excellent guide to tone
production in practice. The work suffers, however, from its incomplete understanding of Helmholtz
motion. The ideal sound that Fischer describes, pure and pretty though it may be, limits the player to a
certain set of tones, and a certain range of the tone-producing area illustrated by the Schelleng diagram.
The exercises encourage the player to develop a set of instincts, a sort of default sound, which would
ensure consistent tone quality. A mature artistic practice, of course, will inevitably lead the player
beyond this default sound, and a thorough understanding of the vibration of the bowed string would
make it easy to understand how this could be accomplished.
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ConclusionUnderstanding the physics of the bow-string interaction provides the player with an objective way to
understand the fundamental problems that violin technique is designed to solve. Chapter 3 explored
how this can be used to analyze key pedagogical texts, and to shed light on the different approaches to
violin technique that they advocate. A working model of the physics of tone production makes it
possible to distinguish between real and apparent disagreements between the authors, to separate
physiological issues from physical ones, and to identify mistaken assumptions about tone production.
This allows the informed player to resolve both real and apparent contradictions between the different
authors, and to effectively combine ideas and exercises from different sources.
Unsurprisingly, string players have developed a very sophisticated understanding of tone production,
over centuries of playing and teaching, but their ideas about it are often confused. Modern teachers
have at their disposal a virtually unlimited supply of writings and exercises available to them, but the
lack a unifying conceptual framework for their thoughts makes it more difficult for them to engage
critically with this body of work. A physical model of tone production offers the modern string player
exactly this; a coherent, consistent model they can use to analyze this material, to understand the
purpose of exercises which were sometimes written centuries apart, to understand the aims and biases
of the teachers that wrote them, and forge a synthesis to suit their own individual needs.
Every complete bow technique needs to address the following fundamental problems:
• How to compensate for the uneven weight and resilience of the bow
• How to manage and control the bow speed, contact point, and bow force, and to understandtheir effect on the sound
• When, how, and how much to tilt the bow
• How to compensate for factors the player cannot control, such as the length and thickness of thestrings, by adjusting the factors which they can control
Historically, pedagogical methods have focused primarily on helping players to overcome these
difficulties, without necessarily understanding them. Skilled teachers and talented students have
successfully navigated these problems for as long as the violin has existed, and their solutions have
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been embedded in the muscles of countless string players over the years, but understanding the
problems has its benefits too, as discussed in Chapter 2.
Ideas for Future ResearchExploring new approaches to the technique of string playing, based on an accurate understanding of the
bow-string interaction, would be the logical next step for this research to take. Historically, teachers,
and players have addressed the interaction between bow and string indirectly, through ideas about
posture and movement. Understanding the physics of the bowed string makes it possible to turn this
around, and reconsider ideas about technique and posture in terms of the problems they were designed
to solve. Four centuries of violin playing have yielded a vast repertoire of solutions to almost every
conceivable issue of posture and technique. What the model outlined in this paper offers is a new way
to compare and evaluate these approaches, and a framework around which to develop new teaching
strategies.
The model described in this paper offers a conceptual frame-work that can be used by teachers and
players to understand and communicate about tone production. Since the shape of the corner correlates
with spectral content, it is possible to predict the effect of any given action on the tone by thinking
about its effect on the shape of the corner. Given a basic understanding of the interactions which cause
Helmholtz motion, and those that limit it, this can be used as a conceptual tool for understanding and
conveying information about tone production.
One obvious approach would be to expand Cheri Collins' model (See Chapter 1) to include the effects
of corner rounding on timbre. Her exercises and ideas are mostly aimed at inexperienced teachers,
working with beginners, but introducing these ideas and following a similar methodology could form
the basis for work with intermediate players and more experienced teachers. These applications, of
course, need not be limited to the classroom. Private teachers can also exploit the language and
understanding Collins describes to teach more clearly and effectively, and are even better positioned
than classroom teachers to explore the more sophisticated model outlined in Chapter 2 of this paper.
Further scientific research could also enrich this program. The tilt of the bow is still poorly understood,
and understanding its effects in more detail would be of great interest to players. Also interesting would
be the inclusion of live players, and an analysis of the physiological effects of tilting the bow.
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Evaluating the trade-offs between flat and tilted bows would be much easier with more data. Similarly,
attempts by Schoonderwaldt, Guettler, and Askenfelt (2008) to update the Schelleng diagram could be
further explored, using both machines and live players. The Schelleng diagram maps out the limits of
bow force at a series of contact points, for a given bow speed. Fully mapping the limits of Helmholtz
motion on the violin would involve testing a more complete range of bow speeds, and should
ultimately yield a three-dimensional map of the possible combinations of bow force, contact point, and
bow speed. Machines could be used to map the possible limits and live players to map the practical
limits. Although the numerical limits would be interesting to learn, it is the shape of the limits that
would ultimately be interesting to players, and could be used as a visual aid in the studio. Ultimately,
this could be expanded to include the effects of tilting the bow, playing on different strings, and in
different positions, to construct a full model of the tonal possibilities of the violin.
Also interesting to players would be understanding the effects of various regimes on perceived volume.
Both the amplitude of the fundamental, which is largely determined by the bow speed, and the
harmonic content of a note, largely determined by the contact point and bow force, affect the listener's
perception of loudness. Teachers and players have disagreed for years whether a tone produced with
stronger partials projects better because it sounds richer, or a tone produced with a louder fundamental
projects better because it is more pure and clear. This would not be difficult to test, using a large set of
listeners and a bowing machine. It would also be interesting to perform the same tests in a variety of
sonic environments, to see whether this depends on context. It is conceivable, for example that a richer
tone would project better when it is played in isolation, but a purer tone would cut through an ensemble
better. Also interesting would be to test how vibrato affects the apparent volume.
Finally, measuring the bowing parameters used by live players to execute a variety of technical tasks
could be used to analyze different playing styles. Askenfelt's study of live players (1988) used only two
players, and assigned a limited set of musical tasks. Expanding this to include a much larger group of
players, from a variety of national, musical, and technical backgrounds, would make it possible to
identify and compare different playing styles, and to analyze the strategies they use to exploit the tone-
producing factors in more detail. Collecting information from the players, about how they were taught
to produce different dynamics and tone colours, and then comparing this to the results of the test,
would allow the experimenters to evaluate how complete and realistic the players' understanding of
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their own technique was, and to some degree to determine how their use of the bowing parameters has
been affected by their training. Similar studies with students, who are being trained with different
approaches, could also be used to demonstrate the effects of teaching strategies. It would also be
possible to collect information from video recordings. Although it is not possible to measure the bow
force from a video, the contact point and bow speed are relatively easy to monitor visually, and can be
fruitfully analyzed in reference to the audio. This would make it possible for a trained observer to
compare different approaches to tone production in performance. An historical survey of video
recordings would also yield information about past performance practices.
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Woodhouse, Jim and Paul Galuzzo. 2004. Why is the Violin So Hard to Play? Plus Magazine. Http://www.plus.maths.org/content/why-violin-so-hard-play [Accessed Feb. 4, 2014].
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Appendix: Exercises and Examples from Pedagogical SourcesMany of the pedagogical texts included in the paper contain exercises which either explain, illuminate,
or exemplify their approach to tone production. Performers and teachers may find many of these
examples to be interesting, or even better useful. What follows is intended to be a representative sample
of the most important, interesting, and illustrative exercises, not a comprehensive catalogue.
I Capet
Capet's work is more thoroughly summarized here than any of the others, partly because his work is so
little known in North America. In the author's opinion, the exercises for even bow division and for
“spun sound” (son filé) are still extremely useful in the studio, and Capet's ideas about smooth string
changes (see below) are vitally important. The roulé is also of potential interest, although more for
sensitizing the player to the bow than for its own sake (as discussed in the paper, Capet's claims for its
direct effect on the tone are somewhat dubious).
Explanation of the role of eachfinger, of how they work together, and of the vertical and horizontal motions they can create (11-12).
The thumb and middle finger forma ring, which functions as the fulcrum of a lever. The thumb andindex finger work on one side and the ring and pinky fingers on the other.
Establishes the independent role of each finger, explains the mechanics of the left hand.
Explanation of Capet's notation for even bow division (14-21).
Capet divides the length of the whole bow (A) evenly into halves (B1 and B2), quarters (C1, C2, C3and C4), eighths (D1, D2, etc.), and thirds (E1, E2 and E3).
Demonstrates the concept of evenbow division.
Exercises for even bow division and straight bow (21-27; 75-77).
Capet suggests placing marks on the bow stick, to make the divisions visible, and then practicing scales and select examples using different even bow divisions.
Teaches better bow control, and better awareness of the bow.
String crossing exercises (27; 78-82).
Expands on the exercises above toinvolve string changes, increases rhythmic complexity.
Integrates string changes into the model developed above, developsleft/right coordination.
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Explanation of roulé (28-29). Explains the roulé stroke and its relationship to the horizontal and vertical bow movements.
Capet claims that the roulé can improve a player's tone, as well as sensitizing them to the function of their left hand and therelationship between the bow andthe string.
Roulé exercises (29-32; 82-86).
Introduces a notation to indicate rolling the bow towards and away from the bridge. These basic exercises involve rocking the bow back and forth during long notes.
Teaches the roulé, sensitizes the left hand to the bow.
Exercises for the vertical movement of the bow (32-34; 88-93).
Using the fingers to execute various string crossings in different dynamics.
Develop the fingers of the left hand, illustrate their potential rolein string crossings and bariolage.
Exercises for the horizontal movement of the bow (34-35; 93-94).
Using the fingers to guide the bowacross the surface of the string, changing the contact point.
Although this can be used to change the tone colour, Capet uses it only to execute uneven bow divisions without altering the volume.
Smooth string change exercises for right arm and wrist (35-36).
Using the arm and wrist, change strings by moving in a smooth, curved line.
Teaches smooth string changes. Since the bridge is curved, the curved motions Capet calls for represent the simplest and most direct pathway for the bow arm totake.
Balancing the bow in double stops (35-40; 95-99).
Various examples and exercises for how the bow can be balanced differently in double stops to balance the notes appropriately.
Teaches flexible application of bow force.
Bow stroke explanations and exercises (41-72).
Capet catalogues and explains a whole series of bow strokes including various types of detachéand staccato, martelé, and lancé, as well as off-the-string bowings like spiccato, sautillé, ricochet, and jeté.
These are mostly of historical interest. Modern players use slightly different definitions of some of these strokes, and stylistic practice has also shifted somewhat since Capet's day.
“Deeper penetration of the bow” (87-88).
Double-stop exercises with varying bow divisions.
Develops better control of bow, especially at slow speeds and high bow forces.
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“Spun sound” exercises (100-105).
Exercises with very slow bow speed in different dynamics.
Teaches the son filé (spun tone), a term which refers to a sound which uses a very slow, legato bowing. A very important aspect of violin technique.
II AuerViolin Playing as I Teach It does not contain very much in the way of practical advice and exercises, as
discussed in the body of the paper. Auer did produce, along with Maia Bang, a series of exercise books
which bear his name. These books are not widely available, and they do not contain very much in the
way of explanations, but a thorough study of them might yield more information about Auer's practical
approach to technique. The series was left out of the current study because it is not clear to what extent
Auer was actually responsible for the content of the books, and to what extent he merely oversaw and
endorsed their production.
Basic Tone Exercise (20-21).
Begin with slow, whole-bow strokes, ten to twelve seconds in length. Introduce a crescendo into every down-bow and a decrescendo into every up-bow. Practice scales and double-stops in this way. Avoidforcing the tone.
Learn to compensate for the uneven weight of the bow.
Scales and Double-Stops (37-52).
Auer focuses exclusively on the left hand in his discussion of scales and double-stops.
Developing general tone-production.
Three and Four Voiced Chords (53).
“Attack the chords from the wrist, using no more than three-quarters of the bow-hair”, use light pressure, move towards thebridge during each chord. Auer also instructs the player to always attack two notes at a time.
Learn to play resonant chords without crushing the tone.
III FleschThe extent and complexity of Flesch's work makes it impossible to comprehensively summarize his
exercises. Rather, a few interesting examples will serve to point the curious reader to some key
passages. Where possible the examples have been chosen from Problems of Tone Production in Violin
Playing (referred to below as Problems), because it is so much more compact and focused than The Art
of Violin Playing (referred to below as The Art).
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Explanation of the point ofcontact (Problems 7, The Art 62-63).
Examples of various situations which call for different contact points, and the counter-examples which demonstrate the pitfalls of choosing a less suitable point of contact.
Demonstrates the importance of the point of contact.
Exercise for mastering different contact points, and changing contact points (Problems 18).
Long tones with a variety of contact points and dynamics marked. Executing these will require the player to adjust their bow force appropriately, in order to maintain tone quality.
Player becomes more comfortable playing at different contact points, changing the contact point and adjusting the bow force properly, thuslearning to produce a variety of tone colours.
The five areas of the string(Problems 8).
Divides the area between the bridge and fingerboard into five smaller areas: at the bridge, “neighbourhood of the bridge”, “central point”, “neighbourhood ofthe fingerboard” and on the fingerboard.
Organizes the theoretically infinite set of possible contact points into fiveareas, each of which will be typically used in different ways. Precursor to Fischer's five soundpoints.
Flesch's discussion of tonecolour and his description of three characteristic colours (The Art 76-77).
The “flute”, “clarinet” and “oboe” colours describe a soft, quiet sound, over the fingerboard a moderate sound in the middle and a loud, bright sound close to the bridge.
These three typical timbres map someof the extremes of the tone-producingarea, and provide a starting point to discuss the relationship between tone colour and contact point.
Discussion of the tone production in double stops, and in fast passages with many large intervals and radical position changes (Problems 12-15).
Examples demonstrate some of thepotential problems caused by double stops, in which each note calls for a different contact point.
Interesting contrast with Capet, who believed this should be managed withbow force.
Exercises for adjusting contact point during a shift(Problems 18).
Each shift calls for a prescribed change of contact point, due to the change in position. Variations on the exercise include a variety of dynamics.
Develops the player's understanding of the relationship between contact point and position. Increases control over dynamic and tone colour.
IV HavasMost of the exercises below are taken from The Twelve Lesson Course, which contains several useful
examples in spite of Havas' mistaken ideas about the mechanics of tone production. A New Approach is
a much more polemical text, and contains fewer exercises.
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String change exercise (A New Approach 33-35).
Very slow exercises involving note and string changes. Each note must be prepared and led by the base knuckles of the left hand. The bow arm follows.
Develops coordination between thehands, establishes the base knuckles as the focus of the player's attention.
Double stop exercise (A New Approach 45-47).
Double stops, practiced separately and together, focusing on tone qualityand the independent use of the base knuckles.
Develops tone quality in double stops, as well as strength and independence in the fingers of the left hand.
Exercises 1-3 (The Twelve Lesson Course 1-6).
A series of exercises which establish the balanced posture Havas advocatesfor the player's body and arms.
Establish and develop the fundamental balances which underlie Havas' approach to posture and technique.
Exercises 4-5 (The Twelve Lesson Course 6-9).
Exercises to establish a bow hold, which uses the thumb and middle finger as a fulcrum.
Works to establish a relaxed bow-hold without actually playing the instrument.
Exercises 6-10 (The Twelve Lesson Course 10-15).
Exercises which illustrate how the motions of the bow arm can be connected to those of the upper arm and shoulder, one of the fundamentalbalances.
This is fundamental to Havas' physiological approach. The motions of the bow arm are explained in relation to the upper arm, shoulder, and ultimately the spine, so that the maximum amount of work is done by the strongest muscles.
Exercises 11-14 (The Twelve Lesson Course 16-17).
Slow tones on open strings, establishing the different positions the upper arm must take to play on each string while maintaining the fundamental balance in the back.
Teaches the four basic positions for the bow arm, one for each string, establishes the responsibility of the upper arm andshoulder for string changes.
Exercises 15-21 (The Twelve Lesson Course 18-26).
A series of simple string change exercises using exclusively open strings.
Further explores the role of the upper arm in string changes and the balance of forces in the bow arm.
Exercises 22-82 (The Twelve Lesson Course 27-67).
A series of exercises to explain and explore the posture of the left hand, and to establish the role of the base knuckles.
Havas' ideas about the leadership of the base knuckles are further explored in this section.
Exercises 83-94 (The Twelve Lesson Course 68-75).
Exercises which apply the same approach to legato, detaché, martelé bowing styles, and to double stops.
Further explores the role the base knuckles can play in leading the right arm.
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V RollandBasic Principles of Violin Playing contains very few exercises, because it is so compact. Readers
interested in Rolland's pedagogical approach to very basic tone production will find examples in The
Teaching of Action in String Playing, but little that is relevant to the issues discussed in this paper.
Illustration of bow hold (Basic Principles 3).
A description of the role played by each finger in the bow hold, accompanied by an illustration which demonstrates the role and placement of the fingers.
Demonstrates the different kinds of leverage which can be used to apply force to the bow. Rolland speaks of positive pressure, which applies bow force, and negative pressure, whichreduces the bow force or helps to lift the bow off the strings.
The “Teeter-Totter” and the “Press-Release” (Basic Principles 4).
The “Teeter-Totter” involves pushing and pulling with the pinky and ring fingers, while the bow is held horizontally, and the “Press-Release” involves using the pronation of the forearm to push down on the bow, which is transmitted through the index finger and thumb.
The “Teeter-Totter” aims to develop the strength and sensitivity of the pinky and ring fingers, which apply negative pressure, and the “Press-Release” to develop the habit of generating bow force through pronation, and thestrength and sensitivity of the thumb and index finger, which apply positive pressure.
“Straight Bowing” (Basic Principles 7-8).
Each of these exercises consists of a series of quarter notes and rests, with various instructions for the player during the rests. These involve releasing tension from the bow arm, adjusting the distribution of weight in the arm, and aiming the bow so that the angle remains parallel to the bridge.
Aims to develop a straight bow stroke without building the habit of excess tension into the bowarm. Encouraging the student to relax between strokes, and to think about how they manage theweight of the arm, help them to understand the balance of forces at play during a bow stroke.
“Other Bowing Patterns” (Basic Principles 14-15).
Expands on the straight bowing exercises described above, to include playing in different parts ofthe bow, a variety of bow speeds, and strokes of different lengths.
Further develops basic bow control, while minimizing tension.
VI GalamianGalamian's work contains few exercises, as such, which are aimed at developing tone production. The
three which are most closely relevant to the subject matter of this paper are described and discussed on
pages 81-82, as well as summarized below. Also included is a brief explanation of the famous
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“Galamian Square”, and the two accompanying positions (The “Triangle” and the “Frog”) which
illustrate Galamian's approach to a whole bow stroke.
The GalamianSquare (Galamian 1985, 51-52).
With the violin held in its normal playing position, the player placesthe bow on the strings, parallel to the bridge, with their elbow held at a right angle. Players with longer arms will find that the bowtouches nearer to the tip, and those with shorter arms that it touches closer to the middle or thebalance point.
The square represents the mid-point of a player's bow stroke, where their arm transitions from opening to closing. Depending on the angle of the violin and the length of the player's arms, the square will naturally produce a contact point which is either nearer to the bridge or far over the contact point. Adjusting the angle of the violin, or the height of the player's arm, can change the location of this contact point, which may make it easier to play in the most common areas of the string.
Playing at the point (Galamian 1985, 51-52).
The extreme end of a player's bowstroke is found when the bow touches at the tip. As the player moves from the square to the triangle, they must apply increasing amounts of bow pressure, in order to maintain a constant bow force.
If the player finds it difficult to keep the bow parallel to the bridge when playing at the tip, it may be necessary to adjust the angle of the instrument.
The Triangle (Galamian 1985, 51-52).
When the bow touches at the frog,a triangle is formed between the forearm, the upper arm and neck, and the strings.
The frog is the heaviest part of the bow, and bowing at the frog uses the largest and clumsiest muscles. Galamian recommends tilting the bow at the frog, arguing that it reduces the bow force. As discussed above, in the body of this paper, the main effect of tilting the bow in this way is on the player, rather than on the string or the bow. When the bow is tilted the player must work harder to control it, and they are less likely to thoughtlessly crush the sound.
Bow speed and contact point exercise (Galamian 1985, 60).
The player is asked to gradually increase the bow speed as the contact point moves away from the bridge, keeping the bow force constant and the dynamic steady.
Demonstrates the relationship between bow speed and contact point, teaches the player to compensate for changes in the contact point by slowing the bow speed (as they approach the frog),or by speeding up the bow (as they approach the fingerboard).
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Contact point and bow forceexercise (Galamian 1985, 60).
The player is instructed to begin far from the bridge, using fast bow strokes, then to gradually increase the bow force and notice that the contact point must be moved towards the bridge.
Illustrates the relationship between bow force and contact point. Whereas Flesch encourages the player to conceive of this relationship in terms of the contact point, this exercise conceives of bow force as the active element, instructing the player to adjust their contact point accordingly.
Uneven bowing pattern exercises (Galamian 1985, 60-61)
Exercises which show how the bow speed, contact point, and bow force can be altered to compensate for uneven bowing patterns.
Like Capet before him, Galamian includes exercises to teach players how to compensate for uneven bowing patterns. A further exploration of even and uneven bowing patterns can be found in Galamian's Contemporary Violin Technique (1966).
Dynamics anddynamic changes (Galamian 1985, 61)
Expands on the above exercises toinclude different dynamics, and versions in which the dynamics change.
Illustrates the variety of ways that dynamics can be achieved with different combinations of bow force, bow speed, and contact point.
VII FischerFischer's exercises represent a valuable resource for violin students and teachers, in spite of the
apparent limits of his understanding of the physics of tone production. Players looking exercises which
will help them to develop their control over the main factors which influence tone production, the bow
speed, bow force and contact point, would be well advised to begin with Fischer's work.
Exercise 56 (37). Play a pizzicato note, then an arco note.Try to imitate the ring of the pizzicato withthe bow.
Establishes ringing tone.
Exercise 57 (37) Start with a very short note, played at the frog. Lift the bow off the string immediately. Listen for the ring after the note.Gradually lengthen the notes, trying to make the same ring during the note as afterit.
Learn how to play longer notes with ringing tone.
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Exercise 58 Play several strokes, each in the same bow direction, lifting the bow off the string between each one. Listen for ring.Do the same without lifting in between. Try to get the same ring during and after the note.Repeat at each soundpoint.
Establishes relationship between bowspeed and resonance.Pressure likely to be adjusted unconsciously, in order to establish ringing tone.
Practice method for Soundpoints (41).
Repeat for each soundpoint, starting at number 5.At each soundpoint, adjust the speed and pressure to find the maximum resonant dynamic; “Make the strings vibrate as widely as possible”.
Shows relationship between contact point, bow speed and bow force for loudest “Resonant” dynamic.
Exercise 64 (42). Repeat for each soundpoint, starting at number 5, use whole bow for each stroke.Dynamic changes for each soundpoint: mp, mf, f, ff, then f.Metronome marking suggested for each soundpoint: 80, 75, 70, 56, 40.
Guides the player's choice of bow speed for each contact point.Suggests tone very close to the bridgeis too noisy to attain maximum dynamic.
Exercise 65 (43) Repeat for each soundpoint, starting at number 5, and in different parts of the bow.Use the same amount of bow for each soundpoint, but change the bow speed to find ideal resonance.Dynamic changes: mp, mf, f, ff, f.
Shows relationship between bow force and bow speed for each contact point.Amount of bow stays constant.Bow force changes from light to heavy.Bow speed changes from fast to slow.
Exercise 66 (43) Repeat for each soundpoint, starting at number 5.Begin at the middle of the bow, using shortstrokes. Establish balance of speed and pressure for maximum resonance.Gradually lengthen stroke (increase bow speed), until using the whole bow. Adjust pressure to maintain tone. Then gradually return to short strokes.Gradually increase tempo from 56 to 75.Play in high and low positions, note changes in soundpoints.
Teaches the player to use a large range of bow speeds, at each contact point, and how to adjust pressure.Demonstrates how soundpoints movein different positions.Produces a range of dynamics and tone colours.
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Exercise 67 (44-45)
Play whole bow strokes, starting at soundpoint 5.Move the bow towards the bridge, and crescendo, then away from the bridge and decrescendo, then do this multiple times per stroke.Adjust other factors to keep sound pure.
Practice changing the soundpoint, first by sliding the bow along at an angle, then by keeping it straight and dragging it along the string.Likely to lead to non-ideal combinations of bow speed and pressure.
Exercise 68 (45) Repeat on different soundpoints, starting atnumber 5, moving to number 2 and back.Use the whole bow on each stroke, each stroke should be one beat at 80 beats per minute.Maintain this bow speed as the soundpoint moves toward the bridge. Increase the pressure to maintain a good tone, and get louder at each soundpoint.
Establishes the different dynamic effects of the same bow speed at different contact points.Demonstrates how minimum bow force increases as contact point decreases, while bow speed is held constant.
Exercise 69 (46) Repeat for each soundpoint, starting at number 1.Start with very little bow, each bow stroke 76 beats per minute.Keep the pressure the same, move the soundpoint and use more and bow for eachsoundpoint. By soundpoint 3-4 whole bow will be used. Return to soundpoint 1.
Shows how the same pressure requires different bow speeds at different dynamics.
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Copyright Acknowledgements
Figure 1 was previously published in John C. Schelleng's “The Physics of the Bowed String”.
Permission to reproduce it here was granted by Scientific American Magazine.
Figures 3, 21, and 22 were previously published in James Beament's The Violin Explained:
Components, Mechanism, and Sound. Permission to reproduce them here was granted by Oxford
University Press.
Figure 20 was previously published in “Why is the Violin So Hard to Play?” by Jim Woodhouse and
Paul Galuzzo. Permission to reproduce it here was granted by Plus Magazine.
Figure 27 was previously published in Paul Rolland's Basic Principles of Violin Playing. Permission to
reproduce it here was granted by the American String Teachers Association.
Figure 28 was previously published in Simon Fischer's Basics: 300 Exercises and Practice Routines
for the Violin. Permission to reproduce it here was granted by C. F. Peters Ltd.
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