a hidden order: revealing the bonds between music and ... · explored in architecture, particularly...

A Hidden Order: Revealing the Bonds BetweenMusic and Geometric Art – Part One

Sama Mara and Lee Westwood

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Harmony of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Harmony of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

A Mapping between Music and Geometric Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Color to Pitch Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

A Relationship Between Rhythm and Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A Unit of Time and a Unit of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Hexagons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Pentagonal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Octagonal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Abstract

The following chapter describes a method of translating music into geometricart and vice versa. This translation is achieved through an exploration of themutual foundations – in mathematics and its role in harmony – of both musicand geometric art. More specifically, the process involves the implementationof principles derived from traditional Islamic geometric art and contemporarymathematics, including fractal geometry and aperiodic tilings.

S. Mara (�)Musical Forms, London, UKe-mail: [email protected]

L. WestwoodUniversity of Sussex, Brighton, UKe-mail: [email protected]

© Springer International Publishing AG, part of Springer Nature 2018B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_18-1

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http://crossmark.crossref.org/dialog/?doi=10.1007/978-3-319-70658-0_18-1&domain=pdf

mailto:[email protected]

mailto:[email protected]

https://doi.org/10.1007/978-3-319-70658-0_18-1

2 S. Mara and L. Westwood

The method was discovered by Mara in 2011 and was subsequently developedduring his collaboration with composer Lee Westwood on the project A HiddenOrder. Examples from this project are used to illustrate parts of this chapter.

Also discussed are the implications of establishing such a connection betweenmusic and geometric art. These include the possibility of unique creativeprocesses that combine practices from both visual arts and musical composition,as well as facilitating the application of developments, practices, and creativeprocesses from one discipline to the other.

KeywordsMusic · Geometry · Visualization · Sonification · Islamic · Art · Fractal ·Aperiodic · Tilings · Harmony

Introduction

The exploration of the relationship between music and visual art recurs repeatedly inthe arts and sciences over the centuries and includes a variety of different approachessuch as the use of theoretical, intuitive, and experiential methods, alongside theexploration of physical phenomena.

The theoretical studies considered include Isaac Newton’s relation of the sevencolors from the visible spectrum to the seven notes of the musical scale, detailedin his book Opticks (1704). Also of interest are the various systems of proportionexplored in architecture, particularly in the Renaissance period, by scholars such asAlberti, who relates the musical intervals (1:2, 2:3, etc.) to rectangles of the sameratios (Padovan 1999). According to the art historian Wittkower, Leonardo Da Vincitheorizes on the shared harmonic principles at play within music and the practicesof perspective in art, stating “The same harmonies reign in music and perspectivespace” (Wittkower 1953). Recent theoretical approaches include “A Geometry ofMusic” by Dmitri Tymoczko (2011), which views Western musical tonality throughgeometric space.

Physical explorations of this relationship between music and art may be seen inexperiments conducted in Cymatics, whereby liquids or particles on metal plates areexposed to the vibrations of sound waves, creating symmetric forms dependent onthe sonic frequencies. In a similar manner, the harmonograph may be used to createspecific geometric shapes via the use of pendulums swinging in relative frequenciesanalogous to the musical ratios.

Experiential and intuitive approaches in this field include the phenomenon ofSynaesthesia, in which the stimulus of one sense may cause an impression inanother, most pertinently where a visual sensation is experienced in response toa sound stimulus. There are many examples of composers relating pitch to color, theFrench composer Olivier Messiaen being perhaps the most well documented, whoseexperiences directly informed his approach to harmony (Messiaen 2002). Similarly,Scriabin’s later works devised elaborate relationships between color and key centers(Galev and Valechkina 2001). In practice, the intuitive and experiential approaches

A Hidden Order: Revealing the Bonds Between Music and Geometric Art – Part One 3

of relating music to art may be seen in the works of the abstract painters of the 1900ssuch as Mondrian, Kandinsky, and Klee, as well as animated pieces by the likes ofOskar Fischinger and John Whitney.

Wassily Kandinsky wrote “The sound of colors is so definite that it would be hardto find anyone who would express bright yellow with bass notes, or dark lake withthe treble” (Kandinsky 1914). This exemplifies the shared intuitive understandingof a relationship that seems to be common to the human experience of colors andsound. If we omit the references to hues from this statement by Kandinsky, it maybe interpreted as relating the pitch of the sound with the brightness of color, in thatthe higher the pitch, the brighter the color. Within a branch of research called cross-modal correspondence that “refers to consistent associations between features intwo different sensory domains” (Griscom 2015), there are many studies exploringthe relationship between color and sound which support Kandinsky’s observation.Ward et al. (2006) found that “both synaesthetes and non-synaesthetes associatehigher pitches with lighter colors” (Griscom 2015).

The sonification of art is perhaps less explored, although the use of the Goldenratio through the implementation of the Fibonacci series is one area in which wefind numerous examples of the influence of geometry and the visual arts on music,as seen in compositions by Joseph Schillinger (Livio 2002), Béla Bartók (Lendvai2000), and Debussy (Howat 1983).

The approach documented in this chapter is primarily a theoretical one whichexplores the common root of mathematics at the foundation of music and geometricart. Throughout this method, geometric art is considered in relation to color andpattern and music in terms of sound and rhythm. From these two aspects of musicand of visual-art, we relate notes to color, and rhythm to pattern.

The reason for this is due to the relative frequencies of sound, color, rhythm, andpattern and how they are perceived. In the realms of color and musical notes, whenconsidered as light and sound waves, the human sensory systems are not able todistinguish individual cycles of the waves, but rather a general sensation of color orsound. Rhythm and pattern are also of a cyclical nature. However, each unit of thecycle may be experienced individually.

In essence, the translation from sound to color is achieved through understandingboth as waveforms and drawing upon the physical properties of waves, such asamplitude and frequency. The relationships derived are supported through researchin cross-modal correspondence.

In establishing a relationship between rhythm and pattern, the approachdemanded study of the mathematical roots of both. Rhythm is here consideredas a division of time and pattern as a division of space. Hence, if both rhythm andpattern are rooted in mathematics, the question of their relationship develops intofinding a meaningful and logical association between their respective mathematicalroots. By defining rhythm as divisions of time, the theory assumes a pulse in themusic. As such, arrhythmic music (e.g., Boulez’s “smooth time” (Boulez 1971))does not carry a meaningful analogy in the method described here.


Harmony

At the root of this approach are the mathematical foundations of “harmony of time”(as exemplified by the musical rhythms of many cultures around the world) and“harmony of space” (as understood and applied in Islamic geometric art and manyother traditions). We refer to “harmony of time” and “harmony of space,” relatingto aural and visual harmony, respectively.

Harmony of Time

Systems of harmony governing rhythm and pitch in Western music are often basedupon simple divisions of time. Concerning rhythm, the bar is divided into a givennumber of beats, each beat being subsequently divided further into equal parts, thuscreating a structure in which the rhythmical aspects of the music may be conceived.

In regard to pitch, and more specifically the intervals between two pitches, it isunderstood that “when the frequency ratio of the two notes is a ratio of low integers:the simpler the ratio, the more consonant are the two notes” (Bibby 2003). Theoctave is a ratio of 2:1, meaning that when a note is played at twice the frequencyof another note, the interval created is an octave. Likewise, in just intonation,relationships of 3:2 (a fifth), 4:3 (a fourth), 5:4 (a major third), and 6:5 (a minorthird) all form intervals found within common musical scales and produce consonantsounds. The tempered scale, now standard in music in the Western world, is alsorooted in these harmonies and although it does not employ them precisely apartfrom the octave, it achieves a harmonious sound by closely approximating theseratios. This is possible because there is a certain tolerance in deviating from theseratios that still creates a harmonious result (Bibby 2003). It is this understanding ofharmony of both rhythm and pitch that is used in this method.

Harmony of Space

In regard to harmony of space, we draw upon the principles applied in Islamicgeometric art, which is rooted in Euclidean geometry. These principles are notunique to Islamic art and are true for other systems of proportioning that originatefrom the regular polygons.

We start with the circle and the equal divisions of its circumference whichproduce the regular polygons (the equilateral triangle, square, regular pentagon, andso on). The forms and ratios that are revealed through the natural subdivisions ofthe regular polygons form the myriad of different patterns of Islamic geometric art.The numerical ratios that are at play are different to those in the harmony of timeand are far more complex.

Within systems of proportion implemented in art and architecture, the artistJay Hambidge defined two general approaches as static versus dynamic symmetry.


Fig. 1 The ratio between theedge of a regular pentagonand its diagonal is the goldenratio, represented by theGreek letter φ (phi)

1

Within static symmetry is included Aberti’s system of proportion as well as the“musical ratios” applied in renaissance art and architecture, consisting of rectangleswith edge lengths in whole number ratios. Dynamic symmetry, on the other hand,involves the use of ratios with irrational numbers such as

√2:1 and the golden

ratio. Although these two systems are not entirely exclusive of one another, it isthe approach regarding dynamic symmetry that is explored here.

To serve as a brief introduction to this system of harmony, we shall look at theratios and forms that appear from the subdivisions of the regular pentagon, hexagon,and octagon. The intent of this is to familiarize the reader with the forms andnumbers at play and to illustrate the harmonious interactions between these forms.In each of the examples to follow, we shall derive a dynamic rectangle from theregular polygons – illustrating the inherent harmonic properties they contain – andshow an example of them applied in Islamic geometric art.

The PentagonThe regular pentagon contains, within its subdivisions, the well-known golden ratio,as seen in Fig. 1.

The fascinating properties of the golden ratio are well documented, withnumerous publications devoted to it observing its occurrence in the visual-arts,architecture, nature, and music, reaching back from Ancient Egypt through theMiddle Ages, the Renaissance, and Modernism, up to the present day.

A few of the numerical properties of the golden number may be seen in Fig. 2.The geometric properties of the golden ratio include the golden rectangle and

accompanying spiral, and the golden triangle formed by two diagonals and an edgeof a regular pentagon. The golden triangle also has a related spiral, seen in Fig. 3.

From the edge of the pentagon and diagonal of the decagon is a dynamicrectangle whose ratio is related to the golden ratio. Subdivisions within thisrectangle display a harmonious arrangement of decagons and pentagons whose


= 1 +1

11 +11 +

1 + ...

= 1+ 1+ 1+ 1+

+ 1=2

1 + = 1

Fig. 2 Unique properties of the golden number. Top left shows the golden ratio expressed as acontinued fraction; top right as a nested radical

1

1

Fig. 3 The golden rectangle with edge lengths 1: φ, with related spiral (left). The regular pentagonwith golden triangle (created by the edge of the pentagon and two diagonals) and related spiral(right)

interplay suggests the extra levels of subdivisions that may continue indefinitely –see Fig. 4.

Within Islamic geometric art, the regular pentagon and related golden ratio standas one of a series of harmonious forms among the other regular polygons, each ofwhich possessing their own unique properties of number and form.


Fig. 4 The regular decagon and subdivision revealing a harmonious arrangement of decagons andpentagons (left). A Classic Islamic pattern developed from the same subdivision of the decagon(right)

Fig. 5 The regular hexagon,with edge length 1 anddiagonal

√3. Two parallel

diagonals and edges form theroot-3 rectangle shown inorange

1

√3

The HexagonThe regular hexagon with edge length of 1 has diagonal of length

√3. A parallel

diagonal reveals a dynamic rectangle referred to as the root-3 rectangle – see Fig. 5.As the root-3 rectangle originates from the hexagon, it may be subdivided

indefinitely with combinations of hexagons and triangles. Figure 6 illustrates someof these harmonic subdivisions.

The root-3 rectangle is often used as a repeat unit in Islamic arts, as shown inFig. 7.

Interactions between the square and the root-3 rectangle create a series ofrectangles with edge lengths in the ratios of 1:

√3, 1:

√3–1, and 1:

√3 + 1 that


Fig. 6 Harmonic subdivisions of the root-3 rectangle

Fig. 7 An example of a geometric pattern from Islamic art, with a root-3 rectangle as the repeatunit

may be used together at various scales and may be used to create endless possiblearrangements and subdivisions within the root-3 rectangle – see Fig. 8.

The OctagonThe regular Octagon with edge length 1 has diagonal of length

√2 + 1 (Fig. 9). The

ratio of 1:√

2 + 1 is known as the silver ratio.The silver rectangle is another dynamic rectangle and may be subdivided by

octagons, squares, and related forms, as shown in Fig. 10.As with the golden rectangle and root-3 rectangle, a repeating sequence of

rectangles may be created by the use of squares inside of the silver rectangle. Thistime, at each generation two squares are placed inside the silver rectangle, leaving asmaller silver rectangle at the next generation, as shown in Fig. 11.


√3

√3 - 1 √3+1

Fig. 8 Sequence showing the interaction between the square and root-3 rectangle. If a square (withedge length equal to the shorter edge of the root-3 rectangle) is placed inside the rectangle against ashorter edge, the remaining area is a rectangle of 1:

√3–1. This process may be repeated, leaving a

rectangle of 1:√

3 + 1. Applying this process again returns the area to the original root-3 rectangle.This process may be repeated indefinitely with a series of rectangles of ratios 1:

√3, 1:

√3–1 and

1:√

3 + 1. Bottom right: A possible subdivision of the root-3 rectangle into squares and smallercongruent rectangles (bottom right)

The silver number has interesting numerical properties (see Fig. 12) and is usedwithin Islamic arts to subdivide the square, resulting in interesting interplays of formand harmony – see Fig. 13.

We have seen here a variety of patterns and geometric forms derived from theregular pentagon, hexagon, and octagon, with the intention of appreciating theirunique – and at times delightful – properties. We have also seen examples ofhow these ratios and forms are at the foundation of harmony for geometric art,particularly in the Islamic tradition. These principles also extend to other traditionsthat are based upon the regular polygons and Euclidean geometry.

Comparing the systems of harmony of time and space, as described here, we findboth similarities and differences. In much the same way as the musical bar is dividedinto beats, which in turn are further subdivided (with each subdivision having itsown relevant and defined place in relation to other parts and to the whole rhythmical


1

√2 + 1

Fig. 9 The relation between the edge length to the diagonal of the octagon describes the silverratio of 1:

√2 + 1. The shaded area is the silver rectangle

Fig. 10 Harmonious subdivisions of the silver rectangle


√2 + 1

Square

Square

Fig. 11 Interactions between the silver rectangle and the square. A silver rectangle may besubdivided into two squares and a smaller silver rectangle (left). Repeating this process (center)reveals the framework for the related double spiral (right)

= 2 +1

12 +12 +

2 + ...

s 2s + 1=s

s = 2 + 1s

Fig. 12 The silver number expressed as a continued fraction (left)

structure), so we see an equivalent with the elements of a pattern, where each unitis further subdivisible (again, with each subdivision having its place relative to thewhole, with no areas left unresolved).

The common root of mathematics is clear in both systems. However, when weobserve the numbers at play, two different systems appear. The harmony of timeis based upon whole number ratios both in pitch and rhythm: 2:1 resulting inan octave (C to C) and 3:2 in a fifth (C to G), while integer divisions of a barcreate a certain number of beats. Visual harmony is also based upon simple integerdivisions, but this time of the circumference of the circle. The numbers that unfoldover the two dimensional interaction of the intersecting lines are relatively complexand involve irrational numbers. These two number systems are an outcome of therespective dimensions at play within the disciplines of music and art, rhythm beingone dimensional (that of time), while pattern is based in two dimensions (those ofspace). The act of bringing together music and pattern becomes essentially aboutbridging the gap between one and two dimensions.


√2 1 1

√2+1

1

Fig. 13 A Geometric design from the Alhambra in Spain, showing the repeat unit as a square,with the silver ratio at the foundation of the design

A Mapping between Music and Geometric Art

Color to Pitch Relationship

The relationship between pitch and color described here is based upon the waveproperties of sound and light. The mapping is not new, but is described for the sakeof completion, in order to present the whole method together.

Sound and light may both be understood as waves, though they are of differingnatures. Light is part of the electromagnetic spectrum and is a transverse wavewith the ability to travel through a vacuum, whereas sound waves are longitudinal,requiring a medium such as air or water in which to travel. Studying the wavenature of these two allows for simple correlations to become apparent between theproperties of color and sound, these correlations being supported by experiments incross-modal correspondence.

Loudness and BrightnessThe amplitude of a wave is defined as the distance between a peak or valley fromthe equilibrium point – see Fig. 14.

The amplitude of a sound wave corresponds to the loudness of the sound, whereincreased amplitude results in increased loudness. In regard to light waves, the


Amplitude

Fig. 14 The amplitude of a wave

amplitude relates to the brightness of light. From this we deduce that the amplitudeof the wave form relates the loudness of the sound to the brightness of the color.

Studies in cross-modal correspondence support this correlation between loudnessof sound and brightness of color. For example, experiments conducted by Stevensand Marks (1965) found consistent correlations between loudness of sound andbrightness of color.

The inverse of the above-stated relationship is also of relevance in this context.If the color presented is based upon a white ground rather than black, one mightexpect a louder sound to have a more intense color, so creating a darker result.

An experiment was conducted that asked subjects to match the loudness of thesound to samples of neutral grey paper of different values, presented on eithera white background or dark background. On the dark background, 10 of the 12participants matched the increasing loudness to increasing brightness of the cards,while 2 matched this with increasing darkness. With the light background, of the10 subjects involved, 4 matched increasing loudness to increasing brightness, 5 toincreasing darkness, with 1 subject matching identical sound pressures to all thegreys (Marks 1974). In this experiment, there is a consistent relationship displayedbetween the increase or decrease in both loudness and brightness: on a darkerbackground the tendency was towards the sequence increasing in brightness, whileon a white background it tended towards increased darkness. It is the relationship ofincreased darkness with increased loudness that is applied in this chapter and waspredominantly applied in A Hidden Order. Starting from the white background ofthe paper, louder notes would leave darker marks, with quieter notes leaving lessdark marks. Ultimately silence would be the white of the background.

Hue and PitchThe frequency of a wave is defined by the number of waves passing any given pointper second and is measured in hertz (Hz). The frequency of light determines thehue of the color, and in sound it governs the pitch of the note. It is based upon thiscommon root of frequency that we relate the pitch of the sound to the hue of thecolor.

The human eye is sensitive to wavelengths between 390 nm and 760 nm(Jacobson et al. 2000), a nanometer being 1/1000,000,000 of a meter, the frequency


range being approximately 400–770 THz. The wave lengths of sound are muchlarger, at around 1.3 m for middle C, the frequency range of audible sound beingroughly 30–18,000 Hz (Taylor 2003). From these frequency ranges, we can see thathumans are able to hear up to around ten octaves (doubling of frequencies), whereasthe range of visible frequencies is under one doubling of frequency. This sets up amismatch in terms of the ranges of the two sets to be mapped to one another.

The auditory experience of the interval of an octave in music is such that twonotes an octave apart (with a frequency relationship of 1:2) posses a similar quality,referred to as octave equivalency. This property leads to the formation of cyclicalscale systems whereby each note that is double the frequency of another has thesame name attributed to it, with a key being defined by the collection of notes withinone octave.

The perception of hue is also of a cyclical nature, with violet and red at oppositeends of the visible spectrum, bridged by magenta (not in itself a spectral color,but a combination of the opposite ends of the spectrum) that completes the cyclicgradation of hues.

The means of mapping hue to pitch makes use of the property of octaveequivalency and the cyclic nature of both pitch and hue. Specifically, the visiblespectrum of colors (plus magenta) is mapped onto the frequencies within one octaveof sound, meaning that any pitch an octave apart from another is mapped to the samehue. This creates a continuous mapping between hue and the frequencies of pitchwithin an octave, independent of any formal system of frequencies, notes, or scales.As each subsequent octave maps to the same range of hues, it is consequently amany-to-one mapping from pitch to hue. Let us now consider how our starting pitchis mapped to a particular hue, from which point all the other frequencies may bemapped relative to this.

C Is GreenTo map a particular frequency of hue to a particular frequency of pitch, we startfrom a wavelength of light. For our purposes, we shall choose a mid-green light at awavelength of 520 nm, though it should be noted that each hue is related to a bandof wavelengths on the visible spectrum and is not assigned a particular wavelengthas such.

When converted to frequency, a light wave of 520 nm comes to 576 THz. Thisfrequency considered as a sound wave is way beyond the human hearing range.However, recognizing the property of octave-equivalency, the frequency may behalved and still retain the same quality and so the same pitch-to-color mapping.Halving the frequency repeatedly (41 times, in fact), stepping down one octave eachtime, eventually brings the frequency within audible range to 262.17 Hz, very closeto the note of middle C (261.63 Hz) in equal tempered tuning. The authors do notmaintain that this relationship between the note of C and the color of green holdsany weight, but it is a seemingly logical way of establishing a mapping betweenpitch and hue.

Once this relationship between the frequency of light and sound is established,the other frequencies are derived in relation to these, resulting in a continuous


B

E D# D

C

# C

F

A#

A

G#

G

F#

Fig. 15 Hue-to-pitch relationship within one octave

mapping. Applied to the notes of the chromatic scale, an approximate relationshipis derived, as shown in Fig. 15.

Brightness, Loudness, and PitchA remaining concern is that all the scales are now matched to the same hue, leavingno differentiation between high-pitched notes and low-pitched ones. This brings usback to Kandinsky’s quote and the correlation between an increase in brightnessand higher pitch which, as was stated earlier, is supported by studies in cross-modalmappings. As described above, brightness has already been attributed to loudness,meaning that now both the loudness of the note and height of the pitch contribute tothe brightness of the color. This relationship of both loudness and pitch to brightnesshas also been noted in cross-modal studies: Marks states that “visual brightness hasat least two structural and functional correlates in the auditory realm – pitch andloudness” (Marks 1989).

The resultant affect of both loudness and pitch relating to brightness is that thesame hue and brightness of a color may be achieved by two notes an octave apart,but with a counterbalancing change in loudness.


Timbre and SaturationThe “timbre” of a sound is a quality that allows us to determine the differencebetween two different instruments playing the same note at the same loudness. Itmay be defined as “the way in which musical sounds differ once they have beenequated for pitch, loudness, and duration” (Krumhansl 1989). Timbre is a complexsubject and various aspects contribute towards it, one of these aspects being therelationships of the overtones of a sound.

In musical sounds, the overtones typically consist of the harmonic series,whereby each overtone is a multiple of the fundamental frequency. It is the relativeamplitudes of these overtones that have an affect upon the timbre of the sound.

In general terms, a tone consisting of a single sine wave and little or no overtoneswill have a very “pure” sound, where the pitch of the sound is clearly discernible. Asound wave with complex interactions in its harmonic series and less order amongthem will have a less pure sound, leading ultimately to white noise, where noparticular pitch is defined.

Likewise with color, a light source containing just one wavelength will output acolor with high saturation, such as the light emitted by a laser. More complex inter-actions of different frequencies of light will reduce the saturation until ultimatelythe color is a shade of grey, white, or black.

By the complexity of the interaction of frequencies in either sound or light,this mapping relates aspects of the timbre of sound to the saturation of color. Thiscorrelation has also been noted in studies in cross-modal mappings (Caivano 1994).

A Relationship Between Rhythm and Pattern

A Unit of Time and a Unit of Space

To explore the relationship between rhythm and pattern, we shall start from a basicpremise where one unit of time relates to one unit of space.

For our unit of time, we shall choose a “beat” in music. This assumes that themusic in question does in fact have a pulse whereby a beat may be defined, therebyexcluding arrhythmic music.

As a unit of space, we shall choose the square. We shall see later that we mayequally choose other regular polygons. The decision to start from a polygon is basedupon the approach to harmony of space described above.

We now have a beat as our unit of time, represented by a square as our unit ofspace. The next question follows, what would two beats look like? One option is toplace another square next to our original square, creating a double square. This is avalid step and is explored later in this chapter. We shall first look at the approach thatrepresents two “beats” also as a square but twice the area of the original square –see Fig. 16.

The process of doubling the area may be repeated indefinitely, each steprepresenting double the number of beats, and thus creating the sequence 2, 4, 8,16, 32, 64, and so on (see Fig. 16). We shall refer to each square as a “generation,”


11

22 4

816

Fig. 16 A square representing one beat in music (left). Doubling the area of the square represents2 beats (center). The process of doubling the area is repeated three more times, resulting in a seriesof nested squares with the largest representing 16 beats (right)

so that the first square may be known as the “first generation,” the square of areatwo as the “second generation,” area 4 as “third generation” and so on.

Beats 1 and 2 are already located within the diagram, the first beat being theoriginal square and the second beat being the second generation minus the firstgeneration square. Beats 3 and 4 are located somewhere in the area defined by thethird generation square minus the second generation square. Beats 5–8 are locatedsomewhere within the fourth generation square minus the third generation square.

We may now ask, where are beats 3–8 located? A solution presents itself whenwe reveal a natural subdivision of these nested squares into a grid of equally sizedand shaped cells. Each cell is now the visual representation of a beat in music – seeFig. 17. The convention here shall be that the cell number is correlated to the beatnumber directly (e.g., cell-3 is the visual representation of beat-3). The grid displaysfourfold symmetry about the origin, so from here only the top-left eighth of the gridneeds to be considered, as this area is reflected and repeated around the origin.

Within a given generation, there is a choice as to the location of a particularcell number. A logical choice within the third generation square is to place cell-3neighboring cell-2, and cell-4 neighboring cell-3. This indexing sequence may becontinued indefinitely, whereby any two consecutive beats are visually representedby contiguous cells – see Fig. 17.

This sequence describes a version of the Sierpinski space-filling curve fromfractal geometry, which may be created by joining the centers of each of theconsecutive cells in the grid by a continuous curve, as shown in Fig. 17. A curveis inherently 1- dimensional, the term “space-filling” referring to the fact that thiscurve will eventually fill a two dimensional space at its limit. As Peitgen describes,“given some patch of the plane, there is a curve which meets every point in that


12

345 6

78

910111213

141516

Fig. 17 The nested square sequence broken down into a symmetric grid of equally sized cells.Each cell represents one beat and is repeated eight times around the origin. The cells in the top leftsection of the grid are indexed according to which beat they represent

Fig. 18 Three stages of a version of the Sierpinski space-filling curve. The curve is created byjoining the centers of each of the cells with a continuous line in the order shown in Fig. 17. Thethree stages illustrate how the curve becomes tighter and more dense. When the cells are infinitelysmall, the curve will cover every part of the defined area

patch” (Peitgen et al. 2004). Successively subdividing the grid and creating thecurve creates more and more dense versions (Fig. 18), tending towards its limitof covering every part of the grid. The principles behind space-filling-curves are aperfect concept to meet our aim of crossing the dimensional gap between 1 and 2dimensions, from line to plane, and from rhythm to pattern.

By implementing this indexing sequence with a selection of rhythms, we seehow each rhythm is represented by a unique pattern that may also be read back from


2 31 4 5 6 7 8 9 10 11 12 13 14 15 16 2 31 4 5 6 7 8 9 10 11 12 13 14 15 16 2 31 4 5 6 7 8 9 10 11 12 13 14 15 16

12

345 6

78

910111213

141516

12

345 6

78

910111213

141516

12

345 6

78

910111213

141516

2

Fig. 19 Three 16-beat rhythms and related patterns. A rudimentary rhythm where every other beatis accented (left). A selected rhythm (center) and a palindromic rhythm where beats 9–16 are thereverse of beats 1–8 (right). The rhythms applied are shown in the row of boxes numbered 1–16above each pattern. Each box relates to a beat, the shaded areas represented by accented beats andsilence represented by white

pattern to rhythm. In Fig. 19, the shaded areas represent an accented beat, and thewhite areas represent silence in the music.

The visualizations reveal an inherent problem with the method and indexingsequence so far described, in that simple rhythms do not necessarily relate to simplepatterns. For example, one of the most simple rhythms in music – where every otherbeat is sounded, leaving the intermediate beats silent – creates a relatively complexpattern (see the first pattern in Fig. 19). Within this pattern shapes are formed onthe horizontal and vertical axes that differ to those on the diagonal axes, which inturn are different from those not lying on the axes at all, while at the origin only is asquare formed. Consequently there are four different forms to represent a two beatrepeated rhythm.

It turns out that palindromic rhythms create simple patterns (Fig. 19). Whilstpalindromic rhythms are not a standard approach to rhythm in music, this translationbetween rhythm and pattern is also not consistent in terms of relating the perceivedcomplexity of the rhythm and pattern.

When considering the qualitative aspects of rhythm and pattern, there are basicrelationships which ideally should hold true, for a successful visualization of musicand vice versa. Two of these aspects are that:

1. A sparse rhythm should create a sparse pattern, and they should increase indensity together.

2. The complexity of a rhythm should be reflected in its visual counterpart, so asimple rhythm creates a simple pattern.

Of these two requirements the first is already met, but not the second. A solutionto meeting both of these requirements would be to re-order the cell indexing, asshow in Fig. 20. In a sense, we have embedded the palindromic aspect within thecell order itself, so relieving the rhythmic counterpart of that restriction.


12

3 4

56

7 8

9 10

1112

1314

1516

Fig. 20 Alternative cell indexing

2 31 4 5 6 7 8 9 10 11 12 13 14 15 16 2 31 4 5 6 7 8 9 10 11 12 13 14 15 16 2 31 4 5 6 7 8 9 10 11 12 13 14 15 16

12

3 4

56

7 8

9 10

1112

1314

1516

12

3 4

56

7 8

9 10

1112

1314

1516

5

12

3 4

56

7 8

9 10

1112

1314

1516

Fig. 21 The same 16-beat rhythms as visualized in Fig. 19, but this time applying the newindexing system. The simple rhythm on the left now creates a visually simple pattern more inline with what one would expect of such a simple rhythm

This cell order no longer has the property of the original space-filling curvewhereby consecutive beats are represented in contiguous cells. However, on visual-izing various rhythms, the results satisfy the requisite where simple rhythms createsimple patterns and the complexity increases together – see Fig. 21.

The new indexing sequence is created through a series of reflections, wherebyeach reflection line runs along the edge of a generation square (see Fig. 22). Thesereflections govern the ordering of the cells.


rl-3

rl-2

rl-1rl-

4

9 10

1112

1314

15165

67 8

3 421

rl-1

rl-2 1

23 4

rl-112 rl-1

rl-3

rl-2 1

23 4

56

7 8

Fig. 22 Series of reflections used to locate and index the cells

Table 1 This table shows the reflection-lines applied to locate three cells within the grid. Notehow the reflection-lines applied relate to the binary numbers themselves

Binary Number Reflection-line-1 Reflection-line-2 Reflection-line-3 Reflection-line-4

1000 � x x x1011 � x � �110 � � x x

Through this process, cell 2 is located via a reflection of cell 1 through reflection-line-1 (rl-1) that runs along on the edge of the original square. Cells 3 and 4 arelocated by reflecting cells 1 and 2 using rl-2 to map them onto the new cells 3 and4, respectively. Rl-3 maps each cell from cell 1 to cell 4 to a new cell, as follows: 1→ 5, 2 → 6, 3 → 7, and 4 → 8. The reflection-lines determine the indexing of thecells as each generation is a reflection of all the previous generations, reflecting theorigin of the grid out to the vertex of the new generation square and preserving therelative order of cells through the reflection.

Binary Counting GridAn interesting property becomes apparent if the cells are numbered with binarynumbers, with the first cell as 0 – see Fig. 23.

Each cell may be located within the grid through a unique combination ofreflections. For example, to locate the cell with binary number 1000 (cell number 8)only one reflection is necessary: rl-4 and rl-1, rl-2 and rl-3 are all omitted (Fig. 24).Figure 25 shows another example, locating the cell with binary number 1011,applying rl-1, rl-2 and rl-4, while leaving out rl-3. As a final example, cell number110 (number 6) is located by applying rl-2 and rl-3, leaving out the first reflection-line. Rl-4 is not applicable as it relates to cell numbers beyond this generation – seeFig. 26.

Table 1 displays the transforms that are applied to locate the cells in the examples.On observing the binary numbers and the reflection lines applied, it becomesapparent that in the above examples the binary expression of the cell numbersencode the instructions as to which reflections to apply in order to locate that givencell within the grid.


01

10 11

100101

110 111

1000 11000001101

11111110000

1001

10101011

Fig. 23 Indexing system using binary numbers

Fig. 24 Locating cell withbinary number 1000. Only thefourth reflection-line (rl-4)shown in orange is applied

1000

rl-1

rl-3

rl-2

rl-4

Reading the digits of the binary numbers from right to left, each digit correspondsto a particular reflection-line in order, whereby the first digit relates to the firstreflection, the second digit to the second reflection and so on. If the digit is “1”then we apply the associated reflection-line; if it is “0” then this reflection-line isomitted. It turns out that this may be extended indefinitely: for example, to locatethe cell with binary number 111000110110011 within the grid, we apply reflection-lines 1, 2, 5, 6, 8, 9, 13,14, and 15, and omit the others. In a sense this grid andindexing sequence may be considered a visual form of the binary counting system.


1011

rl-1

rl-3

rl-2

rl-4

1011

rl-1

rl-2

1011

rl-1

Fig. 25 Locating cell with binary number 1011 using three reflections: rl-1, rl-2 and rl-4

rl-1

rl-3

rl-2

110

rl-1

rl-2

110

Fig. 26 Locating cell with binary number 110 using two reflections: rl-2 and rl-3

An Alternative Square TilingIn the derivation of the previous grid and mapping, we explored the route thatrepresented two beats as a larger square with twice the area of the original square.Now we shall look at two options pursued from an alternative approach, wherebytwo beats is represented by two squares placed next to each other.

Hilbert Curve TilingWe start from one square representing a beat and the double square representing twobeats. A generation may be completed by adding two further squares to the first two,forming a larger square comprised of four squares. To continue the process, we mayrefer to another space-filling curve known as the Hilbert Curve – see Fig. 27.

As this grid resolves to the next generation at four times the original area, itrelates to a time signature based upon the powers of 4 (groups of 4 beats or 16, 64and so on). On visualizing rhythms using this grid with the standard indexing forthe Hilbert curve, we find a similar issue to the Sierpinski space-filling curve, in thatsimple rhythms result in complex patterns (see Fig. 28). As before, this is overcomeby indexing the cells in a different sequence, based upon reflections, as in Fig. 29.In contrast to the first grid explored, we begin to see here how each grid has its owndistinctive quality and enables a different language of form and pattern.


Fig. 27 The Hilbert Curveand cell indexing of the first16 cells in the grid

11 22

3344

55

66 77

88 99

100 1111

122

1313144

155 1616

11 22

3344

55

66 77

88 99

100 1111

122

1313144

155 1616

2 31 4 5 6 7 8 9 10 11 12 13 14 15 16

Fig. 28 Visualization of a simple 4 beat repeated rhythm on a grid based on the Hilbert Curve.The white areas represent silence while darker cells represent accented beats, with the intermediateshades representing varying degrees of loudness

The Dragon CurveWhen starting from a square and a double square, we may alternatively draw uponthe space-filling curve called “The Heighway Dragon.” As opposed to the previous


1 2

43

7

5 6

8 16

14 13

15

1112

10 9

2 31 4 5 6 7 8 9 10 11 12 13 14 15 16

Fig. 29 Visualization of the same simple repeated rhythm as in Fig. 28, using a grid based on the“Hilbert Curve” with an alternative indexing sequence

grids that are created by the use of reflections, this grid is based upon rotations,creating a distinctively visual quality.

Each generation doubles the area of the grid. Unlike the other grids, eachgeneration does not resolve to the same congruent form, but tends towards aparticular form known as the Heighway Dragon (Fig. 30). The grid is indexed usingthe same principles as before, where each rotation maps the existing cells onto thenew cells in the same order – see Fig. 31.

The use of rotations rather than reflections lends the grid a distinctive look andfeel that is no longer so reminiscent of traditional forms of geometry (such as thosefound in Islamic art), but is distinctive of fractal geometry. This grid may lend itselfto certain styles of music, as opposed to the hard crystalline quality of the reflection-based grids.

Hexagons

The above examples all explored possibilities on the basis of a square representinga beat. What happens when we choose a hexagon to represent a beat? Whatimplications does this have upon the system for translating rhythm to pattern?


123 4

5 6

78

999 10

1112

1314 1516

Fig. 30 The first six stages of the Heighway created through a series of 90◦ rotations, and a higherorder of the curve showing its distinctive visual quality

2 31 4

Fig. 31 Heighway Dragon grid, visualizing a simple 4-beat repeated rhythm. The grid iscomprised of four Heighway Dragon curves meeting at the origin, creating the fourfold symmetry


1

23

6

9

1

2

1

23

1

Fig. 32 The regular hexagon, representing 1 beat of music (left); doubling the area results in ahexagram; returning to another hexagon three times the area of the initial hexagon; repeating thisprocess results in a hexagon nine times the original area (right)

1

45

87 9

6

23

Fig. 33 The hexagon divided into a grid representing 9 beats of music

Following a similar process as implemented in the first of our grids, we startby assigning a regular hexagon to represent a beat. Two beats is represented by ahexagram, so doubling the area. At the next step the area of the original form istripled and the sequence returns to a regular hexagon, so completing one generation(see Fig. 32). Continuing this process creates a sequence of nested hexagons thattriples the area with each generation (1, 3, 9, 27, etc.).

This series of nested hexagons will naturally subdivide into a grid of triangles,as shown in Fig. 33. As the grid resolves to each new generation at powers of 3,it relates to musical time signatures based upon the same numbers of beats per barsuch as 3, 9 or 27 and so on.


87 94

56

32 rl-1

rl-2

rl-3

rl-4

rl-1

rl-2

rl-3rl-1

rl-2

rl-11 1

456

23rl-1rl 1

rl-2

rl-3

123

rl-1rl 1

rl-21

2 rl-1rl 1

Fig. 34 Reflections used to locate cells within the hexagonal grid. The resulting grid is a visualrepresentation of the ternary counting system

The grid may be created, and cells located, by a series of reflections in the samemanner as the first grid, though now there are three possible transformations (noreflection, 1 reflection, or 2 reflections) – see Fig. 34.

As with the square grid and its relation to the binary counting system, this gridand indexing system relates to the ternary counting system. Here, each cell numberwritten as a ternary number contains the information as to which reflections to applyand which to omit to locate any given cell within the grid.

More HexagonsReturning to the sequence from hexagon to hexagram, then to a larger hexagon, wemay continue another step to arrive once more at a larger hexagon which is fourtimes the original area, again continuing this sequence to quadruple the area in eachgeneration – see Fig. 35.

Because the hexagon resolves with an increase of three times or four times theoriginal area, hexagonal grids may be created that resolve after groups of 3 cells,4 cells, or any combinations of these (e.g., 9, 12, 16, 18, 36). This enables thepossibility of exploring a variety of different note groupings, time signatures, andaccordingly, musical styles.

Rhythmic MotifsWithin music a rhythmic motif may define an entire style of music. A Flamencocompás, the clave from Central American music, the waltz and so on, all haveunique qualities defined by the arrangement of beats, accents, and rests. Theinstrumentation and performance of these motifs is of vital importance to the music,but the DNA of the style, as it were, is encoded within the rhythm itself.

Figure 36 shows examples of a variety of traditional rhythms translated into theircorresponding patterns. These examples are all 12 beat motifs based upon a hexagondivided into a 12 cell sequence.


1

2

1

234

1 1

234

8

12

16

1

5

87 6

91011

12

4

23

13141516

Fig. 35 The hexagon may also follow a sequence where it resolves to a larger hexagon at fourtimes the original area (top). The related grid displays the first 16 cells (bottom)

As with the rhythmic motifs, each visual pattern has its own unique character andexpression, created purely by the different arrangements of the shaded cells withinthe grid.

RotationsFigure 37 shows an example of a hexagonal grid created through rotations. Theaesthetic quality is in line with the Heighway dragon, but this grid has sixfoldsymmetry rather than fourfold, and relates to a time signature with 6 or 12 beatsper bar. Further grids created through rotations are possible, exploring differentsymmetries and related time signatures.

Grid Symmetry, Time Signature, and Structure of the CompositionThe symmetry of the grid relates to the time signature of the music. The growthof the areas between subsequent generations of the grid determines the number ofbeats in a bar. For example, in the first square grid described, a square resolves to


2 31 4 5 6 7 8 9 10 11 12 2 31 4 5 6 7 8 9 10 11 12

2 31 4 5 6 7 8 9 10 11 12 2 31 4 5 6 7 8 9 10 11 12

Fig. 36 Four 12-beat rhythms visualized using a hexagonal grid, showing the rhythm repeated16 times: a simple rhythm accenting the first beat of each triplet, results in a simple pattern ofuniformly spaced hexagons (top left); an African bell pattern (top right); a Flamenco compás(bottom left); and Arabic rhythm (bottom right)

a larger square with a doubling of the area, relating to either a 2-beat bar, a 4-beatbar, 8-beat bar, or any number in that doubling sequence. The hexagon resolves toa larger hexagon with a tripling of the area, relating to a 3-beat bar, but may alsoresolve to the next generation at 4-times the original area, so relating to a 4-beat bar,or a combination of these.

The structure of the grid also determines the structure of the musical piecebeyond a bar length. It determines the arrangement of the bars into sections andultimately of the sections into the whole composition. For example, with the squaregird, we may choose a bar length of 4 beats. These bars themselves would also bestructured into sections relating to the grid sequence, such as 16 bars. These sectionswould then also be structured according to the grid and could have four sections,


Fig. 37 A grid created through a series of rotations, displaying sixfold symmetry. Successivegenerations of a section of the grid are shown below

creating a macro level pattern over the whole piece, where the juxtaposition of onesection against another will create an overall pattern. Just as when exploring rhythmswithin a bar and the related visual motifs, the macro level of the pattern and overallstructure of the piece is open to creative exploration.


type a tile type b tile

Fig. 38 The two substitutions of the Penrose tiling. The type-a tile is substituted by one type-aand one type-b tile. The type-b tile is substituted by one type-a and two type-b tiles

Fig. 39 Creating a Penrose tiling starting from Decagon (left), recursively applying the substitu-tions from Fig. 38

Pentagonal Symmetry

The grids explored so far involve the square and regular hexagon. These two regularpolygons, as well as the equilateral triangle, create the three regular tilings that usejust one type of regular polygon to tile the plane, leaving no gaps or overlaps. Thesetilings translate to time-signatures commonly found within music, such as 4/4, 3/4,and 12/8.

In recent decades, there has been progress in tiling theory − particularlyregarding aperiodic tilings – the Penrose Tiling being the most well known of these,which tiles the plane indefinitely with fivefold symmetry and has two cell types.Building a grid based upon the Penrose tiling and other aperiodic tilings revealsinteresting implications for the rhythmical and structural counterpart in music, aswe will see.

To derivea Penrose tiling, we shall use a similar approach to the other grids andstart from a regular decagon. This time, rather than expanding outwards, the areasshall be subdivided to reveal the tiling. This process is known as substitution tiling,whereby each cell of a given shape is replaced by a specific grouping of cells toreveal the next generation of the tiling. The two substitutions applied here are asshown in Fig. 38. The cell which forms one tenth of the decagon (hereby calleda type “a” tile) is substituted with one smaller version of itself and a new, widertile called type “b.” The type-b tile, in turn, is substituted with two smaller type-btiles and one type-a tile. These are known as Robinson Triangles. Figure 39 showsthis substitution applied to the decagon four times to create a Penrose tiling. There


Number of Cells 2 5 13 343 8 211

Fig. 40 Number of cells in each generation of the Penrose tilings, also showing the intermediatesteps revealing the Fibonacci sequence

Sections

BarsWhole

Fig. 41 An example of the Penrose tiling governing the structure of a musical piece. Each cellrepresents a beat of music. They are grouped into 5(blue) and 8 (orange) cells, representing a 5beat bar and an 8 beat bar. These groups are arranged into sections of 13 (green) and 21 (purple),finally these sections are arranged into a structure of the whole piece of 3 (red) and 2 (yellow)

are particular orientations in which this substitution must take place. For furtherinformation, see Grünbaum and Shephard (1987).

The amount of cells in one segment of the decagon in each generation increasesin the sequence 1, 2, 5, 13, 34. These are alternate numbers from the Fibonacciseries, the remaining numbers of the Fibonacci series being revealed where the gridresolves as a pentagonal shape comprised of type-b tiles – see Fig. 40.


1

2

34 6

12

5 78

910

111213

23

4

1

5

Fig. 42 Indexing the Penrose tiling using the tiling substitution order. An issue with this indexingcan be seen in the second level of subdivision (center), where cells 2 and 3 together create a type-atile shaded in orange, as do cells 4 and 5 shaded in blue, though the order of the tiles within thisshape is reversed, shown by the arrow

03

57

6 81011

2126

2930

31

32

33

28

2722

232524

1819

2015

1617

14

13

3435

403941

36

3738

50

5149

5452

4847

53

45 4644

4243

12

94

1

2

Fig. 43 The Penrose Cartwheel tiling indexed as by F. Lunnon

Fibonacci, Bar Length, and Structure of CompositionThe implications of the generations of this grid being based on the Fibonacci seriesare that, when explored as music, the Fibonacci series will govern the structure atevery scale from beat to bar, to section, to the whole piece – see Fig. 41.

The two cell types, although different in size, are both considered to be thesame beat length in music and so represent equal lengths of time. This seems


Fig. 44 The Ammann-Beenker tiling, an aperiodic tiling displaying eightfold symmetry madewith two tile shapes. The two tiles, with substitution arrangements, are shown below

acceptable given that both cells may be considered as projections from the samehigher dimensional cubic structure (Senechal 1995). This does raise a concern,however, in that at larger scales of the grid – for example, bar lengths – the twoareas which are congruent to these two forms contain different amounts of cells,and so correspond to different lengths of time.

The shapes congruent to the type-a and type-b cells on any given scale containconsecutive numbers of cells from the Fibonacci series (Fig. 41), meaning thatin musical form there will be two bar lengths within a piece using consecutiveFibonacci numbers. For example, we may choose a bar length of 5 beats and abar of 8 beats. These bars shall then also be grouped into sequences based upon


Fig. 45 Aperiodic octagonal tiling with four tiles, shown with substitutions

the Fibonacci series, and so on, up until the level of the whole piece (Fig. 41). Tomake meaningful use of the grid and the relationship of the forms that are naturallyoccurring within them, the composer and designer must consider the order of “a”type cells and “b” type cells within the bar. What this all adds up to is a rhythmicalstructure tightly governed by the Fibonacci series at every scale of the piece.

Indexing the Penrose TilingOne form of indexing the Penrose tiling would be to follow the sequence of type-aand type-b tiles created from the substitution itself. Figure 42 shows this appliedto the first 13 cells of the Penrose tiling. This sequence presents an issue, in that


congruent shapes within the tiling sometimes have a different route through them,as shown in Fig. 42.

Other indexing sequences have been explored, though an entirely satisfactorygrouping of cells which works at every level of the substitution with the Penrosetiling remains illusive.

However, the Penrose cartwheel tiling (Fig. 43) has been indexed by Fred Lunnon(Grünbaum and Shephard 1987) in a manner that works at every level of thesequence figure and may be applied in this translation method.

Octagonal Symmetry

Another aperiodic tiling which may be applied in this method is the Ammann-Beenker tiling, with eightfold symmetry – see Fig. 44.

Another eightfold aperiodic tiling discovered by Mara, inspired by the aestheticsof Islamic arts, is shown in Fig. 45.

Summary

In summary, we have seen a process for translating rhythm to pattern and vice versa,derived from a simple premise and following logical steps in geometry. This resultsin a geometric grid standing as the visual counterpart of the temporal structureof music, whereby each cell in a grid represents a particular beat, allowing forcreative explorations in either rhythm or pattern to be represented in the other. Wesubsequently explored a selection of tilings, looking at their different symmetriesand related time signatures. We have observed how each tiling has its own visualqualities and vocabulary of forms and have examined the contrasting quality oftilings created through reflections or rotations, as well as the regular tilings versusaperiodic tilings.

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