Contents

p. 3 Introduction

p. 4 Ancient (pre-500 CE) and Early microtonal practices, systems, stylistics: Medieval (500-1400),

Renaissance (1400-1600), Baroque (1600-1760), Ingram, Dumbril, Plato, Pythagoras, Heptagrams,

Babylonia, Assyria, and Greco-Arab Texts

p 7 Ancient practices and oral traditions

p. 10 Al-Farabi, 17TET system, kitab al-Adwa, al-Andalas and barzok

p. 14 Common microtonal practices, systems, stylistics (1600-1900):

Baroque (1600-1760), Classical (1730-1820), Romantic (1815-1910) Bach, equal temperament,

Glarean

p. 17 Jamaica and Africa, Koromanti and Angola, Ethiopian bowl lyre (krar), Quadrille music in

Carriacou

p. 18 Post-romantic and Pre-modernism, experimental, Carrillo, Ives, Rimsky-Korsakov, Russolo,

Experimentalism, polytonality, tone clusters, aleatorics, quarter-tones, polyrhythmic

p. 19 Contemporary & modern microtonal practices, systems, stylistics: Modern (1890-1930), 20 th

century (1901-2000), Contemporary (1975-present), Modernism, Dadaism, serialism, microtonality,

Verèse, Webern, Wyschnegradsky, Hába, Carillo, Villa-Lobos, Ives, Partch, Cowell

p. 22 Yasser, infra-diatonicism, supra-diatonicism, evolving tonality

p. 31 Darmstadt, neotonality, dodecophony, Stockhausen, Boulez

p. 30 22TET, A Just 12-tone scale built on powers of 3 and 5, diminished 7th blue note, 1960s Rio de

Janiero Jazz, Bossa Nova, US jazz, flattened 5th and hexatonics in the Blues, New Orleans

resurgence, Copacabana

p. 34 Pitch and cognitive acculturation, development of musical thought and thought in

sound, schematic and veridical expectancy, mistuning perception

p. 37 Just, Bohlen-Pierce scale, Wusta-Zalzal, Masonic ratios, 22 tone system of India, Ragas,

Messiaen, Babbitt, Cage, Young, French Spectralists, 53TET, 19TET, Bagpipe tuning

p. 48 Midi, scale perception, semiotics, notation, re-creation, Turkish, Eskimo, Indonesian Slendro

in 5TET (Salendro), Thai 7TET

p. 53 Xibeifeng, Xenakis stochastic emulator, fretboards and the 12th root of 2, world Fusion,

evolving timbral domain, microtonality and after the fact of performance, societal technological

status, cultural and logical outset, and aesthetical artistic nuance

p. 56 Conclusion, truth in music, modality of believing, dynamic tonality, Third-stream music,

sound painting, new directions

p. 58 Glossary, p. 62 References

2

Microtonal music and its relationship to

historical practice by Geoff Geer

Introduction

Intonation systems make up a large part of musical performance, often floating beneath the

compositional surface, below the timbres, stylistics, speed and dynamics. It is conscious

organised order of performance and composition that determines what we deem as music. A

clever melody or evocative harmonic line may be altered by taking it out of the underlying

context of intonation systems. Today these systems can be extended through use of an

understanding of previous centuries’ performance stylistics in tonality and microtonality, and

cultural and contextual ideology and application. In the paper we will trace past tonal systems

and practices and musical ways of thinking tonally and microtonally to determine whether any

patterns emerge. Are 12TET,1 24TET or Just intonation (small ratios) the best choices for

today’s musicians? We will look at some of the leading historical musical thinkers and contrast

their ideas with modern microtonal thought and practice, as well as the cutting edge research on

tonality, technology and compositional practice for the 21st century. Are there logical patterns

emerging in human musical thought and practices with regard to some examples of definite links

to past and present practices? Musical practices and their tonal systems and theories build the

sound track to transnational-migrations of peoples, politics, ideologies, capital and mass media

images, acting as boundary-markers even as they cross boundaries, transforming and

reinterpreting them—reconfiguring cultural imagination by expression of desires and memories.

(Shannon, 2007)

Microtonal music, as music that is not 12 tone equal temperament, has occurred worldwide, in

the Americas, in Europe, in Asia, Africa, the Middle East, and Australia.2 Bach wrote pieces (as

1 12 Tone Equal Temperament, 12 equal divisions of the octave.2 Examples include Byzantine liturgical music, Scottish bagpipe, Iranian chamber music, Indonesian Gamelan, Za’atar Jewish music, Bakshish ensemble, and African xylophone. Tonal systems today include equal tunings 5TET (Indonesian slendro), 6TET (Tone Equal Temperament), 7TET (Thai traditional), 12-Equal or 12TET (Western c.1800-present), 15TET, 16TET, 17TET(Arab), 18TET (Wyschnegradsky), 19TET (Guillaume Costeley), 22TET, 24-Equal or 24TET (quarter-tone), 26TET, 31TET (Huygens, Fokker), 34TET, 36TET (Wyschnegradsky),, 41TET, 3

harmonically as possible) using (according to Forkel, his biographer) thirds tuned slightly sharp,

a prerequisite in transpositional functioning. Just intonation was generally used before this in

various systems worked out through ratios, and is defined as small interval ratios. Bach was

limited in composing by meantone temperaments, and today we can hear some of what he was

unhappy with using special software that enables closer approximations, highly accurately, in

Just intonation. Werckmeister, a Baroque era composer notable for his invertible counterpoint,

did away with the unnecessary applicability of enharmonic keyboards of the time, which had

more than 12 notes, of which many were euphonious. While Pythagoras may have developed

whole number ratio tunings, the Harmonists had perhaps thousands of ratio tunings which were

lost after the fall of the Roman empire, with some going to the Arab world for development.

After this, during the early middle ages, consonance was based on a 1/1 unison, 2/1 octave, 3/2

perfect 5th, 4/3 perfect 4th, with 3rds and 6ths being dissonant. In 1300 the English monk Walter

Odington came up with: 5/2 minor 6th, 5/3 major 6th, 5/4 major 3rd, 6/5 minor 3rd, though later

it was realised that it had already been discovered. (Denton, 1996) These various ratios used

throughout history differ markedly from various equal-tone and meantone temperaments that

came later on, including 24-TET (quarter-tone equal temperament). In the 1500s Gioseffo

Zarlino thought that ratios 1 through 6 were consonant, leading to use of major and minor triads

during the renaissance, which developed chordal and harmonic music based on ratios and Just

intonation, yet there was also a growing body of work for fretted and keyboard instruments.

Before this, music was predominantly vocal, and instrumental music then took off in the classical

period. In the 20th century Partch envisioned instruments that could modulate and retain Just

intonation. 24-TET instruments are very complex, and notational systems vary. During the

Baroque period meantone temperament was used: 4ths and 5ths are about 2 cents off, 3rds and

6ths are slightly out, 8 scales are near perfect and 4 are very mis-tuned. With more complex

music and modulations came the need for equal-temperament around 1750. In the middle-ages A

440 varied from 370 to 567 Hz and people had their own tuning forks. The church pitch was

often a whole step higher than the choir pitch, and a compromise chamber pitch resulted from

this at around 420 Hz. Alexander J. Ellis created charts of the pitch of instruments which can tell

43TET, 47-edo (equal division of octave), 50TET, 53TET (Turkish), 72TET. Linear tunings, that temper non-octavenotes via a stack of perfect fifths, include Syntonic (generators P5 and 8ve), Meantone (quarter-comma, septimal), Schismatic (Helmholtz), Miracle (a regular temperament), Magic (generator 5/4 narrows or widens). Irregular temperaments include Well temperament/Temperament ordinaire (Kirnberger III, Werckmeister, Young, Neidhardt, Vallotti, and Young). Other systems include Just intonation, Pythagorean, Partch’s 43-tone, Ptolemy's intense diatonic scale, tonality diamonds, numerary nexus, tonality flux, otonality, hexany, scale of harmonics and non-equal temperament tunings.4

us what we are hearing. (Denton, 1996) For the Russian ancient liturgical styles Joseph Yasser,

whose major work was A Theory of Evolving Tonality, talks about the byways of tonal evolution

and pleads for tonal restoration. Yasser asserts that pentatonic [infra-diatonic] theory precedes

more advanced temperaments, and that quartal harmony, rather than tertian [3rds], is the proper

harmony for the infra-diatonic. This was followed up by his attempt to demonstrate the

pentatonic character of Gregorian chant. Yasser’s letter to Schoenberg criticised his chromatic

12-tone acoustic interpretation. Schoenberg tabulates the harmonic series to the 13th partial. For

Schoenberg these six first partials are founded on the root, fourth and fifth of the harmonic

series, and constitute the diatonic scale, while adding the remaining seven partials forms a

complete chromatic scale. Yasser asserted that the first few notes may sync well enough, but as

the harmonic series evolves (phi ratio) there is greater error between the three ascending series’

pitches microtonally, and he maintained that Schoenberg did not take the trouble to check his

flawed work. For example there is a 38 centitone difference between the Eb at the 7th and 13th

partials, and the C# and Db at partials 11 and 13 are off by almost a semitone, or 43 centitones.

(Yasser, Schoenberg, 1953) A main problematic is how the instruments were built and their

relation to written notation. With the old folk flutes for example, and their recreations, it would

be hard to modulate due to their microtonal nature. Forkel says that Bach tuned his thirds slightly

sharp for modulational functionality. Equal temperament is the aberration and a recent

phenomenon, falling outside of the natural phi ratio phenomenon. While equal-temperament is

practical for modulation and composition we must remember that voice and strings use it for

reference and tonal centricity, along with others like trombone. French music concrète, though

criticised for being overly intellectual at times, delved deeply into microtones, as did the

Spectralists, and this was at odds with the German tradition of notational elektronische music,

although tonal elements existed there too. Interestingly, there was the 1932 Cairo convention on

quarter tones where a canun was tuned to 24TET. Examples where played to Arab musicians

who unanimously agreed that it was out of tune—it may be more accurate to use a Pythagorean

system or divide the octave into 53 commas. (see A.J. Racy's Making Music in the Arab World)

5

Ancient (pre-500 CE) and Early microtonal practices, systems, stylistics: Medieval (500-

1400), Renaissance (1400-1600), Baroque (1600-1760), Ingram, Dumbril, Plato, Pythagoras,

Heptagrams, Babylonia, Assyria, and Greco-Arab Texts

The 4th and early 5th centuries showed enharmonic and chromatic tuning to be more popular

than heptatonic diatonicism, and Aristoxenus records that in the 4th century it was common

knowledge that diatonicism predated Hellenic chromaticism and enharmonicity that either co-

existed with pure diatony or overlaid it.3 Ingram’s popular view that earlier tunings were

defective is cast into some doubt by the discovery of a near-Eastern cyclical diatonic system pre-

existing Aristoxenus’ by two millennia.4 However, Philolaus attests in the 5th century that the

earlier systems were defective, with some heptatonic systems derived from filled in notes.

Arestoxenus names Eratocles as formulating the precept that modulation can only occur at

consonant intersections, and Ion of Chios agrees that this was standard practice around 422 BC.

Enharmonic and chromatic transposition/modulation was restricted to the bounding notes of each

tetrachord, not the inner notes that were often microtonal.5 Ptolemy's διατονικοΰ συνεχοΰς

(diatonic continuous) led to the σύσιημα τέλειον (systima perfect), enabling modulation of the

συστήματα for complete enharmonic and chromatic modulations. (Franklin, 2002)

Plato's term harmonia describes ethnic scales permissible or not in his Ideal State, theorised in

The Republic where different political regimes are discussed—translated commonly as mode we

do not know their exact nature although there is an account by Aristides Quintilianus. (De

Musica I.9, p. 19.1-10, ed. Winnington-Ingram) Although Aristoxenus does not use harmonia in

this sense he seemingly describes it as synonymous with tonos, though this is problematic due to

the concept of eidos (species) of intervals like the octave, akin to the modern and medieval

mode, without the concept of tonic, dominant and polychordia.

A deciphered cuniform tablet, depicting notes on a lyre corresponding to a heptagram (c. 2000

BC) is thought to use thirds in harmony and a diatonic scale. (Kilmer, 1986, cited in Dumbrill,

n.d.) The archeomusicologist Richard Dumbrill argued for over 30 years with colleagues as to

3 In past Greek tragic practices, the chromatic genus did not appear until Euripides, and used predominantly Dorian and Mixolydian, symbolic of dignity and pity. Lydian and Ionian were used and Sophocles was the first to use the Phrygian and Lydian tonoi, although very rare in the tragedy, were the Hypodorian and Hypophrygian.4 Winnington-Ingram, an authority on ancient music, ought to be mentioned for his articles in The Classical World, which accompanied Choudbury and Bogges' medieval discussions on Greek tragedy (Choudbury, 1909; Bogges, 1968). Ingram mentions a work by Robert Browning (Browning, 1963) on Greek tragedy, connected possibly with Psellus, the Byzantium encyclopaedist and philosopher/writer (11th century) (Albert, 1900), of which there is no translation, and based on Aristotle and the music of tragedy most likely derived from Aristoxenus' works. (Feaver, 1969)5 In Just or early Pythagorean tuning the 4th and 5th fell very close to their 12TET counterparts.6

whether or not instead of a heptatonic, with diatonic Assyrian roots, that an enneotonic (9 tone)

scale may have been prevalent, and produces it as archeological evidence. Dumbrill points out

that Occidental diatonicism may have roots not in ancient Babylonia, but stem from a

Pythagorean myth that germinates in mediaeval traditions. In Plato’s Republic (545c-546d) the

[9] muses mention two harmonies,6 or superimposed heptachords, which make up an

enneachord. Babylonian practice would be taught through metaphors and metonymy and by ear,

allowing for wider or smaller non-complex ratios other than Just. Unlike Greek tunings governed

rigidly by ratios,7 in Babylonia there may have been a multiplicity of tonal systems practicably

tuned by ear, and the octave may have been unknown. (Dumbrill, n.d.) All Greek musical

knowledge originates from 10th and 11th century Western adaptations and translations [of Arabic

texts].

Unisons and ‘magadised’ octaves are generally thought to have existed in Greek music, yet

scholars are perplexed as to whether there was simultaneous use of perfect 4ths and 5ths,

indicative of the infra-diatonic scale (5+2)8, yet similar to the sub-infra-diatonic scale (2+3). The

Siamese (5+2) infra-diatonic system lacks the distinct characteristics of Western diatonicism, as

the main part consists of only 5 notes, and 2 subsidiaries (embellishments), and is a closed

system. In European diatonicism this is not the case, and there has never been any standard

indication of temperament historically generally, owing to written melodies often being

converted into other temperaments. The historical point of transition between sub-infra-diatonic

(2+3) and infra-diatonic (5+2) is unclear. (Yasser, 1932, p.152)

Ethics, philosophies and values have always been linked to performance and music, and may

extend to cultural idioms like techniques, gesture and stylistics. For many traditions there

appears to be scant evidence for past musical practices and traditions,9 and Early Music

6 Dumbrill claims there is no evidence that Pythagoras existed, or that he wrote about music if he existed, that he was a fictitious pun invented by the early Greeks, and in light of Near Eastern cuniform mathematical mastery, therewas nothing left for him to discover – and that modern academia is misled on this point. 7 Greek tunings were dominated by ratio and string length, yet Aristoxenus preferred string tension and relaxation, yet many medieval transpositions of Eastern theory, such as al-Farabi, cite their foundation on Greek theory, and may have muddied Aristoxenus’ theories.8 Yasser’s term infra-diatonic encompasses 5 primary notes with two subsidiary, such as 7TET. Diatonic is 7+5 or standard 12 chromatic notes, and supra-diatonic are systems with greater numbers that 7+5 such as 12+7 or 19TET. This is based on the supposition that tonality is evolving from basics like 1, 5, and 4, or that the pentatonic scale cycled in 5ths will make up diatony, and includes progressive use of higher ratios in the harmonic series.9 In breaking down the taxonomy of world instruments into similar attributes one can consider the physical attributes(construction) and culture in the production of musical creation/stylistics heritable and traditional, passed along in instrument making and in cultural gestures that overlay learned implicit tonal understanding. Theoretically one couldask ‘which came first?’ as they are part and parcel of ongoing cultural and human musical development. That instrument creation plays/played a part in the theory behind evolving construction is also a fascinating idea, and has a lot to do with timbres, moods, tonality, pitch, and musical creational thought aspects.7

Performance scholars and performers have looked to living traditions to inspire and bolster

ancient and past European traditions. Often, surface facets are avoided and the larger-scale

structural features are favored in developing new work. Further, Early Music ethnography can be

discerned via original texts and writings from the musicians. Interestingly, Western classical

music generally is not well represented in terms of ethnomusicology, perhaps due to missing

historical gaps and inconsistencies.10 (Shull, 2006)

The Pythagorean comma (diatonic comma) is a small interval (frequency ratio 531441:524288

or 23.45 cents) in Pythagorean tuning,11 and equals 12 Just perfect 5ths. Later Greek ratios were

codified by Ptolemy, expanding Pythagoras’ 3 limit Just 4th and 5th to include a Just major 3rd in

limit 5.

Stemming from 1/1, the ratios for limit 5 Pythagorean Just are:

ratio 1/1 81/80 128/125 25/24 256/243 135.128 16/15 27/25 800/729 10/9 9/8 256/225cents 0 21.51 41.06 70.67 90.22 92.18 111.73 133.24 160.90 182.40 203.91 223.46

ratio 125/108 75/64 32/27 6/5 243/200 100/81 5/4 81/64 32/25 125/96 675/512cents 253.08 274.58 294.13 315.64 337.15 364.81 386.31 407.82 427.37 456.99 478.49

ratio 4/3 27/20 25/18 45/32 64/45 36/25 40/27 3/2 1024/675 192/125cents 498.04 519.55 568.72 590.22 609.78 631.29 680.45 701.96 721.51 743.01

ratio 25/16 128/81 8/5 81/50 5/3 27/16 128/75 225/128 16/9 9/5 729/400cents 772.63 792.18 813.69 835.19 884.36 905.87 925.42 976.54 996.09 1017.60 1039.10

ratio 50/27 15/8 256/135 243/128 48/25 125/64 160/81 2/1cents 1066.76 1088.27 1107.82 1109.78 1129.33 1158.94 1178.49 1200.00

Ancient practices and oral traditions

Ancient practices and oral traditions that passed musical information historically are important

to review - some pitch syllables are:

interval 1 b2 2 b3 3 4 #4 5 b6 6 b7 7Western Do re Re mi mi fa Fa sol la la ti Ti

India Sa re Re ga ga ma Ma pa da da ni NiChina Shàng chě Chě gōng gōng fán12 Fán liù wǔ Wǔ yǐ Yǐ(gongche) 上 尺 尺 工 工 凡 凡 六 五 五 乙 乙simplified ル 人 人 フ フ り り 久 ゐ ゐ

10 See later section on recording and archiving of European folk musics.11 Another definition of the Pythagorean comma is the difference between a Pythagorean apotome and a Pythagorean limma; between chromatic and diatonic semitone: or between twelve just 5th's and seven octaves; or between three Pythagorean ditones and one octave. The opposite in Pythagorean tuning is the diminished 2nd (difference between limma and apotome) equal to a diesis ~ 23.46 cents.12 Fan and Yi are between 4 and #4 and ♭7 and 7. This is a simplified version and there are more characters for otheroctaves and variances for Kunqu and Chinese Opera.8

Balinese Ding dong13 deng dung dangJapan I ro ro Ha ha Ni ni ho hi Hi to ToArabic Dāl rā' rā' Mīm mīm fā' Fā' sād lām Lām tā' tā'

د ر ر م م ف ف ص ل ل ط طByzantine Ni pa pa Vu vu Ga ga di ke Ke zo Zo

Η, η Α, α Α, α Β, β Β, β Γ, γ Γ, γ Δ, δ Ε, ε Ε, ε Ζ, ζ Ζ, ζ

The old Chinese gongshi notation is still used for traditional instruments, and incorporates a

movable do (shang). Like tablature for specific instruments it may have originated with a fixed

do system, later using a movable do.14 Traditional musicians still use the score, yet perform from

memory.

While Western solfege is thought by many to have sprung from Latin roots, there is conjecture

it may have Arabic solmization system origins from an influx of Islamic contributions in

medieval Europe. The syllables are: dāl, rā', mīm, fā', ṣād, lām, tā'. Masonic sources site ancient

solfeggia frequencies in hertz as 396, 417, 528, 639, 741, and 852 (in cents: 0, 89, 498, 828,

1084.8, and 1326.4 or 126). In 1935, due to poor music (and sight-singing) standards in

Hungary, Kodály revised the curriculum that incorporated a movable-do solfege system of

syllables, showing relative, and not absolute, pitch.15

Particular cultural facets and idioms do impact on aesthetic stylistics indicative of time and

genre, yet there are musical elements that lie outside the bounds of standard notation – these

devices carry microtonality and timbre and in the attributes of African Vocality may be

categorized: shouts (intoned or non-intoned), head-voice or falsetto, microtonal utterance like

blue notes and glissandi, interpolated vocality, Afro-melismas (form of recitative), multiphonic

sounds (same generator), guttural sounds (from the throat), and vocal rhythmization

(predominantly rhythmic). All these qualities are speech derivative and imbue emotional

emphasis much the same as language. (Duran and Stewart, 1997)

Microtonally passionate speech as a type of musical iconography triggers recognition and

emotional response to the listener – specific expressions of the human voice. The spiritual Go

down Moses begins with a melody going up and continues up with ‘way down to Egypt land.’

13 The graph approximates equivalents in 12TET.14 The pitch notation was skeletal, making room for improvisation, and evolving offshoot variants make historical determinacy of pitch, system and practice hard to imagine how it may have sounded – and the variant systems of notation became harder to learn. 15 Kodály was first exposed to this in England – a moveable-do system was already in place by Sarah Glover and amended by John Curwen for choral training, which was felt to bolster a grasp of tonal function. Kodály even felt that moveable-do solfege should come before an understanding of the staff.9

Monteverdi’s opening of the opera Arianna employs a similar irony of a falling vocal contour

‘Lasciatemi morire’ (Let me die!).

In 1584 Zhu Zaiyu (Chu-Tsaiya) and then Simon Stevin in 1585 are accredited with the exact

calculations of the equal temperament, both independently though Stevin's less accurately. Fritz

Kuttner was critical that either achieved equal temperament.

Al-Farabi, 17TET system, kitab al-Adwa, al-Andalas and barzok

After c. 872 Al-Farabi had logically divided the octave into 25 units, which he demonstrated on

the Oud.

Fract

ion

1/1 256/

243

18/1

7

162/

149

54/4

9

9/8 32/2

7

81/6

8

27/2

2

81/6

4

4/3 3/2 18/1

1

19/9 2/1

C D E F G A B CCent

s

0 90 98 145 168 204 294 303 355 408 498 702 853 996 1200

Consisting of limma and comma intervals this system is still valid in the Arab world.

C D E F G A B C4/4 1/4 3/4 4/4 4/4 1/4 3/4

These ratios add to 24/4.

The simplest way to describe quarter-tones is: 50 cents or, E = the note exactly in the middle

of (half way between) E and E♭, and E‡ = the note exactly in the middle of (half way between)

E and E♯. The quarter-tone is half way between the natural and the sharp or flat (50 cents in

equal temperament).16,17 Please note that a standard half-flat is a mirrored flat, and that the

alternative strike-through flat is used in this paper.

Safi al-Din al-Urmawi’s 17TET system (13th c.) was the main system until replaced by 24TET

(quarter-tone scale), and kitab al-Adwa (KA) is one of the most influential Arab treatises on

music. (Wright, 1995)

17TETInterval Fundamental Cents

16 The E in maqam rast is usually taken generally to be higher than the E␢ in maqam bayati.17 note. A ¼ tone = half a semitone (50 cents), a ½ tone = a semitone (100 cents), and ¾ tone = a semitone + ¼ tone (150 cents). It must be stressed that the ¾ tone is not, as its name suggests, ¾ of a tone (three quarters of a tone), but a ‘three quarter tone’. Thus two three-quarter tones constitutes a minor third. 10

1 0√2 1 02 17/1√2 1.0416160106505838 70.5882352941176268003 17/2√2 1.084963913643637 141.1764705882350870004 17/3√2 1.1301157834293298 211.7647058823528980005 17/4√2 1.1771466939089177 282.3529411764706080006 17/5√2 1.2261348432599308 352.9411764705883370007 17/6√2 1.277161683956088 423.5294117647059930008 17/7√2 1.330312058198122 494.1176470588234900009 17/8√2 1.3856743389806951 564.70588235294111600010 17/9√2 1.4433405770299566 635.29411764705901400011 17/10√2 1.5034066538560549 705.88235294117647700012 17/11√2 1.565972441175087 776.47058823529406800013 17/12√2 1.63114196696555 847.05882352941155200014 17/13√2 1.6990235884354028 917.64705882352944700015 17/14√2 1.7697301721873238 988.23529411764724000016 17/15√2 1.8433792818817307 1,058.82352941176461000017 17/16√2 1.9200933737095864 1,129.411764705882310000

The 18th degree is 1200 cents.

Al-Farabi extracted the intervals 8ve, 4th, 5th, 7th, whole tone, and quarter-tone on the Oud.18

Also defined was Wusta-Zalzal, greater than a tempered minor 3rd and less than a tempered

major 3rd, with the ratio 27/22.19

In past and present) Arab musical practice there is a similar idea to the Western cadence that is

a template for development and is modulation in the Maqam. One or more notes are incorporated

into the scale of the Maqam producing a second compatible maqam. This modulation can

proceed, transitioning into a Maqam or Maqamat,20 and finally will return again at the end to the

original Maqam. During the Taqasim or tahmelah (free rhythmic forms) it is common for

soloists to modulate many Maqams. Further, this is commonly done by replacing the maqam’s

upper Jins with a compatible Jin ‘of the same size’.

The Maqam is built upon the diwan. One diwan is usually eight notes, and sometimes extends

scalar-wise upwards comprising two diwans. Maqam is more than a scale for the following

reasons:

18 Also, the gambus, an oud offshoot, came to Southeast Asia from Yemen traders in the1500s, and is still in use in Malay folk and religious musics. (Al-Jawharah, 2010)19 In many films depicting the music of the Middle or Near East, a wolf 4 and/or wolf 5, for example, may be heard -E, F♯, G, A , B – the wolf 5th resting 50 cents between the tritone and 5th. The wolf 2 or 3 may be heard, E, F‡,

G‡ and part of specific maqamat, and are just some of the colourful Mid-Eastern nuances in practice [from 24TET perspective], and in Gypsy music from India through to Turkey, Greece, and Spain. One contemporary example of microtonality in practice is in Gypsy music, such as in the band Taraf De Haidouks. 20 This style of evolving compatible scales is prevalent in Gypsy and many European folk musics, as well as jazz.11

-A Maqam can incorporate microtonal variations that are very subtle: so that tones, semitones

or quarter-tones are slightly altered.

-A Maqam has rules defining the starting note (Qarar) and ending note (Mustaqar), which can

in some instances be different to the tonic or dominant (Ghummaz). The second jins starting note

begins on the dominant.

The Samaie genre is composed of four sections (Khana, plural Khanat) each being followed by

the Taslim (refrain).21

1 Structure A T B C D2 Sections/Khanat First

Khana

Taslim Second Khana Third Khana Fourth Khana

3 Start 3rd Dominant Dominant 2nd Tonic4 End Dominant Tonic Tonic Tonic Tonic5 Range 9 9=1/2 9+1/2 12 116 Modulations (outside

the maqam)

Farhafza

Ajam and

Nahawand

Farhafza

Nahawand

and Hijaz

Hijaz

Bayati and

Nahawand

Hijaz

Nahawand

and Ajam

Hijaz

Nahawand

7 Time Signature 10/8 10/8 10/8 10/8 6/88 Length 8 4 8 8 249 Sections Farahafza F Farahafza Hijaz Hijaz Hijaz10 Repeats 1 1 1 1 2

Examples of transposing melodic development:

Bb C D Eb F G A Bb1 tone 1 ½ 1 1 1 ½

Indeterminacies abound within geopolitical and cultural areas, for example the distinctly

European sounding Levantine and North African ‘Andalusian’ musics that, though different,

claim a common al-Andalus commonality. These indeterminacies are likened to the Sufi idea of

barzok, the wonder of the imaginable and indeterminable, which are bounded by constriction, yet

also have potentiality and horizon. Moroccan Andalusian and European musicians perform well

together due to a shared musical commonality, whereas European musicians performing with

Levantine musicians (East Mediterranean) may avoid microtonal modes.22 (Shannon, 2007)

Far Eastern music also abounds with microtonality. In the 8th century the shakuhachi flute came

into Japan from China, with later resurgence, and does not use tongue articulation for pitch

21 Although the tempo is 3+4+3 modern musicians may regard the 10/8 time as 5+5 and is largely regarded as one ofthe important instrumental Arabic forms.22 Syria and Morocco sound strong musical ties to medieval Spain. Andalusian music and heritage help bolster pan-Arab ideologies that coincide with Syria’s Ba’thist ideologies. Heritable and proven historical practices with Andalusian links help authenticate Syria’s heterogeneous pasts tied to Christian, Muslim and Jewish histories which counter what some deem vulgar and unauthentic. (Shannon, 2007)12

reiteration but grace-note articulations, with shaking of the head from side to side. There is no

diaphragmatic vibrato, and whilst the holes produce pitches roughly in sync with equal-

temperament, since there is no valve or fixed-key system microtonal inflection is of relative ease:

glissandi may be produced. (Lependorf, 1989)

This can be contrasted to today’s modern composers. Frank Denyer wrote The tender sadness

of tyrants as they dance (1991) for the shaku-hachi and Western bass flute, a combination which

creates a previously unheard sonority, one that can be both delicate and ruthless. They play

together the whole way through, employing ancient techniques like vibrato, microtonal

inflections and modern techniques like ghost tones whereby the player breathes into the flute

while fingering notes as well as vocal sounds and tap dancing shoes used to knock heavily

against the floor. (Gilmore, 2003)

Common microtonal practices, systems, stylistics (1600-1900):

Baroque (1600-1760), Classical (1730-1820), Romantic (1815-1910) Bach, equal

temperament, Glarean

Standard equal temperament is defined thus: each semitone ratio is exactly the same as it

ascends to the octave,23 regardless of how many intervals there are. Generally it is in the 12-

semitone octave (12-tone equal temperament, 12TET), although others exist such as 17TET,

19TET, 24TET 31TET, 53TET and others. Prior to this, temperaments had narrowest 5ths

throughout diatonic notes producing purer thirds, with wider 5ths between the chromatic notes

(sharps/flats) indicative of the writing period style and treatises, enabling transposable modes

[well temperament]. One possibility of a very early circular temperament was described by the

early 16th century organist Arnolt Schlick, though well temperaments only phased in during the

Baroque, persisting into the Classical period. Some were closer to meantone and others nearer

equal temperament, with no wolf 5th. Keys with greater sharps and flats sounded further out of

tune because of the 3rds, and modulations were used sparingly (i.e. interchange, ornaments,

transitions). The period temperaments include Werckmeister, French Temperament Ordinaire,

Neidhardt, Kimberger, Vallotti, and Young.

23 Non standard divisions in place of an octave include the tritave, stretched octave, and other non-octave scales.13

Meantone (averaging between notes), Helmholtz, Pythagorean, schismatic and miracle

temperament are examples of regular temperaments, where ratios are calculated via powers of a

limited number of generators. Meantone intervals are calculated by the width of the 5th and an

8ve for the syntonic comma [unison].24 [Easley Blackwood attributed the label ‘R’ to the ratio of

the whole tone to diatonic semitone.] In the past, small ratios were used to achieve musical

scales, such as the Just system, however, serious harmonic problems were encountered after the

Middle Ages as music became more complex, with greater polyphony and key changes, and

these perfect intervals no longer sounded harmonic – due to wolf intervals. (Enevoldsen, 2010)

Commas include the Pythagorean comma (23.46 cents), the syntonic comma (21.5063 cents)

Mercator’s comma (21.8182 cents, or 55√2), and Holder’s25comma (22.6415 cents).

Table of commas

Name alternative cents RatioSchisma Skhisma 1.95372078

7934159400

32805:32768 8 perfect 5ths +

major 3rd

5 octaves

Septimal

kleisma

7.71152299

1319534110

225:224 2 major 3rds +

septimal major 3rd

Octave

Kleisma 8.10727886

2071810140

15625:15552 6 minor thirds Tritave [8ve + 5th]

Small

undecimal

comma

17.5761311

5728168290

0

99:98

Diaschisma Diaskhisma 19.5525688

0878068610

2048:2025 3 octaves 4 perfect 5ths +

2 major 3rdsSyntonic

comma

Didymus'

comma

21.5062895

9671485360

81:80 4 perfect 5ths 2 octaves +

major 3rdPythagorea

n comma

Ditonic

comma

23.4600103

8464900870

531441:5242

88

12 perfect 5ths 7 octaves

Septimal

comma

Archytas'

comma

27.2640918

0010023040

64:63 Minor 7th Septimal minor 7th

24 21.5 cents, the difference between four Just 5ths - and two octaves and a Just 3rd - gives a chromatic diesis, or syntonic comma, of ratio 81:80, as a Just 5th [3/2] is 701.96 cents, and a Just 3rd [5/4] is 386.31 cents. It is also the diatonic comma.25 Holdrian comma, or Holder koması in Turkish. Holder’s comma (22.6 cents) is equal to one step of 53-et, or the 53√2, an irrational number that does not describe the compromise of intervals within a tuning system and approximates a syntonic comma (21.5 cents).

14

Diesis Lesser diesis 41.0588584

0549554760

128:125 Octave 3 major 3rds

Undecimal

comma

Undecimal

quarter-tone

53.2729432

3014412520

33:32 Undecimal tritone Perfect 4th

Greater

diesis

62.5651480

0221040120

648:625 4 minor 3rds Octave

Tridecimal

comma

Tridecimal

third-tone

65.337340

826851658

20

27:26 Tridecimal tritone Perfect 4th

19 tone equal temperament (19TET) naturally came about during the music theory of the

Renaissance. The ratio of four minor 3rds to an octave was almost 19th of an octave (648:625 or

62.565 cents), and goes back to the 16th century, used for example in Seigneur Dieu ta pitie

(1558) by Guillaume Costeley, thought to have been written for/in 19TET. In 19TET, due to the

powers of syntonic tuning, the perfect 5th rests at 694.737 cents: each division is a frequency ratio

of 21/19th or 63.16 cents. Some of the ratios in 19TET are closer to Just intonation than 12TET

(like 5/3 major 6th, and 5/4 major 3rd), and this is a good starting case in support of its use.2627

19TET is also a sensible equal temperament as it gives a purer major 3rd and minor 3rd (6/5), and

their inversions, major and minor 6ths, over 12TET - although it has a limited amount of

accessible pitches per octave. Tim Perkins (Tune Up, Antelope Engineering) describes 19TET as

harmonically usable.28 (Sethares, 1991) The 19TET step is 1200/19 or 63.16 cents, slightly more

than half a standard quarter-tone. 19TET can be extended into standard notation without too

much complication. Although the notes are written on the staff as C, C♯, D♭,D, D♯, E♭, E, E♯,

F, F♯, G♭,G, G♯, A♭, A, A♯, B♭, B, (B♯, C♭), the notated enharmonic equivalents are not the

same and each note in succession is 1/19th higher than the previous note.

During the 16th and 17th centuries a particularly dissonant form of a diminished 6th was used,

popularly arising out of the quarter-comma meantone temperament and spanning seven

semitones, called a wolf fifth (procrustean/imperfect 5th). The quarter-comma is a variant of

Pythagorean tuning in which its P5 is diminished by a ¼ of a syntonic comma as opposed to the

Pythagorean Just intonation of frequency ratio 3/2. The quarter-comma's purpose was to obtain26 There is an interesting 19ET from Woolhouse (1835) dividing the octave into 730 parts.27 All notes are within 8 cents of Just intonation on a major C triad in 19TET, as opposed to 14 cents for 12TET.

28 In 19TET there is a perfect minor 3rd. A septimal 3rd may also be produced. A major and minor scale, as well as whole tone, may be fairly well approximated, though slightly and noticeably out. The septimal minor 3rd is 2 2/3 semitones, Just interval 7:6. The septimal major 3rd is 4 ½ semitones, just interval 9:7. 15

Just intoned 3rds of ratio 5:4, and described by Pietro Aron in Toscanello de la Musica (1523) as

'sonorous and Just as united as possible'.29

Modern equal temperament was invented in the 1500’s,30 in order to accommodate increasingly

complex polyphonic music, and to increase the sense of harmony during modulation and key

change. The 12TET system breaks the octave into 12 equivalent parts, resulting in a semitone of

non-simple ratio – approximately the 12th root of 2 (12√2 or 21/12) or 1.059.31

Semitones Interval32 Just intonation Equal Temperament Difference0 Unison Consonant 1/1=1.000 20/12=1.000 0.0%1 Semitone Dissonant 16/15=1.067 21/12= 1.0594630943592953 0.7%2 Whole tone Dissonant 9/8=1.125 22/12=1.122462048309373 0.2%3 Minor 3rd Consonant 6/5=1.200 23/12=1.189207115002721 0.9%4 Major 3rd Consonant 5/4=1.250 24/12=1.2599210498948732 0.8%5 Perfect 4th Consonant 4/3=1.333 25/12=1.3348398541700344 0.1%6 Tritone Dissonant 7/5=1.400 26/12=1.4142135623730951 1.0%7 Perfect 5th Consonant 3/2=1.500 27/12=1.4998261905048882 0.1%8 Dim 6th Consonant 8/5=1.600 28/12=1.5874010519681994 0.8%9 Major 6th Consonant 5/3=1.667 29/12=1.683985480334983 0.9%10 Dim 7th Dissonant 9/5=1.800 210/12=1.7817974362806785 1.0%11 Major 7th Dissonant 15/8=1.875 211/12=1.8887492632848886 0.7%12 Octave Consonant 2/1=2.000 212/12=2.000 0.0%

Holder’s comma of 22.6415 cents, or 53√2 (Arabian Comma), was used widely in the 17th

century. Mercator’s comma of 55√2, or roughly 21.8182 cents, was close to the syntonic comma

of 21.5063 cents. Further, Mercator thought the 53√2 would be of use due to the fact that a cycle

of 53 Just 5ths approximated 31 octaves. 53√2 is closer to Just intonation.

Maqam rast,33 in Holdrian commas:

C D E F G A B C9 commas 8 commas 5 commas 9 commas 9 commas 8 commas 5 commas

29 Zarlino and de Salinas later described the theory more exactly.30 In full use by the 19th century.31 The table corresponds to Seeger’s early 20th century dissonant counterpoint, and the Just tuning systems of Pythagoras and Ptolemy, with dissonance increasing in larger ratios. The Just inverse ratios add to give an octave, for example 5/3 x 6/5 = 30/15 or 2. 32 The chart shows how the only perfect interval is the octave in equal temperament, and how the difference is spread out overall for transpositional functionality.33 The illustration is not using half flats or sharps and is approximate. Nihavend uses medium 2nds (somewhere between 8-9 commas). The medium 2nd or neutral second (n2) is larger than a minor 2nd and smaller than a major 2nd,Just interval = 11:10 or 165 cents (greater undecimal neutral 2nd ). The intermediate neutral 2nd ratio is 12:11 or 150.64 cents. The lesser undecimal neutral second is derived as the interval between the 11th and 12th harmonics (from the harmonic series), and the greater undecimal neutral 2nd is derived as the interval between the 10th and 11th harmonics.16

Maqam nihavand in Holdrian commas:

C DE♭ F G

A♭ B♭ C

9 commas 4 commas 9 commas 9 commas 4 commas 9 commas 9 commas

The 4th century saw the split of the Roman Western Empire and the Greek Eastern which later

became the Byzantine [Roman] Empire. The collapse of the Western Roman Empire in the 5 th

century (Christian takeover) was steady thereafter, due to the extent of Roman culture and art,

into the beginnings of Europe’s Renaissance.34

The first half of 16th century music theory witnessed Henry Glarean as the prominent musical

theorist. Glarean, author of the Book of the Twelve Modes and the Dodecachordon (1547),

proposed 12 modes, eight plus an additional four: Aeolian (modes 9 and 10) and Ionian (modes

11 and 12), and comments that Ionian was the main mode frequently used by composers during

this time.35 According to Ronald Turner-Smish and Mark Lindley, schismatic tuning was used

briefly in the late medieval period.36

Jamaica and Africa, Koromanti and Angola, Ethiopian bowl lyre (krar), Quadrille music

in Carriacou

At the end of the 1600s, in and around Jamaica, many African traditional musics used

microtones in much the same way as blues and rock guitarists accent notes - by bending the

string. Sir Hans Sloane observed slaves playing music in Jamaica and notated it in 1687. In the

‘Koromanti’ first two sections seven notes are used, and the third section eight: the extra note

was likely the result of the French musician Baptiste’s attempt to record microtones not

representable in standard European notations, which would have been somewhere between the34 Invasions following through from Late Antiquity through to the Middle Ages and the formation of new kingdoms in the Western Roman Empire began, whilst in the 7th century Northern Africa and the Middle East dissolved from the Byzantine Empire (Eastern Roman Empire) becoming part of an Islamic Empire, generally thought of as a pseudo-completion with antiquity. Migratory tonal systems are accountable.35 In Isogage in musicen (1516) Glarean addresses the basic elements of music, perhaps used for teaching. Dodecachordon comprises a massive body of work with over 120 compositions, music theory and philosophical andbiographical text. A chronology of modal use beginning with Boethius (16th Century) is discussed in plainsong and monophony ending with a study of modal use in polyphony. Later theorists like Zarlino accepted the twelve modes and although the difference between plagal and authentic is no longer of interest today, the six condensed modes remain.36 The schisma is the ratio of Pythagorean comma and a syntonic comma: 531441:524288/81:80 = 32805:32768, bearing in mind that the pythagorean comma is the distance of roughly a quarter-tone (between 75:74 and 74:73) and that eventually the syntonic ratio of 81:80 later used by Ptolemy raised or lowered the original pythagorean tonal system to produce just major and minor 3rds.17

standard semitones, falling between the keys of a piano. Modern musicologists think that the

mode Baptiste transcribed was a heptatonic scale with the 3rd and 7th partially flattened.37 (Rath,

1993) (Burton, 2012) In 20th century (and perhaps earlier) practice it is possible that European

harmony influenced blues and jazz with the idea of tonic, subdominant and dominant as triadic 1,

3, 5.38

African Jamaican music: Koromanti and Angola

Pitch-class Koromanti Angola (Upper) Angola (Lower) Both3rds 33 14 9 23Intervals 316 26 45 713rds/Intervals 0.10 0.54 0.20 0.32

Farther east, the Ethiopian bowl lyre (krar) is used for music that is highly chromatic with

microtonal embellishments and slides. Some krar tunings (Kignet) are fairly exotic like the

Anchihoy with strings 3, 4, 5 comprising a minor 3rd and nearly tone-and-a-half, and its use is as

an accompaniment to embellish vocal melodies [much like ancient Greek music]. (Kebede,

1977)

Quadrille music in Carriacou is similar to European quadrille dance music, with two sections

of eight bar phrases which are instrumental and in the major key. However, the last remaining

quadrille violinist in Carriacou, Canute Calliste, borrows from African microtonalism in which

some notes are slightly flatter or sharper than heard in European or North American fiddle

playing. (Miller, 2005) (Cultural Equity, N.D)

Contemporary microtonal practices across genres have been affected by the blues.

37 The Akan in Jamaica (from the Kwa speaking West African Gold Coast region to Cameroon, around Ghana) on the other hand had no common use of microtones and preferred notes from the natural harmonic series, yet microtones were in common use slightly south around the Angola region, perhaps not causing Baptiste to misrepresent in notation – use of heptatonics with slightly lowered 7th.38Another rare early American account of African music was made in the late 1700’s by De Bercy of nearly free slaves in Santo Domingo, though sadly the transcription lacked the accuracy of Baptiste’s. Lyrics are often an indicator of a music’s origins.18

Post-romantic and Pre-modernism, experimental, Carrillo, Ives, Rimsky-Korsakov,

Experimentalism, polytonality, tone clusters, aleatorics, quarter-tones, polyrhythmic

The late 1800s encompassed experimentalism, which later led to the expanded tonality of early

20th century works.39 Rimsky-Korsakov’s Oriental sounding Scheherazade may be considered

late Romantic, and a precursor to experimentalism.40 Ives,41 who experimented with quarter-tones,

and Korsakov, are a midquel between Romantic and later Expressionist (and microtonal and

tonal) practices.

Partch created a family of microtonal string, keyboard and percussion instruments tuned to his

Just 43-note scale. Instruments like this were built before in the Low Countries in the 17 th

century, a time when Huygens talked about use of a 31-note octave capable of diatonic scale

transposition in Just intonation.42 Partch extended Just tuning ratios into 7, 11 and 13 limits.

Partch's Daphne of the Dunes, for example, sounds like notes extend past the 12 notes we know,

yet all is beautifully harmonic and based on phi. Ben Johnston extended Just intonation further

(high prime limit) that contained hundreds of pitches per octave.

In 1895 Carrillo wrote quarter-tone string quartets, later using a 96 division system and created

a harp-zyther. Helmholtz wrote in 1863 in On the Sensations of Tone: ‘ the system of scales,

modes and harmonic tissues does not rest solely upon unalterable laws, but is at least partly also

the result of aesthetic principles, which have already changed, and will still further change…’

(Wood, 1986)

Quarter-tones began to be used in western music around the beginning of the 20th century

with Charles Ives: Alois Hába's first work for quarter-tones was Op.no. 9a: Fantasy in quarter-

tones for violin solo (1921) and Ivan Wyschnegradsky's first was Quatre fragments, for 2 pianos

39 A short list of 20th century microtonal composers include: La Monte Young, Alois Hába, Harry Partch, Walter Smetak, Easley Blackwood, Ivan Wyschnegradsky, Terry Riley, Wendy Carlos, Michael Harrison, Per Nørgård, Warren Burt, Giacinto Scelsi, Harry Partch, Ben Johnston, Syzygys, Chico Mello, Tony Conrad, Arnold Dreyblatt, Bent Sørensen, The First Vienna Vegetable Orchestra, Sei Miguel, Pascale Criton, Georg Friedrich, John Cage, James Tenney, Julián Carrillo, Ron George, Bosty, Piotr Kurek, Burkhard Stangl & Kai Fagaschinski, Blues for Spacegirl, Bertrand Denzler, Antoine Beuger, and Ivor Darreg.40 Korsakov jusxtaposed keys by a major third, as in C major and E major,with distinct and easily comprehensible rhythms and had an Eastern feel that was absent in late 19th century work.41 Ives’ 12TET Central Park in the Dark may be regarded as one of the first Experimentalist pieces, with the strings in 3rds, 4ths, and 5ths representing the park’s woods, and ragtime quotes from Hello My Baby and Washington Post March (Sousa) finally ending in tensions of cacophony, with similarities to Experimentalists of the time like Varèse,Ruggles, and Hovhaness The microtonalist Harrison, who studied under Schoenberg at a dance school in California where he worked, helped Ives to come to public attention, conducting the acclaimed Symphony No. 3.42 A 31-tone organ still rests in Haarlem at the Teyler Museum19

in quarter tones (2nd version), Op. 5 (1918). Prior to this it is doubtful if there was a developed

24-tone equal-tempered system with pairing of technology and notation.

Contemporary & modern microtonal practices, systems, stylistics: Modern (1890-1930),

20th century (1901-2000), Contemporary (1975-present), Modernism, Dadaism, serialism,

microtonality, Verèse, Webern, Wyschnegradsky, Hába, Carillo, Villa-Lobos, Ives, Partch,

Cowell

In 1912 Henrey Cowell used tone clusters in The Tides of Manaunaun. In 1913 Russolo wrote

The Art of Noises: Futurist Manifesto and in 1914 conducted intonarumori (noise instruments).

1916 saw Dadaism (anti-art) rise in Zurich with noise music and sound poetry at the Cabaret

Voltaire. Prior to tape slicing and analog and digital sequencing, repetition and form lay more in

the performance domain. This craft has been handed down to modern producers,

In 1917 Verèse suggested instruments that could ‘open up a whole new world of unexpected

sounds.’ Satie’s ballad Parade utilized typewriters, revolvers, sirens and ships’ whistles.

Webern, like Verèse, was not exposed early on to Eastern musics, yet both drew interesting

parallels – Webern’s tendency to clarify structures of motifs with variegated textures in high

definition of timbre, register, duration, articulation etc. is comparable to Asian musics, whereby

whole structures would seem static/erratic without motific definition, which derive

meaning/coherence from differing devices like timbral changes, vibratos, pitch inflections,

articulation. Coherence played a vital role in 20th century composition, as overarching structure

of the whole greater than (and related to) its constituents. At this time Villa-lobos was torn

between European classical and Brazilian folk.43

As neoclassicism and serialism began, a third movement soon sprang up: microtonalism.

Stravinsky and Bartók had exposure in their youth to Eastern and folk musics, and some of

which Stravinsky had assimilated was likely folk of Asian origin, whilst some may have come

from the orientalist Rimsky-Korsakov, who would have been exposed to the Asian music that

spilled over into Russian popular musics. In Les Noces’ opening, large intervals greater than a 2nd

are used with sliding attack typical of some singing styles in Asia.44

43 Villa-lobos’ Amazonas and Uirapurú were derived from ancient indigenous Brazilian folk material and legends.44 Bartók’s serious investigation of East-European folk included the Magyars of the Ural Mountains which contained, at the time, uncorrupted ancient musical elements. Bartók also studied Arab and Turkish music, influencing his compositional aesthetic as an ethnomusicologist – covering melody, harmony and rhythm and 20

Hába may well have marked the beginning of microtonalism in the 1920’s which was

followed by a die-down, with a resurgence in the 1960’s till present, many composers taking it

seriously, with multi-tempered compositions being a sign of 20th and 21st century style, ranging

from Wyschnegradsky45 to Carrillo, due largely in part to awareness of non-Western music,

mainly Arab, Indian and Chinese. Hába’s interest in quarter-tones was largely due to influence

from Slovakian folk music. Mildred Couper also began experimenting and composing at this

time, tuning a first piano a quarter-tone higher than a second resulting in176 pitches (from 88).46

Whilst Scriabin pondered new tonal systems, Ives and Couper wrote them down, and Hába and

Carillo had a large amount of microtonal work, yet Wyschnegradsky had an impressive output

and scope including theory, highlighted by 24 Preludes for two pianos tuned a quarter-tone apart.

He described his tonal system as having two divisional heptachords, separated by a semitone,

instead of the standard double tetrachordal division.47 With + and – taken as quarter-tone

adjustments, a basic scale comprises C, C#, D, D#, E, F, F+; G-, G+, A-, A+, B-, B+, C. (Burge,

1978) Here Wyschnegradsly’s deemed diatonized chromaticism is similar to Yasser’s supra-

diatonic system, although not in 19TET, and transpositions total 24.48 Easley Blackwood’s 16-

notes Andantino is certainly as subtle as any of Wyschnegradsky’s work, with rich microtonal

harmonic content and sweeping microtonal phrases that are not heard anywhere else, in nature or

most other musics, and are extremely sensible and exhilarating, enchanting and sophisticated.

In Finland, due to the Kalevala (distinct folklore set apart from Swedish and Russian

hegemony), folklore collectors of the 19th and early 20th centuries sought to record music which

they thought might be disappearing, due in part to publications such as Kansanmusiikki (Folk

instrumental idioms. Bartók did not however delve into microtonal inflection and stylistics.45 Wyschnegradsjy is extremely subtle in microtonalism, in, for example, Two Preludes.46 Couper also studied with Nadia Boulanger, and after experimenting with quarter-tone tuning, resluting in the ballet piece Xanadu (1930).47 Today the tetrachord may be taken to include either the 4 or #4 (traditionally, and for Wyschnegradsky, the 4 is implied).48 One writer describes Wyschnegradsky thus: ‘It reveals a singularly rich variety of mood and texture, this brought about by a balance between the etude or pattern-type piece and the contrasting tone poem. There are languorous dances and a scherzo, Bartokian motor rhythms, hints of fireflies and fireworks, and a haunting peasant song. One finds harsh two-voice counterpoint in bold octaves, a dirge-like passacaglia, and in no. 11, quasi campana, clangorous bell sounds in large clusters, notated as "a vertically striped half-moon" spanning the interval. Almost throughout, the pianos engage in melodic and harmonic hocket. Whenever possible, the composer has scrupulously marked dynamics and use of the pedals for each instrument.’ (Burge, David, 1978) Wyschnegradsky used third-tones (18-tet, 66.666 cents), sixth-tones (36-tet, 33.333 cents), and twelth-tones (72-tet, 16.666 cents). In Quarter-tone Piano Prelude #1 & #2 by Diesel Bodine (Scott Crothers) it is interesting to note that the harmonics and melody are embellished with microtones. It seems the microtones are not that harmonically or melodically functional, but peripheral embellishments, similar to Wyschnegradsky’s usage, although Wyschnegradsky’s microtonal use is very systematic and even, harmonically interconnected, and employs tonal clustering that is consolidated within overall structures.21

Music). Both lower and higher Finnish education systems take folk music seriously. Konsta

Jylha and his band, Kaus-tinen Purppuiipelimann, draw on ancient folk traditions while

incorporating new ingenuity to the practice, as in reinterpretations. Folk music in the higher

sector education has helped revive mass consumption and appreciation and development in the

Finnish arts, which stress teaching it in changing-world contexts.49 (Ramnarine, 1996)

In the U.S. Charles Ives went on to write Choral for Strings in Quarter-tone (1914) and Three

Quarter-tone Pieces for Two Pianos (1924) and Some Quarter-tone Impressions (1925). Ives

uses two pianos normally pitched with one tuned a quarter-tone down (or up) in the upbeat 3

Quarter-Tone Pieces, which works well over-all as the two seem in parallel and phase

interweaving at moments into a seeming fusion.50 (Ives, 1924)

In Prague around this time Czech composer Alois Hába was also working on quarter-tone

pieces, utilizing two keyboards with one tuned a quarter-tone higher. Hába produced many

microtonal compositions with quarter-tones and sixth-tones. A septimal sixth-tone is 34.98 cents

(50:49). It is the difference between 7:5 (lesser septimal tritone) and 10:7 (greater septimal

tritone, inversion of the lesser tritone). The sixth-tone is tempered out of 12TET, 24TET, and

22TET, but fits in to 19TET, 31TET or odd octave divisions. Partch, on the other hand, devised

‘monophony’ with an octave split into 43 unequal parts. He writes in Genesis of a Music (1949)

that all tonalities stem or expand from unity or 1/1, and that modulations to non-dominant and

non-common scale degrees are possible; and that it is ‘not capable of parallel transpositions of

intricate musical structures’; and that it is not tone specific – conversely capable however of

ordinary and extra-ordinary unheard of modulations resulting in expanded tonality.

In The Complete John Cage Edition – Vol. 27: The Works for Violin 5, there is precision

microtonality, and the chorals are derivative of Satie’s Douze petits chorals and Socrate. For

One, the first note F is drawn out at length, followed by a short pause and then another F, and

this keeps happening with introduction of new notes. The effect is hypnotic as one loses a sense

of pitch-relation. Performed by Irvine Arditti, it works through Zukofsky’s idea ‘to make a

49 Researcher Anneli Könt gave classes of Estonian folk songs where one song, Sinimani seele, had a melody range of a tone, whereby a lead singer calls and chorus answers. The lead line may change by microtone or intervals greater than a 5th, while the chorus reply of contemporary folk students adjusted each time to the change. (Ramnarine, 1996)50 George Ives’ son Charles recalls his father’s construction of his ‘Quarter-tone Machine’ consisting of 24 violin strings: ‘One afternoon, in a pouring thunderstorm, we saw him standing without hat or coat in the back garden; the church bell next door was ringing. He would rush into the house to the piano, and then back again. ‘I’ve heard a chord I’ve never heard before – it comes over and over but I can’t seem to catch it.’ He stayed up most of the night trying to find it on the piano. It was soon after this that he started his quarter-tone machine.’22

continuous music of disparate elements, single tones, unisons, and beatings’.51 (Haskins, 1990)

(Dervan, 2003)

Yasser, infra-diatonicism, supra-diatonicism, evolving tonality

Joseph Yasser deems a basic 5-note structure as a structural basis for a denoted 7-note diatonic

set, and the remaining two notes have secondary functional auxiliary filling. This is deemed the 5

+ 2 complex and Yasser terms it infra-diatonic. In the Chinese heptatonic system (7TET) the two

parentheses notes are termed pien-tones (‘becoming’): F G A (B) C D (E) f. Mododic works

from the Song dynasty most commonly contained modes on G(shang), D(yü), and somewhat

F(kung). This may have influenced the early Japanese ryō system in which the prevalent modes

were on G (Ichikotsu-chō = shang) and D (Ōshiki-chō = yü). In the later Togaku court pien-tones

were modified thus: ryō = G A B (C) D E (F) g (derivative of shang) and ritsu = D E (F) G

A B (C) d (derivative of yü). Alternating the pien-tones from E-B and F-C produces a major-

minor shift.52 (Gauldin, 1983) Within the first 10-note set of the harmonic series is 1, 2, 3, 5, ♭7

and a lydian ♭7 diatonic scale in the first 13 notes [1, 2, 3, #4, 5, 6, ♭7], after which

microtonality becomes increasingly greater. Just intonation is the older way of viewing [and

teaching] the harmonic series. Yasser views a 5+2 (infradiatonic) [pentatonic 5 + 2] as a

precedent for a 7+5 [diatonic 7 + chromatic 5] tonality, that will one day be followed by a Just

expanded tonality, or supradiatony (Yasser, 1932), perhaps like Partch’s 43-note Just scale,

based on ratios, limits, and tonality diamonds. Perhaps a good instrument to begin this tuning on

would be a harp or zither, although transposition would be non-movable as opposed to voice or

fretless strings, or trombone.53 For the Paris Conservatoire it became dogma that all major or

minor dominant ninth chords were ‘natural’, whilst others were ‘artificial’. This is in line with

51It has been suggested that 432hz tuning would be a close and more natural and harmonious choice, as dividing by 3(resulting in 5ths, that string instruments tune in) won’t give numbers that recur, creating dissonant beating., which is the case with A440hz, A442, and A443. Although this only occurs on the open strings. This theory works becauseit is arbitrarily in base 10.52 Further, Hexatonics, and tetratonics, are two frameworks that are very much overlooked. Nonatonics (9), decatonics (10), undecatonics (11), dodecatonics (12), triskaidecatonics (13), tetradecatonics (14), pentadecatonics (15), hexadecatonics (16), heptadecatonics (17), octadecatonics (18), would be part of either extended or upper-structured scales or part of other temperaments such as 19TET.53 Partch’s instruments for 43-just include the zymo-xyl (uses blocks of wood, much like a xylophone), diamond marimba, and others. Partch’s concepts include expanded Pythagorian Just limit tuning ratios and otonality and utonality.23

dissonant counterpoint’s view that dominants drive forward composition in architectural space.

The fundamental is the first harmonic of which other harmonics are said to be partials. The

human brain perceives higher harmonics as being closer together than lower harmonics, closer to

the fundamental, creating a perceived stretching effect that may account for octave perception

discrepancy. Frequencies in the harmonic series are whole number ratios [of the fundamental]

and directly related to Just intonation. If harmonics are present in a note which constitutes a

harmonic series of any frequency, the human brain perceives the overall note as the fundamental,

even if not present. These combinations of partials or harmonics of the fundamental are

perceived as timbre or colour. Strong high overtones in cymbals often mask their fundamental.

David Cope (1997) forwards the idea of intervallic strength, where consonance results from

lower harmonics in the [harmonic] series and dissonance from higher harmonics in the series.54

Shenker linear progression5 of melody over harmony cannot progress without a passing note

from a sequence within the harmonic series, for example 3, 2, 1 over a

54 In practice this may be subjective to what we’re used to, and very high ratios may approximate small (consonant) ratios.5 The Schenkerian graph may straitjacket work, effectively compounding problems further. This makes it less than welcome in ethnomusicology, and although some music anthropologists have never learned to read notation, understanding a Schenkerian graph requires a high degree of musical literacy and discipline in musicology24

25

Notable small (Just) ratios [truncated] along the harmonic series up to limit 15 and mirrored 2:1

symmetry (Yasser, 1932):

Raito Interval cents Centitones Mirror Mirror in centsTonic 1 0 0 2/1 120012√2 ♭2 100 50 7th 1100

16/15 ♭2 111.73128526978 56 15/8 or 7th 1,088.26871473022000000

010/9 w2 182.40371213405998000

0

91 9/5 or ♭7th 1,017.59628786594002000

012√22 2 200 100 12√210 or ♭7th 1000

9/8 2 203.91000173077483500

0

102 16/9 ♭7th 996.089998269230000000

8/7 w2 231.17409353087507100

0 [1/7, 6th harmonic]

115 7 / 4 o r ♭7th

blue

968.825906469124929000

7/6 w♭3 266.87090560373751100

0 [1/6, 5th harmonic]

134 12/7 7 933.129094396262489000

12√23 ♭3 300 150 12√29 or 6th 900

6/5 ♭3 315.64128700055260000

0 [1/5, 4th harmonic]

158 5/3 or 6th 884.358712999447400000

(11/9) 3 347.40794063398187200

0

174 18/11 6 852.592059366018128000

27/22 3 354.54706023140554600

0 (Wusta-Zalzal)

177 44/27 6 845.452939768594454000

5/4 3 386.31371386483481700

0 [1/4, 3rd harmonic]

193 8/5 ♭6 813.686286135165183000

12√24 3 400 200 12√28 ♭6 800

9/7 3 435.08409526164990700 217 14/9 764.9159047383500930004/3 4 498.04499913461258200

0 [1/3, 2nd harmonic]

249 3/2 or 5th 701.955000865387418000

12√25 4 500 250 12√27 or 5th 70015/11 4 536.95077236546553200

0

268 22/15 w5 663.049227634534468000

11/8 4 551.31794236475670700

0

276 16/11 5 648.682057635243293000

7/5 w#4 582.51219260429011100

0

292 10/7 or #4th 617.487807395709889000

12 26 #4 600 300 #4 60010/7 #4 617.48780739570988700

0

308 7/5 or w#4th 582.512192604290113000

26

13/9 w#4 636.61766003853575200

0

319 18/3 or ‡4th 563.382339961464248000

12√27 5 700 350 12√25 or 4th 5003/2 5 701.95500086538741800

0 [1/2, 1st harmonic]

351 4/3 or 4th 498.044999134612582000

11/7 w♭6 782.49203589563178000

0

391 14/11 or 3rd 417.507964104368220000

12√28 ♭6 800 400 12√24 or 3rd 400

8/5 ♭6 813.68628613516518300

0

407 5/4 or 3rd 386.313713864834817000

13/8 6 840.52766176931059200

0

421 16/13 or 3rd 359.472338230689408000

5/3 6 884.35871299944739900

0

442 6/5 or ♭3rd 315.641287000552597000

12√29 6 900 450 12√23 or ♭3rd 300

12/7 6 933.12909439626249300

0

466 7/6 or w♭3rd 266.870905603737507000

7/4 ♭7

blue

968.82590646912492900

0

485 8/7 or w2 231.174093530875071000

12√210 ♭7 1000 500 12√22 or 2nd 200

9/5 ♭7 1,017.596287865940020

000

509 10/9 or w2nd 182.403712134059980000

13/7 7 1,071.701755300185660

000

536 14/13 or ♭2nd 128.298244699814340000

15/8 7 1,088.268714730222240

000

554 16/15 or ♭2nd 111.731285269777760000

12√211 7 1100 550 12√2 or ♭2nd 100

2/1 8ve 1200 600 2/2 0

27

1

0

2

1

3/2

1/2

4/3 /3

1/3 2/3

1/4

5/4 6/4

2/4

7/4

3/4

1/5

6/5

2/5

7/5

3/5

8/5

4/5

9/5

1/6

7/6

2/6

8/6

3/6

9/6

4/6

10/6

5/6

11/6

1/7

8/7

2/7

9/7

3/7

10/7

4/7

11/7

5/7

12/7

6/7

13/7

From full string board to within the octave fundamental

4

6

3 7

3 4 6 7

3 498 884 7

b

b B b b

2

v

7v3V

4V

6V

435 617 782

b6v

933 1072

5th

4th5

6th

3rd

b3rd B4 b6 7

3

498

4th

884

6th

1049

7

231

2 3rd B4 b6 6th 7267

b7th

28

29

cadence. The lydian ♭7 mode and the dominant 9 (#11) are very low in the harmonic series, and

consonant. The #11 is the sixth harmonic (lydian chromaticism of George Russell), consonantly

low in the harmonic series, corresponding with the ability to produce a pentatonic and heptatonic

scale naturally, working upward sequentially in fifths, starting in a lydian mode. The 14 th

harmonic produces the natural 7th, and the flat 3rd occurs at the 17th or 18th (due to the curve)

harmonic above the fundamental – enabling the dorian mode.

The last figure on p. 27 shows how the harmonic series may represent where ratios fall in

terms of the two primary tetrachords in the octave, although skewed from their actual position.55

The fundamental (first harmonic) is designated 1f; the second harmonic (first overtone) is 2f (an

octave), and includes the set root and 5th; the fourth harmonic is 3f (two octaves) and includes the

set 1, 3, 5, ♭7; the eighth harmonic is 4f (three octaves) and includes the set 1, 2, 3, #4, 5, ♭6, ♭7,

and 7. Thus, each time the fundamental frequency repeats [in multiples 2, 3, 4, 5, etc.] an even

set occurs which is doubled in number from the last. The #4 or ♭5 pivotal tetrachord point is

precisely at 12√26. This mirror technique of Just ratios could be used in music in the future. 2:1

Symmetry and reflection of a dorian 1, 2, ♭3, 4, 5, 6, ♭7 or 2/2, 9/8, 6/5, 4/3, 3/2, 5/3, 9/5, 2/1

would be 2/1, 10/9, 6/5, 4/3, 3/2, 5/3,16/9, 2/2 or 1, w2, ♭3, 4, 5,6, ♭7 where the more dissonant

larger ratios near the octave bounds begin to swap (invert) more microtonally. Note that in the

table above, the wolf 2nd (w) (231.174 cents) is identical to 1/7 in the harmonic series (the 6th

harmonic). This is true for 3/2 (P5th), which is 1.5, and ½, which is 0.5. To convert the

harmonics to cents a one is added before using log2(1200).

Kirnbergers’s well-tempered scale is the same as Just intonation with exception of the 2nd, a

major whole tone, out by -10.061 cents, 5th out by –5.292 cents & Major 6th out by +5.291 cents.

The pentatonic, or infra-diatonic mode (infra-diatonicism), is filled in to achieve a partial [such

as a hexatonic dorian (no 6) mode] or fully diatonically expanded modern mode. However,

tones, modes and intervals change with the system of tonality. There are essentially two ways of

looking at expanded supra-diatonic modes: we can wait for a new system and notational

55 For example, touching a string halfway is ½, producing an octave (first harmonic), yet 2/2 + ½ = 3/2, showing the 5th at halfway between root 1/0 and octave 2/1. Further, touching the string at 1/3 or 2/3 will produce a 5th, yet also 3/3 + 1/3 = 4/3, a 4th, and 3/3 + 2/3 = 5/3, or 6th. The skew is not represented in the diagram, as 3/2 should not be at ½ for example, and thus this diagram is for comparison purposes only, as the upper 5-8 tetrachord is a smaller yet relative image of the lower 1-#4 tetrachord. Looking at the tetrachords, among other divisors as well, is good for mirroring and comparing/contrasting amongst other geometrical and syntactic issues within musical language. The stretching phenomenon between the lower and higher tetrachord is exemplified in this skewing effect.

30

semantics and semiotics to occur, along with the building of instruments, or we can add to the

20th century techniques of microtonal symbols, viz quarter-tones, eighth-tones etc., thus

mimicking the effect of diatonicism filled in from a pentatonic core of the past. Hence diatonics

[and chromatics] would be the base for supra-modalities, and microtones will fill in the gaps.

Lastly, for a fundamental phase x,56 when a complete phase is halved [2x], the first overtone or

partial is sounded. This continues on: for 3x, a third of the original phase [produces the third

overtone], 4x, a fourth overtone, and so on. This is the harmonic series. The series can be heard

on the guqin, an ancient fretless 7-stringed zither.57 (Henryshoots, 2010)

Yasser asserts that just as in Faux-bourdon of the 1200’s, where composers struggled to break

away from infra-diatonicism (pentatonic) and infra-atonality, hypothetically taking the root

pentachord [C, D, F, G, A] combined with the 5th pentachord [G, A, C, D, E] and/or 2nd

pentachord [D, E, G, A, B] to form diatonicism [C, D, E, F, G, A, B] (or hexatonics) - yet

without any triadic harmonic concepts, and yet employing altered triadic inversions - so too do

modern composers helplessly try to break from atonality and 12-tone chromatics and diatonics.

Yasser thinks that expanded tonality (supradiatonisism) in the future will require the same

functionality as equal-temperament, and thus deems a logical derivative system like 19TET

should be adopted, studied and taught, in order to see the full rewards of future endeavors,

symphonies, and progressive works.

56 From full stringboard to within the octave: A good visual aid to conceptualize the harmonic series (0-1) is to convert it into a double tetrachord template (1-2). The root fundamental is one single phase. So, for the second harmonic 1/2(x) [of a fundamental frequency in the series], creating two phases, all we need to do is place a 1 before½ to view the ratio precisely between 1 and 2, thus 2/2+½ = 3/2 = 1.5. From there a conversion to cents is straightforward as log21.5(1200)=700 cents or P5. The second harmonic would be two nodes at 1/3(x) and 2/3(x), creating three phases: 1/3(x) is 3/3+1/3 = 4/3, thus log24/3(1200)=498 cents or P4. 2/3(x) is 3/3+2/3 = 5/ 3, thus log25/3(1200)=884 cents or major 6th. The P4 and major 6th fall exactly on each side of the 700 cent halfway point of a P5, and this process continues on up the harmonics in the series and can be practicably translated in this fashion. One may wonder why 700 cents is half-way along 1200 cents, when 600 cents, the symmetry pivotal point, or #4, would be the logical choice. 700, or 3/2, marks the start of the second tetrachord and is the second occurring overtone (third harmonic) in the harmonic series. This illusion is due to the fact that relative distance and wave length becomes shorter as pitch gets higher. Inharmonicity varies between instruments, and even thickness of strings, occurring progressively more, higher up the [harmonic] series, and generally overshooting the theoretical notes. (Inharmonicity - sound due to natural laws is not fully compatible, only indicative, of pure mathematical, physical, and geometric concepts. In 2:1 scales, a point of interest is that the phi ratio falls at 1200/1.618033 or 741.641239 cents, which is 9 cents short of the quarter-tone between the 5th and ♭6 (in the key of C this would be

G‡, and in the key of E♭ a B ). The 833 cents scale is also attributed to phi.)57 Called ‘the instrument of the sages’.31

Darmstadt, neotonality, dodecophony, Stockhausen, Boulez

Darmstadt’s shadow created by Stockhausen & Boulez dissipated by 1984, yet is still

stylistically diverse. (Dominick, 1984) At Darmstadt in 1984 Halbreich lectured that direction is

essential, as too is tension and harmony, and that stasis and colour are contained in modality: yet

stasis occurs in dodecaphony as the human ear cannot make sense of tension and resolution and

direction at complex levels. In microtonal composition and practice this is a prime consideration.

Halbreich also postulated a ‘neotonality’ where spectral harmony extends to the idea of a richer

complexity of harmony considered consonant at higher levels. Classical hallmarks may be

considered to differentiate past and present minimalist Western practices: 1. A strong

tension/relaxation technique (expectancy, fulfillment), 2. Minimalist motifs are functionally

triadic based melodies in question-answer format and similar to classical technique, 3.

‘Periodicity’, 4. Diatonic triad based, 5. Simplicity, 6. Ostinato bass motif recurrence, much like

Baroque driven pulses, and aural pleasure derived unfettered by emotionalism. Banquart lectured

that too many pitches is an overload and only works with ‘defective’ tone rows. (Dominick,

1984)

Stockhausen’s ideas incorporated transition and transformation not only of musical languages,

rhythms, time signatures and pitches, but extended to transition of process that can expand and

contract, moving non-linearly.58 Pitch, rhythm and time, and timbre are illusively separate. The

fundamental pitch that produces harmonics/overtones is not needed for humans to perceive it, as

long as some notes of the harmonic series are contained within it – timbre and characteristics of

any physical sound phenomena are simply sets of partials or harmonics.

58 Stockhausen states that at one point he tried to contract a national anthem into the pitch-space of a major third –

dividing the pitches into microtonal equivalents.

32

22TET, A Just 12 tone-scale built on powers of 3 and 5, diminished 7th blue note, 1960s Rio

de Janiero Jazz, Bossa Nova, US jazz, flattened 5th and hexatonics in the Blues, New

Orleans resurgence, Copacabana

22TET divides the octave into equivalent ratio parts of 22, or the twenty-second root of 2, 22√2,

or 54.55 cents. It is thought to have come from theorist RHM Bosanquet, and inspired by the

music theory of India, had noted how compatible it was with 5-limit tuning (Just intonation).

The small ratios that form harmonic intervals involving prime numbers 2, 3, and 5 are

considered 5-limit intonation. The following chart for Just intonation shows the primes used in

all but the 2nd and 7th dissonant intervals.

Note C D E F G A B CRatio 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1Decimal 1 1.125 1.25 1.3333 1.5 1.6666 1.875 2Cents 0 204 386 498 702 884 1088 1200Name T T S T T T SRatio 9/8 10/9 16/15 9/8 10/9 9/8 16/15Cents 204 182 112 204 182 204 112

16:15 S semitone 1.0666610:9 T minor tone 1.111119:8 major tone 1.125

Which combine to make-up

6:5 Ts minor third 1.25:4 Tt major third 1.254:3 Tts perfect fourth 1.33333r3:2 TTts perfect fifth 1.52:1 TTTttss Octave 2

Note A B C D E F G ARatio 1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1Cents 0 204 316 498 702 814 1018 1200Name T S T T s T TCents 204 112 182 204 112 204 182

A Just 12 tone scale built on powers of 3 and 5 (i.e. 1/9 = 3−2)

Factor 1/9 1/3 1 3 9

33

5

Note

ratio

cents

D−

10/9

182

A

5/3

884

E

5/4

386

B

15/8

1088

F♯+

45/32

590

1

Note

ratio

cents

B♭−

16/9

996

F

4/3

498

C

1

0

G

3/2

702

D

9/8

204

1/5

Note

ratio

cents

G♭−

64/45

610

D♭−

16/15

112

A♭

8/5

814

E♭

6/5

316

B♭

9/5

1018

The 7/4 (factor 1.75) interval (968.826 cents), or septimal minor seventh or harmonic 7th, is 31

cents lower than its equal tempered counterpart. It is linked with blue notes in jazz, and has been

a contentious issue throughout music history. In context it is slightly ‘sweeter’ then a

conventional diminished 7th (or minor 7th in jazz). It is derived from the harmonic series, the

interval between the 7th harmonic and 4th harmonic. Most often in horns it is corrected to 16:9

Just Pythagorean, yet the pure diminished 7th harmonic was used in Serenade for tenor, horn and

strings, by Britten.

The late 1950s and early 1960s Rio de Janiero Jazz scene had a deep Blues influence (Delta

blues, North Mississippi Hill Country Blues) during the Bossa Nova explosion. US jazz

musicians caught on to bossa nova and although seen as whitened samba, the Brazilian Jazz

musicians viewed it as exciting new territory. Popularized by Luiz Gonzaga in the 1940s the

baiāo is the most similar Brazilian music to the blues, complete with microtonal shading,

flattened 5th and string bending - although the major scale is prevalent over the blues hexatonic

scale C-E♭-F-G♭-G-B♭-C, which was not in Brazilian genres prior to bossa nova, and continued

unchanged throughout the 1960s New Orleans resurgence and innovation as well as in

Copacabana. (McCann, 2007)

In the Blues any inflection microtonally upon any of the 12 chromatic notes is used in

composition, and it is the aesthetic style, feel, attack and gesture which makes a composition

unique according to B.B. King. In his book Blues Guitar Method music making is compared to

singing, in that one must take time with the notes and that every note should mean something. A

distinctive player may be known for his distinctive characteristics or style of bending into certain

notes or use of vibrato. This idea of ‘musemes’ sets particular players apart. Jeff Titon addresses

34

the question of blue notes and concludes from early recordings of downhome Blues that ‘pitch

complexes’ are used - these quarter-tones are used consistently from line to line, stanza to stanza.

(Weisethaunet, 2001)

Pitch and cognitive acculturation, development of musical thought and thought in sound,

schematic and veridical expectancy, mistuning perception

With regard to microtonal past and present practices it is important to mention the harmonic

function of notational systems, time representation, and microtonal function. As music is like a

language, with tonal systems and microtonal inflections that can impart meaning (semiotics,

semantics and context),59 its artifacts are important in cultural, traditional and practical aspects of

music making, thinking, and expression. Musical thought may include timbral information as

well as pitch, duration and ornamental embellishment which may be linked to socio-cultural

heritability, where music and other types of passed knowledge are linked and may involve

microtonal information. For example with the Xavante of Brazil a tradition of ceremonial

wailing, called microtonal rising, is practiced by senior age groups during grief, though not by

youths. (Graham, 1994)

Birdsong is microtonal - birdsong pitch and timbre variety are remarkably complex, as are their

structures.60

The phi ratio 833 cents scale (Heinz Bohlen) is based on the golden section, or Fibonacci

sequence. The convergence of any interval and its closest combination tone approximate the phi

ratio (833 cents). The scale has 12 steps of .8333 and is close to 36TET.

Interval Base Closest Combination Ratio Tone Cents2:1 3:2 701.9550008653874180003:2 5:3 884.3587129994474030005:3 8:5 813.6862861351651830008:5 13:8 840.52766176931059200013:8 21:13 830.25324556520174900021:13 34.21 834.174502165894946000

59 Contextual, geometrical and mathematical.60 Ornithological writers like Thorpe, Armstrong, and Hartshorne often compellingly viewed birdsong as a form of music. (Preston, 2004) Things to consider from sonographs are structure, dynamics, timbre, and rhythm. For the Oriole, most of the pitch takes place between 3 and 9 khz. (Oehlkers, 2009) Many rhythms in nature are hypnotic and microtonal, from cricket noises to the sound of translated cymatics from the cosmos, stars and planets - signals shifted into the audio domain, as well as the sound from the microcosm – the natural world contains microtonality.35

34:21 55:34 832.67624672918423300055:34 89:55 833.24846093008577900089:55 144:89 833.029884571097529000144:89 233:144 833.113371854361454000233:144 377:233 833.081482337260849000377.233 610:377 833.093663017901213000

In human audition Just intonation is the easiest on the human ear and it avoids ‘beating’,

whereby vibrations are in interference.

There is another problem to consider when examining any patterns that may emerge from past

musical practices and their tonal systems,61 usually with some degree of microtonal implications,

and that is this: to what degree is internal musical thinking influenced by real-world experience

before it is burned into the mind and ready to use imaginatively?

In a study by scientists at Beth Israel Deaconess Medical Centre and Harvard Medical School

findings showed that after testing subjects to pitches and asking if the last or second to last were

the same, the supramarginal gyrus and dorsolateral cerebellum were ‘significantly correlated

with good task performance.’ The SMG and dorsolateral cerebellum could play a critically

responsible role in storage of short-term pitch [information] and unfolding pitch discernment in

pitch memory tasks. (Gaab, Gaser, Zaehle, Jancke, Schlaug, 2003) This at least is a start to

understanding the nature of memory and microtoal pitch classes.62

In another study, mistunings by Western listeners were swayed by past acculturation and

musical sophistication. Whilst non-musicians showed a different threshold for mistunings for the

culturally-familiar and culturally-unfamiliar, musicians’ thresholds across Western and Javanese

did not differ, suggesting that musical skills can be applied.63 The Bohlen-Pierce tritave 3:1 ET

scale was studied on trained and untrained musicians as well. (Pierce and Mathews, 1987)

These studies are important factors in determining true understanding of pitch relation, and

further microtonal pitch relation in past and current practices.

Arab and Western listeners have had responses recorded to improvised modal music (taqsim) –

heptatonic Arabic (maqam) systems of 24 quarter steps (50 cents) to the octave. Intervals in the

61 In order to practice music, much like language, a system and practice needs to be in place, or devised.62 Volume (in itself a paradoxical term) did not seem to correlate: a study for pitch versus loudness (Clement, Demany, Semal, 1999) suggested that pitch and loudness were processed in separate ‘modules of auditory memory.’63 To put interval and modal acculturation into further perspective, Lynch and Eilers (Lynch, Eilers, 1991) tested 6-month-old and 1-year-old Western infants using an operant-head-turn procedure. In a melody, the infants detected randomly placed mistunings in the Western major, Western augmented, or Javanese pelog, recording a performance pattern similar to adults. The older 1-year-olds performed better in the Western major over the Western augmented and Javanese pelog. 6-month-olds did better in the major and augmented over the pelog. The conclusion is that culturally specific perception and reorganizing of musical tuning starts to affect perception between six and 12 months. This is concordant with studies that indicate reorganization of speech takes place by the end the first year. This is also interesting in light of the Chinese lingua-tonal-inflections to elevated incidence of absolute pitch.36

scales are usually 2, 3, 4, or 6 quarter steps, 6 being quite rare. Participants were asked to

identify elements, segments, and use verbal descriptions and performed reductions (generative

simplifications). Common to Arab practice is detection of emblematic melodic figures, and

differences in segmentation identification were found between European and Arabic participants.

Both registered pauses and register changes, whilst the Arabs noted segmentation of modal

changes (subtle) that went unnoticed to the Europeans. The segments show that Arabic modes go

beyond a tuning system incorporating essential rhythmic and melodic configurations signifying

the maqam. (Ayari, McAdams, 2003)

Experimental studies in the last few decades have investigated expectancy in encoding,

organizing and reacting to melodic content and tones. Meyer postulated that a piece of music in a

given genre will evoke and generate expectancies – the violation of these expectancies is

significant emotionally. The results showed that these musical expectancies are molded by

rhythmic patterns, tonal and harmonic structures as well as melodic structures. (Meyer, 1956)

This exemplifies why it can take time for artwork to become socially validated.

This begs the question why, to an extent, a creation out of any cultural context may not be

deemed valid to begin with, as social meaning is ingrained in the repetitions of life-long

decoding of cultural 1) tuning/tonality systems 2) tonal-melodic-harmonic relation and 3)

language/dialectic reinforcement. The lay-musician or casual listener identifies these patterns

too, although perhaps to a lesser extent, and certainly this forms a large basis of understanding

even for the professional musician in practice.

Barucha furthers a distinction of schematic and veridical expectancy. Schematic is automatic

expectancy generic from one’s culture, veridical musical expectancy hinges upon one’s

cumulative musical experience. Barucha and Todd noted that listeners would often remain

surprised by sequences of music already very familiar to them – knowledge of outcome did not

seem to affect re-experience. (Ram, Moorman, 1999)

37

Just, Bohlen-Pierce scale, Wusta-Zalzal, Masonic ratios, 22 tone system of India, Ragas,

Messiaen, Babbitt, Cage, Young, French Spectralists, 53TET, 19TET, Bagpipe tuning

The Bohlen-Pierce scale uses the 3:1 ratio (tritave, or octave + fifth) instead of 2:1, with 146.3

cents per step in the equal tempered (non-Just) temperament. From a 2:1 ratio perspective this

scale is in 8.202087TET, and avoids octaves.

ste

p

Interv

al

Cents Fundamental Just

0 30/13 0√3 0 1 1/1 = 11 31/13 13√3 146.3038434999154360 1.088182 27/25 = 1.082 32/13 6.5√3 292.6084616715978560 1.184140594988857 25/21 = 1.1904761904761904803 33/13 13/3√3 438.9126925073971200 1.2885607692309613 9/7 = 1.2857142857142857104 34/13 13/4√3 585.2169233431959990 1.4021889487005645 7/5 = 1.45 35/13 13/5√3 731.5211541789951190 1.5258371159564499 75/49 = 1.5306122448979591806 36/13 13/6√3 877.8253850147942380 1.6603888560010867 5/3 = 1.66666666666666667 37/13 13/7√3 1,024.129615850593190 1.8068056703447524 9/5 = 1.88 38/13 13/8√3 1,170.433846686392260 1.9661338478579946 49/25 = 1.969 39/13 13/9√3 1,316.738077522191110 2.1395119415112758 15/7 = 2.14285714285714286010 310/13 13/10√3 1,463.042308357990210 2.3281789044302967 7/3 = 2.3333333333333333311 311/13 13/11√3 1,609.346539193789480 2.5334829434069275 63/25 = 2.5212 312/13 13/12√3 1,755.650770029588340 2.7568911531325972 25/9 = 2.777777777777777713 313/13 13/13√3 1,901.955000865387420 3 2/1

Just and notables table:

Interv

al

Ratio

for

Just

Cents

For Just

Ratio

fundamen

tal

Just

Cents 12-

TET to

Just

Pythag

orean

Pythagore

an

fundamen

tal

C e n t s f o r

Pythagorean

Notables

Uniso

n

1/1 0 1.0000 0 0 1/1 1.0000 0

Min

2nd

25/24

or

16/15

(limit

5)

70.67242 or

111.731285

26

1.0416666

6 7 o r

1.0666666

7

100 +11.73 256/24

3 o r

2187/2

048

1.0534979

4 2 3 9 o r

1.0678710

9375

Diatonic

semi tone /

Limma

=90.2249956

7 8 2 7 o r

113.6850060

5771

Chromatic semitone =

113.685

Maj

2nd

9/8

(limit

3)

203.910001

73

1.125000 200 -3.91 9/8 Just 203.9100017

3

8/7 or 7/6 (limit 7)

Min

3rd

6/5

(limit

5)

315.641287

00

1.2000 300 -15.64 32/27 1.1851851

85

294.1349974

0384

7/6 septimal min 3rd

OR 266.87090560374,

Wusta-Zalzal = 27/22

@ 354 cents, 16/13 in

limit 13

38

Maj

3rd

5/4

(limit

5)

386.313713

86

1.2500000

0

400 +13.68

628614

81/64 1.265625 407.8200034

6155

& 9/7 septimal maj

3rd , 14/11 in limit 11,

9/7 in limit 7P 4 4/3

(limit

3)

498.044999

13

1.3333333

3

500 +1.955

00087

4/3 Just 498.0449991

3

11/8 in limit 11 or

551.31794236476

centsTriton

e/dim

5

45/32

or 7/5

(limit

7)

590.223715

5 9 o r

582.512192

60429

1.4062500

0 or 1.4

600 +9.776

28441

or

+17.49

729/51

2

1.4238281

25

611.7300051

9232

25/18 asymmetric Just

a n d 7 / 5 & 1 0 / 7

Septimal tritones in

l i m i t 7 , 1 0 / 7 =

617.48780739398

centsP 5 3/2

(limit

3)

701.955000

86

1.5 700 -

1.9550

0086

3/2 Just 701.9650008

6

Wolf 5th = 678.49,

16/11 in limit 11 =

648.68205763524

centsMin 6 8/5

(limit

5)

813.686286

13

1.60000 800 -

13.686

28613

128/81

or

6561/4

096

1.5802469

1 3 5 8 o r

1.6018066

4063

792.1799965

3 8 1 8 o r

815.6400069

285

13/8 tridecimal 6th in

limit 13, 14/9 in limit 7

= 764.91590473835

cents, 11/7 in limit 11 Maj 6 5/3

(limit

5)

884.358712

99

1.66667 900 +15.64

128701

27/16 1.6875 905.8650025

9616

18/11 undecimal 6th, or

852.59205936602

centsMin 7 9/5 or

16/9

(limit

3)

1017.59628

786594 or

996.089998

26923

1.80000

or

1.7777777

78

1000 17.596

2878 or

+3.910

00173

16/9 1.7777777

77777777

996.0899982

6923

7/4 Septimal min 7th or

968.82590646912,

16/9 symmetric Just ,

12/7 in limit 7 =

933.12909440059

cents, & 7/4 in limit 7Maj 7 15/8

(limit

5)

1088.26871

4

1.875 1100 +11.73

1286

243/12

8

1.8984375 1109.775004

32694

Octav

e

2/1 1200 1200 0 1200

21/1200, or the 1200th root of 2 is roughly 1 cent, or 1.0005777895. If n = cents then n = 1200 ·

log2 (b/a). Further if a and cents n are known then b may be calculated: b = a x 2n/1200.

The human ear can discern a difference of 1Hz for sustained notes. A common major 6 th of C

in equal temperament is 440.00 hz. (also 441hz)

The wolf 5th is almost a ¼ tone flatter than a P5 and thus placing it between a tritone and P5.

The Wusta-Zalzal is 27/22 or 1.22727272727272 or 354.54706023141 cents putting it between

a minor 3rd and major 3rd.

If limits 3, 5, 7, 11, and 13 are graphed against any equal temperament it can be seen that

rarely do all 12 chromatic equal tempered notes fall very near limit tuning, while falling nearer

ET the higher the equal tempered divisions are, as in 53-TET and 72-TET - which are still

39

slightly out by a few cents. Limit 3 and 5 forms Just intonation. The most common equal

temperaments are: 5, 7, 12, 19, 22, 24, 31, 34, 41, 53, 72.64

The differences of the old Masonic ratios are as follows, and can be viewed as d/t = speed.

(Sfakianakis, n.d.)65

Re/do = 9/8: 1 = 9/8 9/8=1.125Mi/re = 10/8 : 9/8 =10/9 10/8=1.25 1.25/1.125=1.1111 or 10/9Fa/mi = 4/3 : 10/8 = 16/15 4/3=1.33333 1.33333/1.25=1.06666 or 16/15So/fa = 3/2 : 4/3 = 9/8 3/2=1.5 1.5/1.33333=1.125 or 9/8La/sol = 5/3 : 3/2 = 10/9 5/3=1.66666 1.66666/1.5=1.11111 or 10/9Si/la = 15/8 : 5/3 = 9/8 15/8=1.875 5/3=1.66666 1.875/1.66666=1.125 or 9/8Do/si = 16/8 : 15/8 = 16/15 16/8=2 2/1.875=1.06666 or 16/15

Comparative Table 1:

Interval 12-TET 12-

TET

Just Pythagorean 19-TET 53-TET 53-TET Scottish Indian

Unison 20/12=1 0 0 0 0 20/53= 1 0 0 0

Min

2nd

21/12=12√2 100 7 0 . 6 7 2 4 2 o r

111.73128526

90.22499567827

or

113.68500605771

63.158 24/53= 53/4√2 99.99957691

0310416400

29.850 9 0 o r

112

Maj

2nd

22/12=6√2 200 203.91000173 203.91000173 189.474 29/53= 63/9√2 203.7735345

7914678100

187.682 203

Min 3rd 23/12=4√2 300 315.64128700 294.13499740384 315.789 213/53=53/13√2

294.3394160

6292929500

256.597 294 or

316

Maj 3rd 24/12=3√2 400 386.31371386 407.82000346155 378.947 217/53=53/17√2

384.9055263

5548759900

343.091 386 or

407

P 4th 25/12=12√32 500 498.04499913 498.04499913 505.263 222/53=53/22√2

498.1128243

0692445500

493.957 498

Aug 4th 26/12=√2 600 590.22371559

or

582.512192604

29

611.73000519232 568.421 226/53=53/26√2

588.6791264

9928285400

548.649 590 or

612

P 5 27/12=12√12

8

700 701.95500086 701.95500086538

7418000

694.737 231/53=53/31√2

701.8866215

7910072000

684.729 702

Min 6th 28/12=3√2 800 813.68628613 792.17999653818 757.895 235/53=53/35√2

792.4528009

3970148200

729.879 792 or

814

Maj 6th 29/12=4√8 900 884.35871299 905.86500259616 884.211 239/53=53/39√2

883.0187369

7003403600

871.949 884 or

906

64 Purity of tritones (25/18 and 36/25) is controversial in 5-limit tuning, and 7-limit tuning gives the septimal tritone (7/5 and 10/7), 582.512 cents and 617.488 cents respectively. These two ratios are considered more consonant than 17/12 (603.000 cents) and 24/17 (597.000 cents) in 17-limit tuning, and closer to an equal-tempered value of 600.000 cents. The undecimal neutral 6th (18/11, 852.59 cents) and tridecimal nuetral 6th (13/8, 840.53 cents) are two of the three neutral 6ths – the last is the equal tempered (18/11, 850 cents). They are approximately a quarter-tone flat of 12-ET minor 6ths and a quarter-tone sharp of major 6ths.65 Here a 2nd/root is similar to 5th/4th and 7th/6th @ 9/8. Similarly the 3rd/2nd and 6th/5th are @ 10/9 and 4th/3rd and 8ve/7th are @ 16/15. Two ratios are harmonic inverses of each other if they combine to make an octave. For example 3/2 x 4/3 = 2. (Enevoldsen, 2010)

40

Min 7th 210/12=6√32 1000 1017.59628786

5 9 4 o r

996.089998269

23

996.08999826923 1010.526 244/53=53/44√2

996.2263076

2672999600

985.799 996 or

1017

Maj 7th 211/12=12√2

048

1100 1088.268714 1109.7750043269

4

1073.684 248/53=53/48√2

1,086.79218

0595960150

1049.363 1088 or

1110

Octave 212/12=2 1200 1200 1200 1200 253/53=2 1200 1200 1200

The powers (of logarithms) show the exact figures of 12TET. This chart shows the Indian and

Pythagorean ratios to be the same, whilst the next chart shows the added 53TET notes for the full

22 shrutis.

The 22 tone system of śrutis (‘tones’/microtones) used predominantly in heptatonic sets

described by Bharata and Dattila, comparable to Western 12TET and 53TET makes a lot of

sense in that if looked at from the perspective of 7 note modes based in a 12TET system, each

note would have one of two inflections with the exception of the root and 5th. The table below

illustrates the 10 notes with slight inflection (20 notes in all) plus the root and 5th, summing to 22

in total.66

22 tone system of India:

Shrutis 12-TET 53-TETName Ratio Cents Frequency

(Hz)

Name Frequency Note Cents Frequency

Ksobhinī 1 0 261.6256 C 261.6256 0 261.6256Tīvrā 256/243 90 275.6220 C# 277.1826 4 90.566037735849019100 223.44424Kumudvatī 16/15 111.73 279.0673 5 113.207547169811002000 279.3053Mandā 10/9 182 290.6951 D 293.6648 8 181.132075471698153000 290.4816Chandovatī 9/8 203 294.3288 9 203.773584905660637000 294.3056Dayāvatī 32/27 294 310.0747 D# 311.1270 13 294.339622641509454000 310.1114Ranjanī 6/5 316 313.9507 14 311.111111111111024000 314.1937Raktikā 5/4 386 327.0319 E 329.6275 17 384.905660377358421000 326.7661Raudrī 81/64 407 331.1198 18 407.547169811320710000 331.0677Krodhā 4/3 498 348.8341 F 349.2282 22 498.113207547169641000 348.8478Vajrikā 27/20 519 353.1945 23 520.754716981132149000 353.4401Prasāriṇī 45/32 590 367.9109 F# 369.9944 26 588.679245283018988000 367.5829

Prīti 729/512 612 372.5098 27 611.320754716981092000 372.4218

66 In Carnatic music, where there are two different ratios on the same note there is a difference of 81:80, the syntonic

comma (21.51 cent diesis), which is one explanation of India’s 22-Śruti tonal system. The 13th swarasthana results

in an octave: or x12 = 2. As x is the twelfth root of 2 we obtain a figure of 1.06, and Pa is a ratio of 1.498 instead of

1.5, and the trained musician is able to hear the difference. Carnatic music is based on rational division. (Sriram,

1990) Higher degrees of harmony are associated with ratios with powers of 2 (2:1, 4:1, 8:1…) as well as small

integers (like 3:2 which is easily identified by the ear).

41

Mārjanī 3/2 702 392.4383 G 391.9954 31 701.886792452830232000 392.4229Ksiti 128/81 792 413.4330 G# 415.3047 35 792.452830188679375000 413.4982Raktā 8/5 814 418.6009 36 815.094339622641527000 418.9415Sandīpanī 5/3 884 436.0426 A 440.0000 39 883.018867924528173000 435.7053Ālāpinī 27/16 906 441.4931 40 905.660377358490696000 441.441Madantī 16/9 996 465.1121 A# 466.1638 44 996.226415094339526000 465.1488Rohinī 9/5 1017 470.9260 45 1,018.86792452830176000

0

471.2721

Ramyā 15/8 1088 490.5479 B 493.8833 48 1,086.79245283018860000

0

490.1298

Ugrā 243/128 1110 496.6798 49 1,109.43396226415078000

0

496.582

Ksobhinī 2 1200 523.2511 C 523.2511 53 1200 523.2512

Ragas may be comparable to 12-tone technique in the sense that ragas use re-ordering of motifs

instead of partitioning of pitch classes as in serialism, the main difference is in transposition. The

sage Matanga defines swara (tone) as ‘that which shines by itself.’ Individual tones are

embellished using Gamakas, which translates as ornaments which are melodically more involved

than simple ornamental devices external to melody, having values which are assigned to specific

notes for example, and have ‘structural relationship with the raga, with volume, pitch and timbral

inflection and structural functionality foreign to Schoenberg’s tonal world-view. The tala

indicates the timing - employment of rhythmic stresses, and influenced Messiaen. However,

Milton Babbitt’s use of operators to influence rhythmic structure after the late 1940’s is unrelated

to the tala and is independently a part of Western music. (Wen-chung, 1971)

Gamaka comes from the Sanskrit gam, to move, leading through the spaces between scale

tones and illuminating the microtones. Gjerdingen described Seeger’s melographs of Carnatic

music thus: ‘if we conceive of movement as a primary phenomenon, then the notes and rhythms

become secondary phenomena.’ (Battey, 2004) This idea corresponds to modernist coherence

and to Romantic gestalten (shape, form) as the sum over parts.

Amelia Cuni performs vocal microtones with precision and emotion on Amelia Cuni – John

Cage Solo For Voice 58: 18 Microtonal Ragas. Cage employs stochastic elements to generate

chance for the ragas. Cuni uses 20 years of study and performance of dhrupad vocalism in a new

context enabling her to ‘step back’ from traditional raga, and connect with her Western origins,

broadening musical vocabulary. In Cuni’s opinion, Cage connected the 18 microtonal ragas to

‘their original meaning, without relying on traditional canon only, but providing strategies to free

their innate generative power…effective even in a de-contextualised framework…that is an

42

eclectic compendium of compositional techniques relating to music and theatre as well . . .’. 67,68

(Cuni, 2011)

The 53TET frequencies69 are very close to the Shruti (22) system. 53TET is compatible with

syntonic and schismatic temperaments, and is arguably close to Just tuning in limit 5. The53√2=1.0131641430249148.

intervals 53TET power Fundamental cents1 1 02 53√2 1.0131641430249148 22.6415094339624213003 53/2√2 1.0265015807114097 45.2830188679243145004 53/3√2 1.040014594335196 67.9245283018870344005 53/4√2 1.053705495203023 90.5660377358490191006 53/5√2 1.067576625048014 113.2075471698110020007 53/6√2 1.081630356430202 135.8490566037735890008 53/7√2 1.0958690931423387 158.4905660377359270009 53/8√2 1.110295270621048 181.13207547169815300010 53/9√2 1.12491135636339 203.773584905660637000

11 53/10√2 1.1397198503489083 226.41509433962265200012 53/11√2 1.1547232854672358 249.05660377358504200013 53/12√2 1.1699242279513258 271.69811320754714700014 53/13√2 1.18532527781639 294.33962264150945400015 53/14√2 1.19686402614609 311.11111111111102400016 53/15√2 1.2167382713357153 339.622641509433932000

17 53/16√2 1.2327555879634662 362.26415094339628600018 53/17√2 1.24898375883818 384.90566037735842100019 53/18√2 1.2654255596753214 407.547169811320710000

67 Legend goes that the first singer of Indian antiquity, Tumburu, tonally expanded the Samaveda chant from a pentatonic chord to six or seven pitches. Knowledge of that style suggests it was originally a pre-filled pentachord and not pentatonic collection, and excavations in the Indus valley recovered lyre-type seven string instruments validating the description of the archaic vina. Historian William Hunter estimates that pitch names (swaras) of the set (Sa Ri Ga Ma Pa Dha Ni) were already prevalent during the time of the Sanskrit grammarian Pānini in the 4th century B.C. Concrete evidence occurs later around A.D. 100-500 in the Nātyaśāstra, yet passages contained therein refer to more ancient practice. (Gauldin, 1983) This system differs from Western 22-TET.68 Cage thought that a recording ‘destroys one’s need for real music. It substitutes artificial music for real music, andit makes people think that they’re engaging in a musical activity…’ (Haskins, 2010) This is an interesting point to note in terms of what music and musical practices are, how they are created (performed/composed), and heard (as noise, veridical expression or schematic frameworks) – and perhaps listening to recordings do not engage but reflect,as in watching a television program or looking at a picture. Reflection may be a form of after-engagement – although after-engagements before technological mediums were committed to memory and notation, aiding musical memory and language, etymologies, semantics, and contextual bases culturally, imaginatively, and scientifically. Music itself encompasses vastly different genres under performance (composition), from hypnotic to meditative, scientific to cultural, synthetic to organic, calculated to aesthetical. A picture can be a personal memory, like a performance, or a connection with schematic and veridical history and cultural identity – but the concept of a remembrance (recording) being part of a new experience (veridical) is also considerable. These are important factorsto address in the practice of musical arts, including microtonal practices.69 It is further believed that 53TET may be used pivotally in temperament modulation, known as dynamic tonality, asin for example shifting maqamat, or in Western terms micro-tonal modal interchange. 43

20 53/19√2 1.2820838027302701 430.18867924528301100021 53/20√2 1.298961337279338 452.83018867924539600022 53/21√2 1.3160610501071177 475.47169811320768800023 53/22√2 1.333385866000247 498.11320754716964100024 53/23√2 1.3509387482476742 520.75471698113214900025 53/24√2 1.3687226991475057 543.39622641509425300026 53/25√2 1.3867407605205309 566.03773584905668000027 53/26√2 1.4049960142305022 588.67924528301898800028 53/27√2 1.4234915827112675 611.32075471698109200029 53/28√2 1.442230629500841 633.96226415094356000030 53/29√2 1.4612163597825027 656.60377358490565200031 53/30√2 1.4804520209330247 679.24528301886780700032 53/31√2 1.4999409030781112 701.88679245283023200033 53/32√2 1.5196863396551512 724.52830188679251300034 53/33√2 1.5396917079833807 747.16981132075475200035 53/34√2 1.559960429841549 769.81132075471686600036 53/35√2 1.5804959720531908 792.45283018867937500037 53/36√2 1.6013018470796005 815.09433962264152700038 53/37√2 1.6223816136206164 837.73584905660372600039 53/38√2 1.6437388772233101 860.37735849056598300040 53/39√2 1.6653772908986904 883.01886792452817300041 53/40√2 1.687300555746526 905.66037735849069600042 53/41√2 1.7095124215883912 928.30188679245281800043 53/42√2 1.732016687609049 950.94339622641501000044 53/43√2 1.7548172030062736 973.58490566037733400045 53/44√2 1.7779178676492289 996.22641509433952600046 53/45√2 1.8013226327455147 1,018.86792452830176000047 53/46√2 1.8250355015169928 1,041.50943396226425000048 53/47√2 1.8490605298845093 1,064.15094339622639000049 53/48√2 1.8734018271616335 1,086.79245283018860000050 53/49√2 1.8980635567575257 1,109.43396226415078000051 53/50√2 1.9230499368890601 1,132.07547169811306000052 53/51√2 1.948365241302321 1,154.71698113207542000053 53/52√2 1.9740138000035974 1,177.35849056603785000054 53/53√2 2 1200

19 Tone Equal Temperament:70

Degree Interval Cents Fundamental Note Closes

to Just

interva

l

Difference

to Just in

cents

Name

1 0 1 A 1/1 0 Unison2 19√2 63.15789473684

1961400

1.0371550444

461919

A# 36/35 +14.388 1/4-tone, septimal diesis

3 19/2√2 126.3157894736

84091000

1.0756905862

201824

Bb 15/14 +6.873 major diatonic semitone

70 19 Tone Equal Temperament makes sense as it contains a ¼-tone (septimal diesis), major diatonic semitone, a minor whole tone, septimal minor third, a minor third, major third, septimal major third, perfect fourth, a septimal and Euler’s tritone, a perfect fifth, septimal minor sixth, minor sixth, major sixth, septimal major sixth, Just minor seventh, classic major seventh, and septimal diesis – octave; which are approximate to Just intonation in cents by roughly +/-0.148 to +/-14.585.44

4 19/3√2 189.4736842105

26522000

1.1156579177

615438

B 10/9 +7.070 minor whole tone

5 19/4√2 252.6315789473

68383000

1.1571102372

827198

B#//Cb 7/6 -14.239 septimal minor third

6 19/5√2 315.7894736842

10474000

1.2001027195

78103

C 6/5 +0.148 minor third

7 19/6√2 378.9473684210

52587000

1.2446925894

640233

C# 5/4 -7.367 major third

8 19/7√2 442.1052631578

94853000

1.2909391979

47405

Db 9/7 +7.021 septimal major third

9 19/8√2 505.2631578947

36871000

1.3389041012

244722

D 4/3 +7.376 perfect fourth

10 19/9√2 568.4210526315

78978000

1.3886511426

146562

D# 7/5 -14.092 septimal tritone

11 19/10√2 631.5789473684

21038000

1.4402465375

38759

Eb 10/7 -14.091 Euler's tritone

12 19/11√2 694.7368421052

63137000

1.4937589616

544857

E 3/2 -7.218 perfect fifth

13 19/12√2 757.8947368421

05346000

1.5492596422

666558

E#/Fb 14/9 -7.021 septimal minor sixth

14 19/13√2 821.0526315789

47567000

1.6068224531

33765

F 8/5 +7.366 minor sixth

15 19/14√2 884.2105263157

89383000

1.6665240127

97089

F# 5/3 -0.148 major sixth

16 19/15√2 947.3684210526

31625000

1.7284437865

632112

Gb 12/7 +14.238 septimal major sixth

17 19/16√2 1,010.52631578

9473650000

1.7926641922

757116

G 9/5 -7.070 Just minor seventh

18 19/17√2 1,073.68421052

6315920000

1.8592707100

168127

G# 15/8 -14.585 classic major seventh

19 19/18√2 1,136.84210526

3157930000

1.9283519958

849902

Ab 35/18 -14.388 octave - septimal diesis

Degree 20 would be note A, completing 1200 cents. The intervals in 19TET ascend in pitch by

63 cents.71

Lindström and Wifstrand created a program that could write in 19TET and import from

12TET, finding that people preferred 12TET over 19TET with the exception that the minor 3rd

was preferred in 19TET. (Lindström and Wifstrand, 2012)

Bagpipe tuning gives very interesting ratios:

Degree Interval Cents1 1/1 02 117/115 29.8503 146/131 187.682

71Joseph Yasser and Joel Mandelbaum have written music in 19EDO. Mandelbaum’s doctoral thesis explains why he thinks 19TET is the really only practically viable system between 12 and 24, and that the next one on is 31 equal temperament. 45

4 196/169 256.5975 89/73 343.0916 141/106 493.9577 81/59 548.6498 150/101 684.7299 125/82 729.87910 139/84 871.94911 205/116 985.79912 11/672 1049.363

Joe Heaney uses a ‘waver’ on certain notes, a device like an appogiatura or unstable flutter and

not as fast as vibrato yet faster than a roll, which he places on 4 th and 7th degrees on ascending

and with a technique of variation. Notation simply marks their place and does not signify what

they sound like.73 (Williams, 2004)

In Ferneyhough’s Renvoi/ Shards for quarter-tone guitar and quarter-tone vibraphone, which

incorporates microtonal techniques in the pitch and time domain,74 there is atonality, change of

time signatures cycling throughout, glissandi, dynamic change, artificial harmonics, half

sharps/flats which seem aleotoric – which is in stark contrast to tonality and tonal systems in

Western styles in previous centuries. If anything, it would be similar aesthetically to some

Chinese musics or certain Nile (Egypt) or Tibetan musics.75 (Incipitsify, 2012)

Somewhat akin to minimalism, the French Spectralists, or Spectral Music starting in the

1970’s, used waveforms of sounds and expanded them out over a symphonic composition that

employed microtonality. The French serialists expanded into 24TET and microtonal serialism.

Franco-American composer Rudhyar’s ideas are similar to Varèse’s of psychic power, indeed

Varèse’s ideas that music was ‘organized sound’ and that sound was ‘living matter’ were of

historic import, 76 and parallels the Chinese idea that a tone is an entity unto itself, with the

further perplexing concept that the meaning lies within the tones: that is, deeper into the music.

As a fundamental feature of Asian music this idea involves a vocabulary of articulations, timbre,

inflections, and intensity fluctuations. The importance of the single tones themselves is the

72 Note. 11/6 is a 21/4-tone, undecimal neutral seventh. (microtonal-synthesis.com)73 The glottal stop used by many male sean-nós singers is a throat technique of stopping the air which draws attention to the line, and an echo effect is created of the word just before the break74 Pitch/time=speed as frequency/time=length, or notes/bar=bpm. Hence, to work out the speed of a song one must divide the amount of notes/pulses in the spectrum of one bar/measure to obtain the ratio or beats per minute (bpm). It is interesting to note the relation between speed and distance, as it is this function that traces the curve between point and wave (rhythm and pitch). This can be useful in music and can have microtonal outcomes in pitch-frequency as well as timing.75 It is in stark contrast to Tchaikovsky, Rimsky-Korsakov, Copeland, Bernstein or Prokofiev, whose stylistics were directly delineated from post 1730 (or earlier) aesthetics. Minimalism in the 60's, starting with La Monte Young, used microtonality76 This corresponds with semiotic theory whereby sign and symbols represent the specific (logic) and context and forms represent the generic, allegory (creative).46

antithesis of Western polyphonic composition, whereby multi-linear harmony and equal

temperament undermine these values to an extent - these ideas are subordinate. Since Varèse this

idea is now common and a hallmark of 20th century music.77

Varèse was more concerned with complex structures of developing sound (tones) over single

line development. There are striking similarities in his works to Asian musics, for example in the

opening of Intègrales and the ha movement of tagaku (Japanese court) style composition: the

ryuteki (transverse flute) and hichiriki (double reed) is similar to the E-flat clarinet and trumpets

and conveys the nuclear ideas linearly. The sho (mouth organ) is similar to the B-flat clarinet and

piccolos contributing to upper registers. The koto (movable-bridged zither) and biwa (lute) use

lower sonorities as do the trombones. Both Togaku and Varèse use a percussion ensemble adding

a fourth dimensional texture and moving with specific timbre, register and function related to the

material. (Wen-Chung,78 1971)

Midi, scale perception, semiotics, notation, re-creation, Turkish, Eskimo, Indonesian

Slendro in 5TET (Salendro), Thai 7TET

Midi tuning and Western instruments are dominated by equal temperament (except fretless

strings, voice and harps/zithers), where tuning is slightly out to accommodate the ability to play

in all 12 keys. Real-time processing, in today’s systems for pitch related functions, including the

ability to extend into other tuning systems, is becoming more widespread. Keyboards are well

suited to midi and historically based microtonal keyboards may serves as models. Midi

keyboards in live performance, using arbitrary tuning systems, and free from the restrictions of

the studio, would have the exciting intricacies and nuances of live human performance. (Keislar,

1987)

Perception lies at the heart of music, and paradoxes remain central to music, art and literature.

In 1986, in Music Perception, Dr. Diana Deutsch described her tritone paradox she discovered

regarding two notes linked by a tritone. When successively played one after the other some will

hear an ascending pattern, whilst others hear it descending – an experience which can be

77 Some of Varèse’s work, like Arcana, use the idée fix, made well known by Berlioz’s Symphonie fantastique, and usually not transposed. Lietmotiv, used by Wagner, however, is transposable.78 Wen-Chung, as well as Tenney, McPhee and others, was a student of Varèse. 47

‘particularly astonishing’ to experienced musicians. Tonally, tritones play an important part in

evolving music.79

Another paradox described by Dr. Deutch in Musical Illusions and Paradoxes (1995) is the

glissando illusion.80

Scale pattern: two lines, left ear and right ear, played simultaneously,

&=Y=S=W=U,===U=W=S=Y! &=R=X=T=V,===V=T=X=R!

and the perceived scale81, left and right ears.

&=R=S=T=U=V=W=X=Y! &=Y=X=W=V=U=T=S=R=!

Semiotics and a plethora of signs and communicative symbology may be utilized in

composition. The Phoenicians had managed the semiotic transition from syllabic to alphabetic

c.1500 BC, and possibly may have advanced musical notation by the Common Era. There is

79 A listener may hear C followed by F# as descending, and as a different tone pair is played, for example G# then D, it seems to ascend – while another listener may hear them the other way about. This is due to the timbre, partials, artifacts and inflections that make up the sound structure, just as one may sometimes hear a singer seem to sing up an octave for some moments and realize the illusion.This idea of how we perceive information is akin to Ingo Swann’s idea that there are levels of senses that can access information ‘achieving perception appropriate to them’. Anthropologists estimate that pre-modern human societies did not ‘think in terms of senses’ as Swann puts it. (Swann, 1994)80 Glissando’s are a facet of microtonality and thus will be given a brief mention here. An oboe plays a tone and a sine wave ‘glides’ up and down in frequency (pitch), and these are switched (panned) left to right repeatedly in a manner that whenever the oboe is on the left, for example, a portion of the glissando is on the opposite right, and vice versa. On stereophonically separate speakers some illusions are produced. The oboe is rightly heard jumping from left to right ear, whereas the glissando seems ‘joined’ together, and the human ear will localize the glissando in‘a variety of ways.’ Right-handers often hear it going left to right as the pitch goes from low to high, and right to leftas the pitch goes from high to low. Yet, lefthanders often gain completely different illusions altogether. Dr. Deutch describes other paradoxes and illusions. The last I shall mention which would be a good setting in a microtonal context for future purposes is the Scale illusion (1973). The top is on the right speaker/ear and bottom is on the left. What effect would be produced if there were glissando marks in between notes in the following passages? Might it not accentuate the paradox more clearly?81 Microtonal music would require more musical thinking, though certainly 24-TET for example should be a natural extension of 12-TET, and any other systems would use the same parts of the brain to recognize pitch and remember pitch group sequences, making it commonly practicable, especially with cultural support. The notation of half sharpsand flats may also approximate other tonal systems well, as chromatic 12TET notation may approximate Just intonation.48

great similarity in the Jewish cantillation (pitch marks to speech) notational system and

Ethiopian – the link extends to musical symbols in Syrian and Armenian, whilst the Egyptian has

faded to oral tradition. Fellasha communities in Ethiopa still practice ecumenical vocal chants in

Hebrew with melismatic vibratos and microtonal slides before and after main tonal syllables.

(Kebede, 1980)

Pining for systematic efficiency in communicative symbol logistics in the deep array of

microtonal notational stylistics, Read states that the notation of Penderecki is ‘commendable’

and Hàba is ’guilty of using different symbology for the same microtonal intervals in several of

his works.’ Read’s cataloging in this regard exemplifies the stylistic aesthetical logic that bridges

inspired creativity with communicable scoring. (Polansky, 1991) Polansky argues that many

composers feel bound by the12-tone canon and the generic use of the ‘microtonal’ in which say a

septimal major 2nd (8/7) which is larger than the12-equal-tempered 2nd is simply not microtonal

per se, but are part of ratio systems implemented into the divisibility of the octave. Pioneer

microtonalists like Partch, Carillo and Hàba82 were as diverse as they were stubborn – composers

tend to cling to a personal developed style of notation and there is some contention over what the

field should be called at all. Polanski argues that the 150-200 year tenor of 12-tone equal

temperament is microtonal as much as any other system since the Greeks, and that it is tenuous

as an absolute since its short inception and life, with suspect respectability in European and

American art musics.

Ben Johnston goes so far as to assert that 12-tone equal temperament is a lie – that the human

ear does not naturally hear these ratios, and whatever advantages of 12-TET may be it has also

seduced us into believing it the only way.83 There is also a link between microtonalism and

indeterminacy in Johnston’s works. (Rapoport, 1988)

Schenkerian note-to-note analysis can predict shape for non-Western musics, although the

criticism from the ethnomusicology bloc is that Schenkerian notation cannot cope with timbral

variations and non-Western temperaments, microtones or slides which may be key musically.

(Stock, 1993)

Influenced by geo- and politico-historicity, west coast America served as locale for

microtonists including Cowell, Cage, McPhee, Harrison, and Hovhaness. It includes an Asian

82 Hàba commissioned specialized quarter- and sixth-tone instruments (trumpets, pianos, clarinets).83 For over 40 years Johnston investigated rational pitch structures and tried to forward its practice in performance. The St. Louis Symphony’s antagonism for Johnston’s Quintet for Groups stemmed from a performance fiasco, yet performers investing time achieve good results as in the Fine Arts Quartet’s recording of his Fourth String Quartet. Johnston had some quirks such as foreshadowing of microtones by double flats in one early work – somewhat akin to the triple sharp in Alkan’s Qausi-Foust.49

and African population with a history of commerce unbound by politico-acculturation with rising

ethnomusicological study (Asian composers and musicians). Prior to this, Carpenter and Griffes

leaned toward orientalism via impressionism. Rudyar’s idea of a note as a ‘living entity’ was

comparable to the idea that in Asian music one is ‘confronted with living tones’.84

Indonesian Slendro in 5TET (Salendro)

Interva

l

1 2 3 4 5 6

Cents 0 240 480 720 960 1200

Thai 7TET

Interva

l

1 2 3 4 5 6 7 8

Cents 0 171.428571

428571429

342.857142

857142858

514.285714

285714287

685.714285

714285716

857.142857

142857145

1,028.57142

857142857

1200

Murman-Hall, Ozgen and Lux Musica performed works by the 17th century Moldovian

Demetrius Cantemir who lived in Istanbul from 1687-1710.85 Scholars have not attempted to

recreate the musical practices from that time, preferring to gain insight into the Ottoman court’s

musical life, yet these skillful musicians attempt the former. These cross-cultural performances

fuse traditional and non-traditional styles resulting in hybrid styles that have particular emphasis

on early music. The performances (recordings) combine Turkish and non-Turkish with historic

European-type renderings of that period’s Ottoman music. The musicians also perform new

works reflective of Cantemir’s compositions and improvisations, experimentally placing

monophonic tradition into a polyphonic frame. The musicians are less comfortable interpreting

non-Western pieces and there is clash of tonality due to the intonation systems of the

instruments, especially in passages where fixed pitch instruments accompany microtonal makam

84 Rudyar, the Scriabin influenced Franco-American, was heavily influenced by Eastern philosophy and mysticism, claiming that Western composers were not interested in the audible single tones but more on pitch relation. This is consistent with Russolo’s ideas, and throughout minimalism and noise-art. Edward MacDowell and others had surmised earlier an oriental idea of value in texture but, misconstrued it as sound without music, and is still at the heart of misunderstanding Asian music as well as contemporary music today. Eichheim traveled and collected instruments, though insincerity to his endeavors and research in the music field led to only a few crude works.85 Featuring pieces from all over the Ottoman Empire like Moldovian dances such as syrba and zhok de nante, and Turkish like prsrev and saz semaisi, and stylistics drawn from his treatise Edvar. Using Western instruments (viols, lutes, flutes, keyboards) and adaptation of Turkish instruments (kemence, kudum, tanbor) they however do not utilize for example Western instruments like the viola d’amore or non-Western’s such as the ney.A new instrument called the kemence is used and the classical tanbor. Lux Musica uses a more usual modal heterophony for harmonization already in high use in Turkish art music, for example the delayed heterophonic patterns combined with pedal tones often in parallel intervals.50

intervals set apart from equal-temperament, such as makam Bestenigar. However some like

Nihavend [close to minor, as Rast is to major] work well due to the close relation in tonality and

pitch class. (O’connell, 2006)

Many of the circumpolar Eskimo musics have been effected over time by the West, for

example in Greenland, where ancient complex compositions comprising microtones and subtle

inflections and interesting rhythmic structures in 5/8 or 7/8, only practiced by a handful now,

have given way to the copying of bland western folk formats.86 American Indian and

Paleosiberian elements are found in North and West Greenland. Vocables and a compact song

are used in Alaska and Siberia, and in Greenland and East Canada a dual call and response

(refrain-chorus) is used. Tetronics and pentatonics are used in Greenland and Alaska, although

the Copper Eskimos use chromatics, hexatonics, and heptatonics, and all use microtones.87

Eskimo music abounds with microtonal accents and embellishments connected to certain

contexts which affect meaning, and glissandi are also used. Westwards of the Copper Eskimos,

abrupt tonal centre change occurs. The melodies are usually arch-shaped, with call and response.

In Alaska the 2nd lowest note is repeated or prolonged, and the descent of the arched melody

slows. In the West intervals greater than an octave can occur, whilst lesser leaps occur in the East

(Siberia) of a 4th or 5th. Melodies often have ascending and descending 4ths.88 (Johnston, 1975)

86 Missionization early on (Moravian, Anglican, Catholic) affected communal musical practice. The acquisition of boats for cod liver oil from shark fishing and the decline in seal hunting effected its associated songs, and later in Alaska socio-politics brought change, for example the need for hunting songs disappeared. In Alaska, where contact between Whites and Eskimos is newer than Greenland, it is thought that musical compartmentalization occurs.87Pentatonics prevail in Alaska and Siberia. Alaskan and Greenlandic tonal range in song is about a 5th or 6th, except a 10th or 12th in the case of the Copper Eskimos; Alaska and Siberia have a range of about a 5th or 6th. 88Copper music plays between two tonal centres. Ethnic symbols like traditional music were forbidden under the old Soviet regime - the hunters and deer herdsmen of Thule and Angmagssalik in Siberia knew nothing of the more free-style expressive song of the West Eskimo, and Alaskan Eskimo music which was influenced from Siberia and the closer American Indian city civilizations enjoys many exciting prospects such as the pan-Canadian Eskimo NorthernGames.51

Xibeifeng, Xenakis stochastic emulator, fretboards and the 12th root of 2, world Fusion,

evolving timbral domain, microtonality and after the fact of performance, societal

technological status, cultural and logical outset, and aesthetical artistic nuance

Xibeifeng in the 1980’s blew the lid off things, ‘the North West Wind’ inspired by Shaanxi folk

with rough vocals, rock instrumentation and beats, arcane melodies (with microtonal inflections),

Turkish instruments, drones and pitch ornaments (arabesques).89 (Huang, 2001)

Exploration of microtones in Xenakis' stochastic Metastasi s is explored well in the visual

Xenakis-Emulator and a 48 tone system is employed, though it is not clear what the intonation

system is. Glissandi within the composition is defined extraordinarily.90 (Kammerbauer, 2009)

The divisor of standard equal tempered guitar fret placement used by all but a vanishing few

makers is 17.817152 arrived at from the logarithmic function the 12th root of 2 (1.0594631),

resulting in the octave or 12th fret at exactly the center of the total length.

(truetemperament.com)

The equation in April 2013 of Premier Guitar showed that longer string scale length gives

higher tension. Longer scale with greater tension increases upper harmonics, whilst low notes are

described as more articulate and defined. (Hoepfinger, 2013)

One of the main factors to consider in how practical applications incorporating microtonality

will be achieved in the future, in light of past practices, is the growing amount of reliance on

technology in problem solving. Past practices are only beginning the process of factoring in from

modern technology. Theoretically the limits to practical music making would seem endless aided

by technology – yet at the same time as endless without specific technologies.91

Notably, in popular context, the world music marketplace is bridging genres. These genre-

fusions incorporate application of musical understanding and tonality and may be regionally

specific. Musics are being made continually with the aid of technological and human innovation

which also brings a new lexicon with each generation.92

89The Shaoshu Minzu, minority peoples of the North West are Mongols, Kazaks, Hui, Uighurs, which call to mind the 'exotic other' in Han China – a place of crossings and possibilities.90 It is based on Xenakis’ strip windows facade design on the monastery La Tourette, and is truly an innovative exploration of microtonal relationship as well as the placement in time of notes. 91 Sonic art, where music is more like a 3D painting than imaginable now, may be a field on the horizon and may have an integral performance factor. However, in many respects most things have not changed drastically in regard to physical performance of music except when the instrument is distinctly from the modern computer age.92 Due to information efficiency and capital flows we now have such cross-cultural genres as Czech bluegrass, Indonesian rap, Japanese salsa, South Asian reggae and Afropop, as well as American shakuhachi or mrdangam players, Chinese lieder, and Philip Glass performing with Tibetan monks.52

Schoenberg may have rejected microtonal experimentation because the time was not yet ripe:

‘whenever the ear and imagination have matured enough for such music the scale and the

instruments will all at once be available. It is certain that this movement is now afoot, certain that

it will lead to something.’ (Perlman, 1994)

Many non-Western musics have evolving timbral non-pitch and time domains which Bret

Battey calls pitch continuum traditions, outside the musical expression via the scalar and metric

pitch lattice. Technology tools today are highly focused on pitch and time musical expression as

opposed to pitch continuum, or timbral-shifting, musicality.93 Battey has written prototype

noncommercial software for personal composition that uses bezier-spline programming to

manipulate the microtonal pitch-time domain that is currently not easily possible to date. In the

future this type of powerful programming seen in applications like Photoshop and visual effects

software may be incorporated graphically into music software for synthesis.94 (Batty, 2004)

Contemporary film musics (e.g. Morricone, Rahman) borrow from past and current tonal

systems, sometimes with borrowing from alternate tonal systems in (modulation) passages (i.e.

written in multi-cultural styles). It is important to note the amount of microtonality going into the

modern production of music.95 Effects use can be like the writing of a symphony and have

become very complex: equalization of tracks is microtonal alteration. One difference between

past practices with microtonality compared with today’s is that microtonality on recordings can

be added after the fact of performance. In the east Levantine, like Turkey and the Nile, music is

still written and performed with microtones. In film music, distinct aesthetic world tonalities are

becoming more fused.

Any patterns emerging from microtonality, including tonal systems and socio-historically

rooted aesthetics, carry the trappings of societal technological status, cultural and logical outset,

and aesthetical artistic nuance, and is often slowly changing and built on previous works. Ethics,

philosophies and values are connected to early performers of music as well as today’s, along

with techniques and stylistics despite cultural change by cross-pollination of thought and ideas.

The fact that fused- microtonal musics are increasingly more commonplace suggests a

departure from standard pitch and rhythmic based musics, and as more emphasis is passed from

93 Pitch continuum may be explored in any musical segment such as the Carnatic alap – an unmetered part where theraag is explored.94Battey says, ‘Picacs can render pitch, amplitude and spectral centroid bezlists into breakpoints for envelopes.’ The software was created originally to write Hindustani music.95 Much of this microtonal post-production is subtle, although also a front piece to modern music. As timbre is explored more, pitch and time may become secondary and subliminal.

53

the compositional to production there appears a link to new styles of production as a

compositional form in which non-pitch and rhythmic facets are factored in, like timbral

elements, via use of sophisticated plug-ins (i.e. the plug-ins may be used like an instrument).

Here we have two important links: the first of musical trends out of simpler pitch and time bases

via multi and microtonal synthesis or means (e.g. stochastical, non-linear or linear), and second,

composition linked to production whereby the two become analogous.

On one hand we have Klangfarbenmelodie, where a musical line is split into several

instruments to colour timbre, discussed by Schoenberg as timbre-structures and also called

Pointillism, as well as Schoenberg and Webern’s idea of emancipation of dissonance where the

ear becomes accustomed to dissonance in context. If we think about noise music we see that

these ideas have been brought forth and used microtonally and timbrally through use of

sophisticated software and equipment that use many of the same classic principles. The key is

context, even if multi-timbral and multi and microtonal systems are in use (e.g. repetition for

contexts). Just as triadic music was [debatably] distinct after the 1400s and as the chordal 7 th was

to the 1600s, as the chordal 9th was indicative of 1750’s and the whole-tone scale was of 1880, so

too is chromaticism, microtonality and twelve-tone technique a feature of the 20th century.

However, microtonality is deeply rooted in the past, though not under the same guise of 12TET

or 24TET standardization.

Today production/compositional softwares96 help define new music through multi-faceted

contexts. The exception to the developmental direction may lie in stochastic musics (Xenakis,

Cage, where microtonality and multi-timbres take place though are hard to notate for re-creation.

However, in the future even this may be possible. Rossolo had defined and performed early noise

music as an aesthetically viable art-form.97

96 A main factor between past microtonal practices and modern practices lies in the realm of technology. In a performance of a hundred simultaneous recordings at different speeds it would be hard to discern signals from noise – brief patterns and colours of perceived non-randomness may be attributed and strung together via pitch, timbre andtime by the mind. These patterns may be so subtle and compounded amongst other tones and frequencies that imagination may alter the performance for each listener, and pareidolia might occur. Equalization morphs timbral, pitch and perception of rhythmic structures, and new musical dimensions are accessed via technology. Roger Penrose believes there may be quantum computation in human brain microtubules, effectively bringing up the question, can humans achieve sonically what standard computers are able to achieve today by non-technological means? That is to say, could we have achieved similar results to computer aided soundtracks, if computers never existed? With specialized instruments and enough time, I believe we could come close. Perhaps the best composers, conductors and performers can approximate, and even allude to standard sounds from modern genres like Glitch, Drum and Bass, and House. Non-algorithmic processes imply non-computability brain functions that are not randomor deterministic. Penrose attributes this idea to thought and consciousness, because of the suggestion that objective reduction and quantum computation might be linked to consciousness. (Hameroff, 1998)97 2,400 years ago Plato said ‘I would teach children music, physics, and philosophy; but more importantly music; for in the patterns of music and all the arts are the key to learning.’54

Conclusion, truth in music, modality of believing, dynamic tonality, Third-stream music,

sound painting, new directions

In A Theory of Musical Semiotics there is a chapter entitled On the Truth in Music (or what

Schoenberg and Asafiev said about the Modality of Believing). It states that the effect of

believing, persuasiveness, and convincingness is imperative to any musical communication as

well as the semiotics of spectacles, outlining its role in past music crisis and change. This would

hold true in microtonality as well. Michael Foucault, on considering epistemes, thought that

quotients of epistemology could alter historical development, whilst stylistic outcome is rooted

in the change of aesthetical thinking. (Tarasti, 1994) In Chapter six of Metaphor and Musical

Thought Spitzer decrees that allegory (Dionysus) overturns symbol (Apollo), an idea first

attributed to Goethe (though Todorov’s study points to Schiller, Kant, Moritz, Meyer) that sees

allegory as ‘the general through the particular’ and symbol as ‘the general in the particular’.

Todorov furthers this exposition thus: symbol is ‘productive, motivated, intransitive’ and

allegory, which is the reverse, is ‘transitive, arbitrary, rational.’ (Spitzer, 2004) This

demonstrates the link between symbol as musical basis and allegory as stylistic sociocultural

semiotics of musical etiquette, akin to grammar vs. linguistics. These musical semiotic and

semantic concepts are crucial to microtonal practices, tonality, and language and syntax.

Today, new progressions are possible with dynamic tuning bends, which allow modulation

between equal temperaments in real time [due to the width of the generator, from meantone

temperament, of the 5th and octave]. (Plamondon, 2008)

In future microtonal practice, jazz, classical, third stream and world fusions may incorporate

stylistics like blue notes, changes that use microtonal maqaamat, Balinese or other obscure ratio

intervals like those in Scottish bagpipes, mixing aesthetical cross-genre nuances and expanding

tonality aesthetics. Western microtonal practice halted early due to standardization of theory,

intonation systems, and instrument making practice, and now lies largely in the electronic

domain with exception to some world musics.98 Partch envisaged expanded Just tonality

instruments with transpositional ability, whereas Stockhausen saw room for expanded rhythm

and pitch, as the two are immediately linked.

98 A large part of music making lies in musical training, practicality and theory. If technology will play a role in future microtonal music, mathematical systems and new concepts will also play a role. (wolfram.com) This may alsoinclude new branches of logic and mathematics or physics and computer sciences, or experimental mathematics that will be a distinct part of future culture. One early example of this is Stockhausen’s phase shifting work in Samstag aus Licht, as well as microtonal and micro-time bases.55

To conclude, although some specialized microtonal instruments have been built, and many new

innovative instruments are springing up,99 instrument performance techniques are very similar

generally in both the past and present, while composition is experimenting more in the direction

of non-standard pitch frequencies and non-pitch and rhythm based aesthetics, including

performance utilizing and incorporating recent technologies and stylistic fusing. It is therefore

likely that technology for practical performance will catch up with compositional

experimenting.100,101 Certain rhythmic and microtonal structures are beyond human performance,

and new genres like chill-house, acid-jazz, glitch artists and noise artists, include technology in

the human equation.

In instrumental teaching and practice one could use 12TET and 24TET as the model, while

encouraging the ear towards Just intonation, thus avoiding problems in transposition.

Digital music producers have been using plug-ins to fine tune, within a cent, using their ears,

which was in the past not generally practicable, although a bulk of theory was known.

For the electronic composition iTET for Sampled Piano (originally sketched in 1200TET) I

employed passing phrases in many tonal systems. iTET for Sampled Piano uses 24TET, 53TET,

31TET, 17TET, 19TET, 7TET, 5TET, Just ratios and Bohlen-Pierce 8.20208TET with 3:1

(tritave, octave + 5th) ratio, and includes dynamic tonality (temperament modulations). To aid

99Ralph Novak pioneered the multi-scale fanned fret-board for modern electric instruments, a principle used by some of the 16th century lute makers. Multi-scale fan frets are becoming more common. Tolgahan Çoğulu has a secured microtonal guitar patent with grooves and removable mini frets that can sweep back and forth for the desired tonal system, which is especially useful for mid-Eastern musics. (Çoğulu, 2010) H-Pi Instruments’ Tonal Plexus microtonal keyboard uses 211 keys per octave arranged in 12 columns. 41 regions of 5 keys each = 205, and a further 6 duplicate enharmonic keys. (7 naturals, 7 sharps, 7 flats, 7 double-sharps, 7 double-flats and 6 triple-sharps, 6 triple-flats) (Hunt,2013) The Eigenharp has 120 keys (each one tilts to give a flexible tone), percussion buttons, recording, playback, looping, and running on sampled sounds is played via keyboard like a fretboard, tap-pad and mouthpiece, and can sound like a band. The electric violin has also become enhanced for the digital age andpickup technology can easily convert signals into midi to use sound samples or other desired processes, and is set to play a role in future music, especially tonal/microtonal. The Tenori-on was one of the better new musical gadgets to come out lately. It looks like a game of minesweeper, responds to touch in real time, looping themes intuitively, creating ‘soaring, rippling compositions that mesmerize beginners and experts alike.’ The other gadget that seemed fairly robust is the Hapi Drum, looking slightly like a steel drum and played like a bongo with a hole in the base. The player controls the amount of noise with their lap, and notes are accompanied by a ‘subtle resonant harmony from other musically compatible notes.’ (webUrbanist.com, n.d.)100 Perhaps in the future there will be colloquial labels like non-standard pitch-time phrases/phrasing, but currently standardization of notational and graphical systems, and technologies, are unraveling. As technology grants the ability to organize and annotate more information, there appears to be departure out of standard pitch and time aesthetics. Sound painting, although live, is inspired by technology related genres, and may incorporate samples. Time is an elusive word, and architectural devices and musical theory [like dominant 7] that can shape time through human expectancies involving consonance and dissonance are part of pitch-class, duration-class, and their relative durations in sequential patterning.101 Many traditional musics are codified now with the aid of the hypnotic, and often highly microtonal, pulse-driven [grid-locked] programmed backing tracks, whereas in the past this hypnotic affect was produced solely by performance instruments.56

aestheticism, uncommon and unfamiliar tonalities are at times grouped as discordant and

balanced with smaller ratio familiar tonalities for tension and resolution.

Glossary

12TET – 12 tone equal temperament; the system breaks the octave into 12 equivalent parts,

resulting in a semitone of non-simple ratio – approximately the 12th root of 2 (12√2 or 21/12) or

1.059.

Chromatic tuning – Traditionally, in 12TET, chromatic tuning consists of all 12 semitones, of

100 cents each. Chromaticism is the expansion of diatony which adds a further 5 notes to the

traditional 7 (diatonic).

Cymatics – Study of vibration, sound, and translation through physical mediums and material

effects of sound.

Dodecaphony – (dodecaphonous)Twelve-tone technique, serialism.

Diatonicism – (διατονική) Diatonic describes scales, modes, chords, and harmony, that is non-

chromatic (χρωματική), non-enharmonic, often heptatonic and built on tetrachords.

Eidos – (εἶδος), from οἶδα, ‘I know’ and Proto-Indo-European weyd- meaning to see or know.In Greek taken to mean essence, species, form, or type.

Enharmonic tuning – Enharmonic, or the equivalent note, in the sense of enharmonic tuning are

n o t e s t h a t r o u g h l y a p p r o x i m a t e e a c h o t h e r .

Enneachord – 9 note chord, enneotonic (9 tone).

Epistemes – quanta or packets of transmittable and interactive knowledge that may be contrasted

with empiricism. In Foucaultian philosophy, the total bounds of knowledge and ideas that define

a given epoch’s episteme (idea of true knowledge).57

Euphonious – sounding pleasant, agreeable.

Hellenic chromaticism – Chromaticism that is not strictly constricted by equal temperament.

Heptatonic – 7 note scale or chord

Hexatonic – 6 note scale or chord

Hypo-mixolydian – 5th up from a mixolydian, the ancient Greek mixolydian however was a

lochrian. Thus, a hypomixolydian is a modern dorian. Practicably, the scale extended slightly out

horizontally below and above the root and 8th, with rules.

Infra-diatonic – Yasser’s term for tonal systems that fall below the standard heptatonic (7 note)

scale base which is expandable to 12 as 7 + 5. This includes pentatonic 5 note bases, expandable

to 7 as 5 + 2.

Inharmonicity - varies between instruments, and even thickness of strings, occurring

progressively more, higher up the harmonic series.

Just intonation – Notes or frequency ratios that correlate to the harmonic series, generally small

ratios to begin and larger ratios higher up the harmonic series (limit tuning).

Log (logarithm) – log216=4 or 2x2x2x2, where 2 is the base, 4 is the exponent, and 16 is the

power.

Melisma – (μέλισμα) or song, recitative form of several notes to a syllable. [melismas,

melismatic]

Metonymy (metonym, Greek, μετά “other” + ὄνομα “name”) – use of term that substitutes for a

thing, such as The Crown in place of British government, or White House in place of US

government.

58

Museme – A small element of music whereby meaning is not further destroyed, broken down

from constituent parts in musical semiotics, and analogous to morphemes in linguistics.

Neoclassicism – (νέος κλασσικός) Art, architecture, music, literature and theatre inspired by

classical Greece and Rome, mainly during the 18th and early 19th century paralleling

Romanticism.

New progressions - new chord progressions that utilize different temperaments (intonation

systems).

Pareidolia- Cognitive process whereby real sounds are misconstrued imaginatively by picking

out certain frequencies and timbres, and associations via unknown time processes.

Polychordia – many-stringed, classically more than 7, and up to 10, 11, or 12 in ancient Greek

lyres and kitharas.

TET – Tone Equal Temperament, the logical division of a string [or other method] into equal

parts.

(e.g. 22TET, or 22EDO or 22ET, also written 22-tet, 22-edo)

Third stream – synthesis of Classical and Jazz, with the element of improvisation.

Tonos – (τόνος) accent or stress. In modern Greek and Latin typography and orthography it is

designated as the symbol ΄ over a vowel.

Schematic – term used to denote implicit acculturated framework of experience and knowledge

that is unquestioned or assumed and may be subconscious to a degree, and may not be a true

representation of logical modes of thought or experience.

Serialism – Musical processes, originally defined by Schoenberg, where notes are shuffled so

that no two notes re-occur in any given phrase.

59

Solfege – (solfeggia, solfege system) spoken syllables for each pitch in a scale or mode.

Solmization – attribution of unique syllables to notes.

Syntonic comma - 81:80, 21.5 cents, German Syntonie, in synergy, harmony.

Ultra-diatonic – Yasser’s term for tonal systems that are beyond standard contemporary

chromatic 12 tone diatonicism (7 + 5), and for Yasser the next logical choice was 12 + 7 in

19TET.

Veridical – (veri,or true) term used to describe flexible and creative use of accumulated

experiences and knowledge.

Wolf 5th - dissonant form of diminished 6th, 16th and 17th centuries, popularly arising out of the

quarter-comma meantone temperament and spanning seven semitones (procrustean/imperfect

5th).

Wusta-zalzal - greater than a tempered minor 3rd and less than a tempered major 3rd, with the

ratio 27/22.

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