**Andrián Pertout**

Three Microtonal Compositions:

The Utilization of Tuning Systems
in Modern Composition

Submitted in partial fulfilment of the requirements
of the degree of Doctor of Philosophy

Faculty of Music,
The University of Melbourne,
March, 2007

*Dedicated to my father,
the late Aleksander Herman Pertout
(b. Slovenia, 1926; d. Australia, 2000)*

**Abstract**

Three Microtonal Compositions: The Utilization of Tuning Systems in Modern Composition encompasses the work undertaken by Lou Harrison (widely regarded as one of America’s most influential and original composers) with regards to just intonation, and tuning and scale systems from around the globe – also taking into account the influential work of Alain Daniélou (Introduction to the Study of Musical Scales), Harry Partch (Genesis of a Music), and Ben Johnston (Scalar Order as a Compositional Resource). The essence of the project being to reveal the compositional applications of a selection of Persian, Indonesian, and Japanese musical scales utilized in three very distinct systems: theory versus performance practice and the ‘Scale of Fifths’, or cyclic division of the octave; the equally-tempered division of the octave; and the ‘Scale of Proportions’, or harmonic division of the octave championed by Harrison, among others – outlining their theoretical and aesthetic rationale, as well as their historical foundations. The project begins with the creation of three new microtonal works tailored to address some of the compositional issues of each system, and ending with an articulated exposition; obtained via the investigation of written sources, disclosure of compositional technique, mathematical analysis of relevant tuning systems, spectrum analysis of recordings, and face-to-face discussions with relevant key figures.

**Introduction
**

The actual term ‘microtonal’ is generally reserved for music utilizing “scalar and harmonic resources” outside of Western traditional twelve-tone equal temperament, with “music which can be performed in twelve-tone equal temperament without significant loss of its identity” not considered “truly microtonal” by some theorists. Most non-western musical traditions (intonationally disengaged from contemporary Western musical practice) almost certainly accommodate this description. In the online

“Strictly speaking, as can be inferred by its etymology, ‘microtonal’ refers to small intervals. Some theorists hold this to designate only intervals smaller than a semitone (using other terms, such as ‘macrotonal’, to describe other kinds of non-12-edo intervals), while many others use it to refer to any intervals that deviate from the familiar 12-edo scale, even those which are larger than the semitone – the extreme case being exemplified by Johnny Reinhard, who states that all tunings are to be considered microtonal.”3

In the West, the concept of microtonality was notably given prominence to during the Renaissance by Italian composer and theorist Nicola Vicentino (1511-1576), in response to “theoretical concepts and materials of ancient Greek music,”4 and later, by music theorists R. H. M. Bosanquet (1841-1912), as well as Hermann L. F. Helmholtz (1821-1894), and his “translator and annotator” Alexander John Ellis (1814-1890).5 With regards to the adoption of microtonality by composers in more recent times, according to

“The modern resurgence of interest in microtonal scales coincided with the search for expanded tonal resources in much 19th-century music. Jacques Fromental Halévy was the first modern composer to subdivide the semitone, in his cantata

Gardner Read offers the following historical perspective:

“The history of microtonal speculation during the first half of the twentieth century displays six names above all others: Julián Carrillo, Adriaan Fokker, Alois Hába, Harry Partch, Ivan Wyschnegradsky, and Joseph Yasser. All six contributed extensive studies on microtones – historical, technical, and philosophical – and all but Yasser composed a significant body of music based on their individual explorations into microtonal fragmentation of the traditional twelve-tone chromatic scale. Later theorist-composers – notably Easley Blackwood, Ben Johnston, Rudolf Rasch, and Ezra Sims – have extended those explorations into various tuning systems and temperaments, and each has devised a personal notation for various unorthodox divisions of the octave.”

Read identifies five essential strategies for the procurement of microtonal intervals, which include: quarter- and three-quarter-tones, or the division of the octave into twenty-four equal intervals; eighth- and sixteenth-tones, or forty-eight and ninety-six equal intervals; third-, sixth-, and twelfth-tones, or eighteen, thirty-six, and seventy-two equal intervals; and fifth-tones, or thirty-one equal intervals; as well as “extended and compressed microtonal scales” with forty-three, fifty-three, sixty, seventy-two, or more equal or unequal intervals in the octave.7 J. Murray Barbour on the other hand pronounces Pythagorean (“excellent for melody, unsatisfactory for harmony”), just intonation (“better for harmony than for melody”), meantone (“a practical substitute for just intonation, with usable triads all equally distorted”), and equal temperament (“good for melody, excellent for chromatic harmony”) as the “four leading tuning systems,” or the “Big Four.” Barbour also makes mention of the “more than twenty varieties of just intonation,” and “six to eight varieties of the meantone temperament,” as well as the “geometric, mechanical, and linear divisions of the line” for the mathematical approximation of equal temperament.8 According to Barbour, tuning systems may be classified into two distinct classes: the first being ‘regular’, where all fifths but one are equal in size; and the second, ‘irregular’, where more than one fifth is unequal in size. The former includes Pythagorean, meantone, and equal temperament, while the latter (as classified by Barbour) excludes just intonation.9

The pitch discrimination threshold for an average adult is around 3Hz at 435Hz, which is approximately one seventeenth of an equal tone, or 11.899 cents, although a “very sensitive ear can hear as small a difference as 0.5Hz or less” (approximately a hundredth of a tone, or 1.989 cents). Tests conducted in 1908 by Norbert Stücker (

“The Just Noticeable Difference (JND) for frequency is the smallest change in frequency that a listener can detect. Careful testing such as that of E. Zwicker and H. Fastl (

The three microtonal works discussed in the thesis include

The aim of the dissertation is to present an articulated exposition of three ‘original’ and unique microtonal composition models individually exploring the expanded tonal resources of Pythagorean intonation, equal temperament, and just intonation. It is also proposed that the thesis outlines their theoretical and aesthetic rationale, as well as their historical foundations, with mathematical analysis of relevant tuning systems, and spectrum analysis of recordings providing further substance to the project. Theory versus performance is also taken into account, and the collaboration with an actual performer is intended to deliver the corporeal perspective. It is anticipated that the thesis will not represent current acoustic and psychoacoustic research at any great depth, and therefore should not be seen to serve as a comprehensive study of physics and music. It will nevertheless provide a foundation for the exploration of tuning systems, and additionally, present a composer’s perspective – as opposed to a musicological or ethnomusicological study – of microtonal music composition.

The work,16

In view of the fact that stretched, as well as compressed octaves are a common occurrence in Piraglu’s tuning of the

Chapter two (the equally-tempered archetype) begins with a discussion about Partch’s notion of two distinct classes of equal temperaments: those that produce equal third-tones, quarter-tones, fifth-tones, sixth-tones, eighth-tones, twelfth-tones, and sixteenth-tones; as opposed to those that divide the octave into nineteen, thirty-one, forty-three, and fifty-three equally-tempered intervals.18 This is followed by a brief history of some important studies of the equally-tempered paradigm, namely by Julián Carrillo Trujillo, Ferruccio Busoni, Ramon Fuller, and Easley Blackwood, with the latter two serving as benchmarks for the establishment of the criteria to properly assess the musical virtues of a particular equal temperament. The deviation of basic equally-tempered intervals from just intonation, Fuller’s eight best equal temperaments, and Blackwood’s concept of ‘recognizable diatonic tunings’ are then discussed. Nicolas Mercator’s fifty-three-tone equally-tempered division of the octave, which is Fuller’s recommendation for a temperament with the capacity to approximate just intervals, is consequently presented, along with an opposing view by Dirk de Klerk.

In order to illustrate the principal evolutionary markers leading up to the adoption of equal temperament in the West – from Pythagorean intonation, meantone and well temperament, to equal temperament – Pietro Aron’s quarter-comma meantone temperament is introduced, as well as Joseph Sauveur’s forty-three-tone equal temperament, which approximates fifth-comma meantone temperament. The origins of equal temperament are then traced back to 1584 China, and Prince Chu Tsai-yü’s monochord. What follows is a discussion of the geometrical and numerical approximations of Marin Mersenne and Simon Stevin, which culminate in Johann Faulhaber’s monochord, and the first printed numerical solution to equal temperament based on the theory of logarithmic computation.19 The mathematical formula for twelve-tone equal temperament, the equally-tempered monochord, and beating characteristics of the twelve-tone equally-tempered major and minor triads are then sequentially presented, which are followed by the equal thirds, sixths, fifths, and fourths in piano tuning.

The work,

Chapter three (the harmonic consideration) begins with a basic outline of just intonation and ‘extended just intonation’, or the incorporation of partials beyond the sixth harmonic.20 A historical and scientific perspective of the harmonic series is then presented, together with examples of the beating characteristics of the first eight partials of the harmonic series, as well as of the mistuned and properly tuned unison, and mistuned and properly tuned octave. Dissonance, with special reference to the theory of beats, is defined according to James Tenney, Helmholtz, Bosanquet, and Johnston. The complement or mirror image of the harmonics series, or the ‘subharmonic series’, is also discussed, together with Partch’s theory of ‘otonalities’ (pitches derived from the ascending series) and ‘utonalities’ (pitches derived from the descending series).21 A comparative table of intonation then provides interval, ratio, and cents data for the twelve basic intervals of just intonation, Pythagorean intonation, meantone temperament, and equal temperament.

In order to illustrate the basic principles of proportions and string lengths, the traditional structure and function of the monochord is explained, with the generation of simple octaves and fifths utilized to demonstrate the theoretical basis for the Pythagorean monochord. A table depicting all the intervals of the harmonic series from the first partial through to the one-hundred-and-twenty-eighth partial is then presented. Combinational tones, or differential and summation tones, are also subsequently explained, together with their implications on the intervals of the octave, just perfect fifth, just perfect fourth, just major third, just minor sixth, just minor third, and just major sixth. This is followed by a discussion of periodicity pitch, and its theoretical significance in relation to JND, or Just Noticeable Difference. The relationship of prime numbers, primary intervals, and prime limits to just intonation principles is subsequently explained.

The concept of just intonation is then illustrated via the construction of a seven-note just diatonic scale, and the presentation of the beating characteristics of the just major triad. This is followed by the construction of a twenty-five-note just enharmonic scale, and its development into Johnston’s fifty-three-tone just intonation scale. Harry Partch’s forty-three-tone just intonation scale, and his rationale for the consequential harmonic expansion to eleven-limit is then explained. The twenty unique triads, fifteen unique tetrads, and six unique pentads made possible via the inclusion of the eleven-limit intervals are additionally presented. The final octave division discussed in the chapter is Adriaan Daniël Fokker’s thirty-one-tone equally-tempered division of the octave, and in view of its capability to approximate the tonal resources of seven-limit just intonation.

The work,

Intervals based on Pythagorean intonation have been simply named according to their cyclical position, and therefore follow an either ascending 3/2 incremental progression from natural, sharp, double sharp, to triple sharp; or a descending 4/3 incremental progression from natural, flat, double flat, to triple flat. The procedure is exemplified via the twenty-seven-note Pythagorean scale, which incorporates fifteen intervals generated by an ascending series of fifths, or the pitches C, G, D, A, E, B, Fb, Cb, Gb, Db, Ab, Eb, Bb, Fbb, Cbb, and Gbb; and another eleven intervals, by a descending series, or C, F, B#, E#, A#, D#, G#, C#, F#, B##, E##, and A##. The method adopted in equal temperament on the other hand is a nomenclature based on the comma approximations to Daniélou’s ‘scale of proportions’, or sixty-six-note just intonation scale, with every interval not characterized by the equal semitones and quarter-tones of 12-et and 24-et further indentified via its origin (for example: 5-et supermajor second, 7-et grave or small tone, and 9-et great limma, or large half-tone). Exceptions to this rule include 31-et, 43-et, and 53-et, which because are not discussed in the thesis with relation to other intervals, do not require a differential prefix with the same conditions. Adriaan Daniël Fokker’s thirty-one-tone equally-tempered division of the octave introduces a further element to intervallic nomenclature. The system, which was developed by David C. Keenan, involves the prefixes: double diminished, subdiminished, diminished, sub, perfect, super, augmented, superaugmented, and double augmented for unisons, fourths, fifths, and octaves; while subdiminished, diminished, subminor, minor, neutral, major, supermajor, augmented, and superaugmented for seconds, thirds, sixths, sevenths, and ninths. Perfect and major, or “the ones implied when there is no prefix,” represent the central position of a range based on comma or diesis increments from -4 to +4 (for example: diminished third, subminor third, minor third, neutral third, major third, supermajor third, and augmented third).22 For intervals beyond five-limit intonation, James B. Peterson’s recommendations for the naming of bases has been adopted, which results in the following additional prefixes for seven-, eleven-, thirteen-, seventeen-, nineteen-, twenty-three-, twenty-nine-, and thirty-one-limit: septimal, undecimal, tridecimal, septendecimal, nonadecimal, trivigesimal, nonavigesimal, and untrigesimal (for example: septimal superfifth, undecimal subfifth, tridecimal subfifth, septendecimal superfifth, nonadecimal superfifth, trivigesimal superfifth, nonavigesimal subfifth, and untrigesimal superfifth).23 The classification of 724 unique intervals incorporated into the comparative table of musical intervals (see Comparative Table of Musical Intervals) includes all the intervals cited in the current study.

The notation symbols utilized in the thesis include the five standard accidental signs of Western music; four common quarter-tone and three-quarter-tone symbols; twenty-three unique symbols based on Daniélou’s division of the whole-tone; Ali Naqi Vaziri’s notation system, or four accidentals of Persian music; Johnston’s system of notation, which contains twenty-three unique symbols for the notation of just intonation up to the thirty-first harmonic; as well as Fokker’s nine symbols for the notation of thirty-one equal temperament. All these symbols have been incorporated into a 211-character microtonal notation PostScript Type 1 font (see Microtonal Notation Font), which was created via the modification of a selection of symbols in the Coda Music Finale’s Maestro font utilizing CorelDraw 13.0 and FontMonger 1.0.8.

For further information on Chapters 1 (Theory Versus Performance Practice: Azadeh for Santur and Tape), 2 (The Equally-Tempered Archetype: Exposiciones for Sampled Microtonal Schoenhut Toy Piano) and 3 (The Harmonic Consideration: La Homa Kanto for Harmonically Tuned Synthesizer Quartet), as well as volumes 2 and 3 (Folio of Compositions), please do not hesitate to contact me.

There are two versions of the microtonal font, with the first, ‘Microtonal Font’, intended for use in Finale, while the second, ‘Microtonal Font Text’, intended for use in Microsoft Word. The ‘Pertout PhD2007 - AppendixB.pdf’ file will provide a graphic display of all the available characters with the appropriate ANSI character shortcut key.

Finale Instructions: click [key signature tool], double click the first measure, select [Nonstandard Key Signature], click [Special Key Signature Attributes], click [Symbol Font], select the "Microtonal Notation" Font, click [Symbol List ID], then specify each microtonal value. For example, the following procedure will result in symbols allowing a pitch to be [1] raised by 25/24, or one small just chromatic semitone (70.672 cents), [2] lowered by 25/24, or one small just chromatic semitone (70.672 cents), [3] raised or lowered to natural tone, [4] raised by 36/35, or one septimal comma (48.770 cents), and [5] lowered by 36/35, or one septimal comma (48.770 cents):

List Element 1 (Alter Amount: 1, Characters: Alt+033, click [Insert], click [Next]; List Element 2 (Alter Amount: -1, Characters: Alt+034, click [Insert], click [Next]; List Element 3 (Alter Amount: 0, Characters: Alt+039, click [Insert], click [Next]; List Element 4 (Alter Amount: 2, Characters: Alt+042, click [Insert], click [Next]; List Element 5 (Alter Amount: -2, Characters: Alt+043, click [Insert], click [OK] x 3.

In the simple entry pallete utilize the "half step up" (click a note to raise its pitch by a half step) and "half step down" (click a note to lower its pitch by a half step) buttons to navigate between accidentals. Take note that Finale 2007 allows for up to 15 elements, and therefore essentially allowing for 7 positive and 7 negative values, as well as a neutral value to activate the natural sign.

All ‘Microtonal Notation Font’ materials in zip archive fomat: Microtonal Notation Font by Andrián Pertout.

Note: The .zip file includes True Type and PostScript Type 1 (PC and Mac versions) of my ‘Microtonal Font’, intended for use in Finale and ‘Microtonal Font Text’, intended for use in Microsoft Word, as well as my ‘Comparative Table of Musical Intervals’ PDF (classification of 724 unique intervals incorporated into the comparative table of musical intervals) and ‘Microtonal Notation Font’ PDF (all the available characters with the appropriate ANSI character shortcut key).

1 Douglas Keislar, ”Introduction,” *Perspectives of New Music *29.1 (Winter, 1991): 173.

2 Lydia Ayers, “Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional Applications,” (DMA diss., U. of Illinois, Urbana-Champaign, 1994, PA 9512292) 1-2.

3 Joe Monzo, “Encyclopedia of Microtonal Music Theory,” *Microtonal, Just Intonation Electronic Music Software*, 2005, Tonalsoft, 17 Nov. 2006, <http://www.tonalsoft.com/>.

4 Accounts of the *arcicembalo* (a two-manual harpsichord capable of producing thirty-six distinct pitches per octave) and *arciorgano* (organ adaptation) were presented by Nicola Vicentino in his treatises *L’antica musica ridotta a la moderna prattica* of 1555 and *Descrizione dell’ arciorgano* (1561). For a further discussion, see Don Michael Randel, ed., *The New Harvard Dictionary of Music* (Cambridge, Mass.: Belknap Press of Harvard U Press, 1986) 47.

5 John H. Chalmers, *Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales* (Hanover, NH: Frog Peak Music, 1993) 1-2.

8 J. Murray Barbour, “Irregular Systems of Temperament,” *Journal of the American Musicological Society* 1.3 (Autumn, 1948): 20.

9 J. Murray Barbour, *Tuning and Temperament: A Historical Survey* (New York: Dover Publications, 2004) x-xi

11 Harry F. Olson, *Music, Physics and Engineering*, 2nd ed. (New York: Dover Publications, 1967) 123.

12 “Pitch discrimination is measured by sounding two pure tones in quick succession and gradually reducing the difference in frequency until the observer is unable to tell which of the two tones is higher. The steps usually employed in such a series are 30, 23, 17, 12, 8, 5, 3, 2, 1, and 0.5Hz, at the level of international (standard) pitch.” For a further discussion, see Seashore, *Psychology of Music* 56-57.

13 William A. Sethares, *Tuning, Timbre, Spectrum, Scale*, 2nd ed. (London: Springer-Verlag, 2005) 44.

15 Alain Daniélou, *Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness* (Rochester, VT: Inner Traditions, 1995) 35-37.

17 Lloyd, and Boyle, *Intervals, Scales and Temperaments: An Introduction to the Study of Musical Intonation *166-67.

18 Harry Partch, *Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfilments, *2nd ed. (New York: Da Capo, 1974) 425.

21 David D. Doty, *The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation*, 3rd ed. (San Francisco: Other Music, 2002) 28-30.

22 David C. Keenan, “A Note on the Naming of Musical Intervals,” *David Keenan’s Home Page*, 3 Nov. 2001, 22 Nov. 2006, <http://users.bigpond.net.au/d.keenan/Music/IntervalNaming.htm>.

23 James B. Peterson, “Names of Bases,” *The Math Forum: Ask Dr. Math*, 15 Apr. 2002, Drexel U., Philadelphia, PA, 22 Nov. 2006, <http://mathforum.org/library/drmath/view/60405.html>.