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

Microtonality

In a Perspectives of New Music article, Douglas Keislar states that the term microtonality “conjures up images of impossibly minute intervals, daunting instruments with hundreds of notes per octave, and wildly impractical performance instructions,” but that “such difficulties in fact characterize only a small percentage of the music that uses tunings other than standard twelve-note equal temperament.”  Keislar then suggests that American composer Ivor Darreg’s proposal of the Greek term ‘xenharmonic’ or ‘unfamiliar modes’ is perhaps better suited to music utilizing “radically different tunings.”1  Alternative language for the term ‘microtonal’ is presented by Lydia Ayers in Exploring Microtonal Tunings: A Kaleidoscope of Extended Just Tunings and their Compositional Applications, with the following list of  expressions: “tuning; microintervals; macrointervals or macrotones, such as 5-tone, 7-tone, and 10-tone equal temperaments; omnitonal; omnisonics; neoharmonic; xenharmonic; ‘exploring the sonic spectrum’; and non-twelve.”  Although in spite of Ayers’s general attraction to the broadness of ‘omnitonal’, ‘microtonal’ is nevertheless espoused for its universality.2

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 Encyclopedia of Microtonal Music Theory, Joe Monzo provides the following discussion about the etymology of ‘microtonal’

“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 New Harvard Dictionary of Music:

“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 Prométhée enchâiné (1847).  The first microtonal piece to use Western instrumental forms is a string quartet by John Foulds (1897); and the earliest known published quarter-tone composition, Richard Stein’s Zwei Konzertstücke, op. 26 (1906), is for cello and piano.”6

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

Pitch Audibility and Discrimination

Although it may be stated that the human ear has a general capacity to hear frequencies between the ranges of 16Hz and 16,000Hz (equal to 16 to 16,000 cycles per seconds, and approximately C0 and B9), it must be noted that numerous factors influence the actual outcomes.  The 16Hz lower limit is dependent on two principal factors, being wave intensity and shape; with the inclusion and exclusion of pure tones displacing the figures for the lower limit to anywhere between 12Hz and 100Hz (approximately G-0 and G2).  The 16,000Hz upper limit is generally reserved for a healthy population under the age of forty, with adolescent capacity as high as 25,000Hz (approximately G10); a supposed ‘normal hearing’ population in some cases not surpassing a 5,000Hz (approximately D#8) upper limit; and another probable large percentage incapable of hearing beyond 10,000Hz (approximately D#9).10  The frequency range of the 88-key pianoforte is between 27.5Hz and 4,186Hz, or A0 to C8, and therefore encompasses pitch material with a range of over seven octaves.  The seven-octave range additionally represents the range embodied within the collection of instruments that constitute the traditional symphony orchestra.11

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 (Zeitschrift für Sinnesphysiologie 42: 392-408) of sixteen professional musicians in the Viennese Royal Opera conclude a pitch discrimination threshold between one five-hundred-and-fortieth (0.1Hz) and one forty-ninth of a tone (1.1Hz), or 0.370 and 4.082 cents, with an average of 0.556Hz (approximately a hundredth of a tone), or 2.060 cents.12  In Tuning, Timbre, Spectrum, Scale William A. Sethares adds the following to the discussion:

“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 (Psychoacoustics, Springer-Verlag, Berlin [1990]) has shown that the JND can be as small as two or three cents, although actual abilities vary with frequency, duration and intensity of the tones, training of the listener, and the way in which JND is measured.”13

Three Microtonal Compositions

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.

The three microtonal works discussed in the thesis include Azadeh for santur and tape, no 389 (2004, Rev. 2005) – composed for Iranian santurist Qmars Piraglu (formerly Siamak Noory) – which features the Persian santur (72-string box zither), and serves as a practical study of Persian tuning systems, with its presentation of both ‘theoretical’ and ‘performance practice’ tunings; an ‘acousmatic’ work entitled Exposiciones for sampled microtonal Schoenhut toy piano, no. 392 (2005), which attempts to explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut model 6625, 25-key toy piano, a complex polyrhythmic scheme, and sequential tuning modulations featuring the first twenty-four equally-tempered divisions of the octave; and La Homa Kanto (or ‘The Human Song’ in Esperanto) for harmonically tuned synthesizer quartet, which derives its pitch material from Lou Harrison’s five-tone scales (presented in Lou Harrison’s Music Primer: Various Items About Music to 1970) and features ten distinct tuning modulations: 3-limit through to 31-limit just intonation systems based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth, twenty-third, twenty-ninth, and thirty-first partials of the harmonic series.

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.

Folio of Compositions

Other works incorporated into volume two and three of ‘Folio of Compositions 2003-07’ include: Symétrie intégrante for Flute, Organ and Electronics, no. 394 (2005-06); Aequilibrium for flute, clarinet, viola, cello, trombone, piano and percussion, no. 395 (2006); Tres imágenes norteñas for shakuhachi and harpsichord, no. 396 (2006); L’assaut sur la raison for symphony orchestra, no. 386 (2003); Digressioni modali for tenor saxophone and pianoforte, no. 387 (2003); La flor en la colina for flute, clarinet, violin, violoncello and pianoforte, no. 388 (2003-04); Bénédiction d'un conquérant for symphony orchestra, no. 390 (2004); and Zambalogy for harp, no. 391 (2004).  These works do not represent the microtonal models of the first three compositions, yet certainly adhere to an exploration of alternative scalar and harmonic materials, and their application in contemporary compositional practice.  Pitch material for these works has been generated via a selection of methods such as multi-octave grouping (pitch material based on multi-octave scales constructed of dissimilar tetrachords), modality (modes generated by the major, in, hirajoshi and kumoijoshi scales), aleatoric formation (pitch material generated via indeterminate means), pitch class set theory (pitch material derived from the 208 basic pitch-class sets of set theory), synthetic symmetry (hexatonic and octatonic major and minor scales), cluster generation (pitch material derived from five-note chords and inversions), physical and psychological concepts of consonance and dissonance (the harmonic language of the twelve primary intervals), polymodal and polytonal juxtaposition (multiple scales and tonalities), as well as cross-cultural abstraction (non-Western music theoretical concepts).

Methodology

Chapter one (theory versus performance practice) begins with a brief history of Persian music, and is followed by the presentation of Safi al-Din Urmawi’s seventeen-note gamut and division of the whole-tone, and an explanation of the significance of the tetrachord in the construction of melodic and harmonic structures.  A discussion of Persian musical scholarship in the twentieth century then introduces the three separate theories on intervals and scales of Persian music proposed in the twentieth century: the twenty-four equally-tempered quarter-tone scale proposed by Ali Naqi Vaziri in the 1920s, the alternative twenty-two-note scale proposed by Mehdi Barkešli in the 1940s based on Pythagorean principles, as well as the theory of the five primary intervals of performance practice presented by Hormoz Farhat in the 1990 publication of his doctoral thesis The Dastgah Concept in Persian Music.14  The division of the octave into twenty-four equally-tempered quarter-tones is given a historical perspective, as well as a mathematical exposition, while the concept of Pythagorean intonation is firstly illustrated via the construction of a twenty-seven-note Pythagorean scale with the necessary intervals to facilitate the general modulations of Western tonal music; and secondly, via Daniélou’s ascending ‘scale of fifths’, or cyclic division of the octave, which presents a series of fifty-nine consecutive fifths, or sixty lü.  The BCE Chinese origins of Pythagoreanism and its philosophical significance according to theorist King Fâng are also subsequently discussed.15  The development of the seventeen-note gamut by Mehdi Barkešli into a twenty-two-note Pythagorean scale is then presented, which is followed by Farhat’s theory of flexible intervals, or of the five primary intervals of performance practice – advocated by Farhat in opposition to both twenty-four-tone equally-tempered, and twenty-two-note Pythagorean scales of Vaziri and Barkešli.

The work,16 Azadeh for santur and tape, is then introduced, together with a brief biography of the artist, Qmars Piraglu; a description of instrument, the Persian santur (a 72-string [or 18 quadruple-stringed] box zither); and a discussion of the essence of the Persian modal system.  Following the establishment of the tuning analysis protocols, a detailed exposition of the tuning process of the santur for dastgah-e segah (on F) is presented.  Spectrum analysis results collected on three separate occasions (with a periodicity of 3-6 months) for each of the twenty-seven sets of strings are then analyzed with regards to the intervallic size of octaves, perfect fifths, perfect fourths, tempered perfect fourths, and neutral thirds.  An analysis of variance is then conducted with the data collected, which in turn produces mean measurements with the capacity to characterize tuning characteristics.  A tuning system comparison then concludes a relationship between Farhat’s and Piraglu’s division of the whole-tone, with Farhat’s theory of flexible intervals accorded as the most plausible hypothesis.

In view of the fact that stretched, as well as compressed octaves are a common occurrence in Piraglu’s tuning of the santur, the theory of the ‘piano tuner’s octave’ is discussed, along with the natural phenomenon of inharmonicity – a factor especially affecting plucked and struck strings (along with other musical sounds with a short decay).17  A comparison is also made between the tuning of a triple-string unison of a piano and a quadruple-string unison of a santur.  Climate and its effects on tuning are then considered, and especially in order to substantiate Piraglu’s claims of the climatic conditions of Melbourne, Australia being “unsatisfactory” for the tuning of the santur in comparison to Tehran, Iran.  The twenty-four gušes for dastgah-e segah according to a prominent radif associated with Musa Marufi are then presented, followed by the pitch organization of the adopted six most prominent elements of the radif of dastgah-e segah.  Finally, the structural scheme of the work and its basis on ‘golden mean’ proportions are explained, as well as the sampling process of the santur and vocals, and digital processing that culminates in the tape element of Azadeh for santur and tape.

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, Exposiciones for Sampled Microtonal Schoenhut Toy Piano, is then introduced, together with a brief history of the toy piano, the Schoenhut toy piano sample, as well as concepts of sound diffusion and polyrhythmic theory utilized in the composition.  In order to illustrate the design of the proposed notation for the twenty-four equal temperaments, Daniélou’s ‘scale of proportions’, or harmonic division of the octave, which presents a series of sixty-six unique intervals is introduced.  Paul Rapoport’s Pythagorean notation then provides an alternative to the system of notation based on Daniélou’s subdivision of the whole-tone.  Sléndro and pélog scales are then discussed from a historical perspective, with the gamelan gedhé sléndro and pélog tunings from Sri Wedhari theatre auditorium in Solo, Central Java serving as the ‘performance practice’ model.  The harmonic characteristics of the sléndro and pélog scales are then presented in accordance to five-limit intonation principles.  What follows is a systematic exposition of the compositional application of each equal temperament between one and twenty-four.

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, La Homa Kanto for Harmonically Tuned Synthesizer Quartet, is then introduced, together with a presentation of Harrison’s five pentatonic scales, which serve as the pitch material, the ‘1967 William Dowd French Double Harpsichord’ sample, and Johnston’s system of notation, which serves as the system of notation utilized in the score.  Compositional strategy is then discussed, together with Harrison’s concept of composing with melodicles, or neumes, which is adopted and developed into a system incorporating three categories of motivic manipulation: melodic transformation of motive, rhythmic transformation of motive, and harmonic transformation of motive.  What follows is a systematic exposition of the compositional application of each just intonation limit between three and thirty-one.

Interval Nomenclature and Notation System

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.

Andrián Pertout, 2007

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.


Microtonal Font

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.


Downloads

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).

Azadeh in PDF format for santur and tape – composed for Iranian santurist Qmars Piraglu (formerly Siamak Noory) – which features the Persian santur (72-string box zither), and serves as a practical study of Persian tuning systems, with its presentation of both ‘theoretical’ and ‘performance practice’ tunings.

Azadeh for santur and tape, no. 389 - 11900K MP3 [13:01]

Exposiciones in PDF format for sampled microtonal Schoenhut toy piano, which attempts to explore the equally-tempered sound world within the context of a sampled microtonal Schoenhut model 6625, 25-key toy piano, a complex polyrhythmic scheme, and sequential tuning modulations featuring the first twenty-four equally-tempered divisions of the octave

Exposiciones for sampled microtonal Schoenhut toy piano, no. 392 - 6410K MP3 [7:00]

La Homa Kanto in PDF format (or ‘The Human Song’ in Esperanto) for harmonically tuned synthesizer quartet, which derives its pitch material from Lou Harrison’s five-tone scales (presented in Lou Harrison’s Music Primer: Various Items About Music to 1970) and features ten distinct tuning modulations: 3-limit through to 31-limit just intonation systems based on the third, fifth, seventh, eleventh, thirteenth, seventeenth, nineteenth, twenty-third, twenty-ninth, and thirty-first partials of the harmonic series (MP3 download).

La Homa Kanto for harmonically tuned synthesizer quartet, no. 393 - 10800K MP3 [11:50]

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