An Application of Calculated Consonance in Computer-Assisted Microtonal Music

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2013-12-23 An Application of Calculated Consonance in Computer-Assisted Microtonal Music

Burleigh, Ian George

Burleigh, I. G. (2013). An Application of Calculated Consonance in Computer-Assisted Microtonal Music (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/24833 http://hdl.handle.net/11023/1225 doctoral thesis

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An Application of Calculated Consonance in Computer-Assisted Microtonal Music

Ian George Burleigh

A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE INTERDISCIPLINARY DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF COMPUTER SCIENCE and SCHOOL OF CREATIVE AND PERFORMING ARTS — MUSIC CALGARY, ALBERTA DECEMBER, 2013

Harmony (the audible result of varied combinations of simultaneously sounding tones) ought to, for the most part, sound pleasing to the ear. The result depends, among other factors, on a proper choice of the pitches for the tones that form harmonious chords, and on their correct intonation during musical performance.

This thesis proposes a computational method for calculation of relative consonance among groups of tones, and its possible practical applications in machine-assisted arrangement of tones, namely the choice of tone pitches and their microtonal adjustment. The consonance of tone groups is calculated using a model that is based on the physiological theory of tone consonance that was published by Hermann Helmholtz in the middle of the 19th century.

Given a group of tones that have ﬁxed pitches, changes in the aggregate dissonance caused by adding another “probe” tone of a variable pitch can be represented as a “dissonance landscape”. Local minima in the “height” of the landscape correspond to local minima of the aggregate dissonance as a function of the pitch of the probe tone. Finding a local dissonance minimum simulates the actions of a musician who is “tuning by ear”. The set of all local minima within a given pitch range is a collection of potentially good pitch choices from which a composer (a human, or an algorithmic process) can fashion melodies that sound in harmony with the ﬁxed tones.

Several practical examples, realized in an experimental software, demonstrate applications of the method for: 1) computer-assisted microtonal tone arrangement (music composition), 2) algorithmic (machine-generated) music, and 3) musical interplay between a human and a machine.

The just intonation aspect of the tuning method naturally leads to more than twelve, potentially to many, pitches in an octave. Without some restrictions that limit the complexity of the process, handling of so many possibilities by a human composer and their precise rendi- tion as sound by a performing musician would be very diﬃcult. Restricting the continuum of possible pitches to the discrete 53-division of the octave, and employing machine-assistance in their arrangement and in sound synthesis make applications of the method feasible.

ii Preface

There was Eru, the One, who in Arda is called Ilúvatar;and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made. And he spoke to them, propounding to them themes of music.

And it came to pass that he declared to them a mighty theme, unfolding to them things greater and more wonderful that he had yet revealed: “Of the theme that I have declared to you, I will now that ye make in harmony together a Great Music. Ye shall show forth your powers in adorning this theme, each with his own thoughts and devices.”

And a sound arose of endless interchanging melodies woven in a harmony that passed beyond hearing into the depths and into the heights, and the music and the echo of the music went out into the Void, and it was not void.

J.R.R.Tolkien, Ainulindal¨e(The Music of the Ainur), The Silmarillion, 1977.

Music is made from melodious strands of tones (voices) that are “woven in a harmony” into greater patterns and structures. The weaving and interplay of melodic strands (counterpoint) bring into existence chords (combinations of simultaneously sounding tones) and thus harmony. The composer may deal with all the voices as equally important, or choose one voice as the primary one that carries the main melody and make the rest to function as an accompaniment. In any case, the proper combination of tones that sound together is a most essential factor in making music.1

Weaving of musical patterns is not unlike solving a combinatorial puzzle: a task to organize a large number of items so that they ﬁt well together. There are many possible combinations of the items; most of the combinations have to be outright rejected since they do not lead to any potential solution. Somewhere among the remaining combinations may lie possible answers; there could be several or perhaps many of those. Some may be but acceptable, some may be good, and occasionally there are some that are exquisite.

1“Harmony”. Oxford Music Online.

iii A common musical puzzle is the problem of creating a melody that fits a given harmonic progression. (This text proposes a computational method that can be used to assist in solving such puzzles.) A famous example is Charles Gounod’s “Ave Maria”,2 superposed over J.S. Bach’s Prelude in C.3 The beautifully clear structure of the Prelude with a mobile, yet firmly grounded harmonic progression makes it a great foundational material for composition of musical variations. In Chapter 4 we shall discuss several melodies and other musical structures that were fitted over the chord progression of the Prelude, to demonstrate the feasibility of our method and the practicability of the experimental software in which the method has been implemented.

Weaving musical patterns is a craft and an art, practised mainly by music composers and performing musicians. Music composers create musical compositions (that include solutions of various musical puzzles) and write them down using musical notation in the form of musical scores. Performing musicians interpret the scores, in a live performance or during a studio recording, turning them into sound. It takes talent and many years of training for both music composers and performing musicians to acquire necessary skills and insight. The composer and the musician can be the same person: some top performers of classical music and many songwriters of popular music write and perform their own compositions and songs. Improvising musicians, in particular the players of Renaissance, Baroque, Eastern, and jazz music, invent musical themes (or create them from prepared and practised melodic fragments) at the time of their performance.

Fitting numerous composed or improvised (that is, invented on the spot in “real” time) melodies over an existing harmonic progression is one of the essential aspects of jazz music. Jazz musicians do often borrow chord progressions from other sources, but almost always create their own original melodic themes. Some of such newly created themes even have become a part of the “jazz songbook”, a corpus of “standard” jazz songs. For example, Miles Davis’s song “Donna Lee”4 is a bebop melody written over the more traditional harmonic progression of the song “Back Home Again in Indiana”.5 The chord progression of George Gershwin’s song “I Got Rhythm”6 became known as “rhythm changes”7 and was used in

2Méditationsur le premier préludede S. Bach, 1853. 3Prelude and Fugue No.1 in C major, BWV 846. 4“Donna Lee (1947)”. Wilson et al., JazzStandards.com — the first and only centralized information source for the songs and instrumentals jazz musicians play most frequently. 5“Indiana (Back Home Again in Indiana) (1917)”. ibid. 6“I Got Rhythm (1930)”. ibid. 7“Jazz Theory: Rhythm Changes”. ibid.

iv so many jazz songs, that it is very frequently used as one of the most common background accompaniments for improvisation at jam sessions.8 Chapter 5 describes an experimental real-time system for a musical interplay of a human and a machine, as if they were two improvising jazz musicians.

Music theorists analyze existing compositions, attempt to discover inner principles and rules that guided their weaving, and then construct theories that attempt to explain how the compositions were built and why their tone patterns do (or do not) sound in harmony. Many musical works can be then subsequently created following the rules of a given music theory, within the limits of the musical genre for which the theory was constructed. Great composers, however, often in their work stretch or even break the boundaries of established theories and thus advance musical practice.

And then there are engineers. Engineers may not have the creative talent or feelings of artists, nor the insight of music theorists; but engineers recognize structured patterns and with that ability they build machines. Computers are the ultimately ﬂexible and adaptable machines brought to action by computer programs, machines that can with remarkable success assist composers and musicians to solve musical puzzles, write musical scores, and render them as sound. This text describes one such attempt to build a suite of software applications that assist composers in creating music with higher level of tuning complexity and precision than otherwise possible. The approach comes not from emulating established musical practices, but from following natural (acoustical) principles of musical sound. A particular attention is given to presenting an argument for the choice of just intonation, and to the musical consequences that result from it.

8Gatherings where jazz musicians play “jazz standards” and take long improvised “solos”.

v To the fellowship of the upper room and my favourite Martians

vi Table of Contents

Abstract ii

Preface iii

Dedication vi

Table of Contents vii

List of Tables ix

List of Figures x

Deﬁnition of terms xii

Online materials xvi

1 Introduction and background 1 Music and machines ...... 2 Computer-assisted music ...... 3 Working deﬁnitions ...... 5 Western music ...... 9 Music theory rudiments ...... 10 Music notation ...... 14 Division of the octave ...... 17 The twelve-tone division of the octave ...... 19 Just intonation ...... 20 Computer-assisted microtonal music ...... 22 Experiments and experimental tools ...... 22 The outline of the work ...... 25

2 Tone sensations 27 The pitch spiral ...... 27 Sonograms ...... 30 Reassigned sonograms ...... 33 The harmonic series ...... 34 Harmonics of the singing voice ...... 37 The intervals of the harmonic series ...... 46 Consonance and dissonance, redeﬁned ...... 49 Helmholtz’s Tonempﬁndungen ...... 50 Experiments ...... 52

3 Octave divisions 61

vii Octave equivalence ...... 61 Division of the octave by cycles of ﬁfths ...... 62 Temperaments — how do they sound? ...... 69 Pythagorean (the ﬁfth) tuning ...... 70 Just tuning ...... 71 The 53-tone microtonal division of the octave ...... 75 Microtonal notation of the 53-tet system ...... 78

4 Calculated consonance and its applications 85 Dissonance curves and landscapes ...... 85 Dissonance landscape displayed on the pitch spiral ...... 87 Sequencing music in 53-tone division ...... 90 The C53 Suite: Variations on Prelude in C, BWV 846 ...... 94

5 Related work: interactive music 107 Qvin Musical Processes ...... 108

6 Conclusions and future work 116 Future work ...... 117

A C53 Suite 119

B The pitch spiral 129

C Included media and software 133

Glossary of terms 134

Bibliography 136

viii List of Tables

1.1 The eight musical intervals ...... 12 1.2 Perfect intervals ...... 13 1.3 Imperfect intervals ...... 13 1.4 Intervals and their inversions ...... 14

2.1 Intervals of the harmonic series ...... 36

3.1 Equal divisions of the octave ...... 81 3.2 Just tuning and 12-tet approximated in 53-tet ...... 82 3.3 The 53-tone equal division of the octave ...... 83 3.4 The 53-tone notation ...... 84

ix List of Figures

1.1 Piano keyboard ...... 11 1.2 C major scale ...... 11 1.3 Enharmonically equivalent altered notes...... 12 1.4 Prelude in C, opening measures ...... 16

2.1 The pitch spiral ...... 28 2.2 Source and diﬀerence tones on the pitch spiral ...... 29 2.3 Various grids on the pitch spiral ...... 30 2.4 Sonograms of a whole-tone scale ...... 32 2.5 The whole-tone scale in Qvorum ...... 34 2.6 Harmonics of a complex tone, shown on the pitch spiral ...... 38 2.7 Tuˇzba(Desire) ...... 39 2.8 A sonogram of the singing voice, linear frequency scale ...... 39 2.9 A sonogram of the singing voice, logarithmic frequency scale ...... 40 2.10 Time-frequency reassigned sonogram of Example 2.2 ...... 41 2.11 Example 2.2, strong harmonics only ...... 42 2.12 Example 2.2, 12-tet grid ...... 42 2.13 Example 2.2, fundamental harmonics ...... 43 2.14 Example 2.2, the ﬁrst phrase ...... 43 2.15 Jabluˇnka (Apple Tree) ...... 44 2.16 A sonogram of a violin/voice duet...... 44 2.17 Two shades of B4 ...... 45 2.18 Jabluˇnka — microtonal notation ...... 45 2.19 Example 2.6A ...... 57 2.20 Example 2.6B ...... 58

3.1 Octave equivalence: C2, C3, C4, and C5 ...... 61 3.2 Octave equivalence with difference tones ...... 62 3.3 The cycle of fifths with moments of symmetry ...... 63 3.4 The Pythagorean comma ...... 65 3.5 Tempering out the Pythagorean comma ...... 66 3.6 Just and equally tempered fifths ...... 67 3.7 Pythagorean, just and equally tempered thirds ...... 67 3.8 Tempering out the syntonic comma ...... 68 3.9 The meantone temperament (unequal semitones) ...... 68 3.10 Just C chromatic scale on a tuning lattice ...... 72 3.11 Continued tuning lattice ...... 72 3.12 Comma drift ...... 73 3.13 Comma shift ...... 73 3.14 The 53-tone system ...... 77

x 3.15 Just, 12-tet, and 53-tet ﬁfths ...... 77 3.16 Just, 12-tet, and 53-tet thirds ...... 78 3.17 12-tet and 53-tet major triads ...... 78 3.18 The 53-tet and 12-tet notations compared ...... 80

4.1 The dissonance curve ...... 86 4.2 Helmholtz’s construction of the dissonance landscape ...... 86 4.3 The dissonance landscape of simple tones ...... 87 4.4 The dissonance landscape of complex tones ...... 87 4.5 The dissonance landscapes on the pitch spiral ...... 89 4.6 The original sequencer Qvinta written in Java ...... 93 4.7 The current C++/Qt version of Qvinta ...... 93 4.8 Rusty Reﬂections, the opening ...... 100 4.9 Farmer’s Fuguetta, the opening ...... 102 4.10 Farmer’s Fuguetta, the ending ...... 103

5.1 The Qvin IDE ...... 109 5.2 The Qvin Musical Processes plugin ...... 109 5.3 ‘qmp’ user interface example ...... 111

B.1 The pitch spiral ...... 131

xi Deﬁnition of terms

This is an interdisciplinary text. We use a number of terms that are, no doubt, familiar to many of the readers, but which meaning often differs depending on the context where they are used. In the back of this text you will find a more complete glossary of many such terms, but there are a few that need to be defined before we proceed further, since they are used frequently from the very beginning. Their meaning could be otherwise vague or confounding. This is how they are used within this text:

Sound: (noun) A physical entity; a mechanical disturbance created by vibrating mechanical “sonorous” bodies that propagates as a sound wave through elastic media, typically through air.9 In this text we restrict the meaning of ‘sound’ to “audible sound”, i.e. such sound as can be sensed by human ear.

Sound: (verb) To create such sound wave (above).

Sound element: A coherent sound with clear beginning and ending. For example, a tone played on a musical instrument and notated by a single note in the musical score is a sound element.

Amplitude: The maximum displacement of vibrations or of a point on a (sound) wave. In this text, the term is interchangeably used to refer to the amplitude of a sound, to sound intensity, or sound loudness. If used in a strict sense, ‘amplitude’ refers to the maximum displacement within a complete cycle of the vibrations, and ‘instantaneous amplitude’ refers to the current amplitude at a given time instant.10

Amplitude envelope: A curve (a function of time) that limits the amplitude of a sound element. A proper amplitude envelope must be a smooth non-negative curve that begins and ends with zero.

Sensation: A physical experience (related to the human body) that results from stimulation of a sense organ, sensory nerve, or sensory area in the brain. Sensations can be in many

9“Sound”. Encyclopædia Britannica Online. 10“Amplitude (physics)”. ibid.

xii cases objectively observed and studied.11

Sound sensation: An experience that results from stimulation of the ear (as a sense organ), carried by the auditory nerve and processed by a respective sensory area of the brain. Periodic changes in air pressure cause the sound sensation of a tone.

Perception: A process in which sensations are transformed into a mental experience (related to the human mind), making inferences about the sources of the sensory stimuli, and forming mental concepts that reﬂect the outside reality. Perception draws not only on the current sensation, but also on past experience of relevant sensations. Perception can be observed only by the perceiver and therefore remains subjective, although it can be studied indirectly.12

Sound perception: A mental experience that is the outcome of a sound sensation. Sound perception also draws on past experience of sound sensations, such as the exposure to various kinds of musical sound, and other types of conditioning. For example, “ear training” in the course of which musicians learn to recognize musical intervals develops listening attention and “musical memory”.

Musical sound: Sound with certain qualities that make it usable in music. Although there are many kinds of musical sound, the only kind of musical sound speciﬁcally referred to in this text is the tone.

Tone: An elementary musical sound. A tone is a sound with a periodic waveform of a given stable frequency. It can be either a sustained sound, or a sound element. (‘Tone’ can also refer to a basic musical interval that has a size of two semitones.)

Note: A graphical symbol that represents a tone (as a sound element) in musical notation; also a name that designates the pitch of a tone (see ‘Pitch’ below).

Frequency: The number of cycles of a periodic wave in one second.

Pitch: A perceived quality of a tone that is related to its frequency. Pitch is a “position” on an imagined scale between “low” and “high” ends of the audible sound spectrum.

11“Sensation”. Encyclopædia Britannica Online. 12“Perception”. ibid.

xiii A slower frequency of vibrations corresponds to a lower pitch, a faster frequency to a higher pitch. The terms ‘low’ and ‘high’, with respect to the pitch of a tone, are probably based on a physical analogy: higher tension of a string, naturally associated with movement upward, causes the string to vibrate faster.13 In the usual Western musical context, pitch is speciﬁed by a note name: a letter from A to G with a subscript that indicates the octave range to which the pitch belongs and an optional accidental

sign (], [, etc.) that indicates a “chromatic” alteration of the pitch, e.g. C4 or G5].

Interval: Used strictly to mean musical interval: a diﬀerence between the pitches of two tones.

Comma: A very small musical interval, signiﬁcantly less than a semitone.

Simple tone: A tone with a sinusoidal waveform; causes a sensation of a plain musical timbre (tone “colour”). Among traditional (non-electronic) musical instruments, the clean sound of the ﬂute is close to that of a simple tone.14

Complex tone: A tone with an arbitrary periodic waveform that is not sinusoidal. By Fourier theorem, a complex tone is a sum of simple tones which frequencies form a harmonic series. Complex tone causes a sensation of a richer musical timbre. Most musical instruments, namely bowed string instruments of the violin family, reed wood- wind instruments, or brass instruments produce complex tones.15

Harmonic: (noun) One of the simple tones which sum forms a complex tone.

Partial: (noun) ‘Partial’, short for “partial tone”, is used in musical contexts as a synonym for ‘harmonic’. In this text, we use the term ‘partial’ only to refer to data items produced by the sound analysis library “Loris”.

Fundamental: (adjective) Applied to ‘frequency’ or ‘harmonic’. The “fundamental frequency” is the basic frequency of a complex waveform. The “fundamental harmonic”

13Goetschius, The structure of music, p.16. 14The flute sound has most energy in the fundamental (first) harmonic, and relatively little energy above the second harmonic. Owing to the plain character of the flute sound, it is easy to experience strong sensations of difference tones when two flutes play together. 15The instrument players do control, among other aspects of sound, the harmonic content of the produced complex tones: by choosing the place and style of bowing, by controlling the vibrations of the reed with their embouchure, etc.

xiv is the basic component of a complex waveform: a simple tone with the fundamental frequency.

12-tet: Twelve-tone equal temperament.

53-tet: Fifty-three-tone equal temperament.

Key: Except when discussing the piano keyboard, where ‘key’ means the lever that is pushed down in order to strike a string and produce a sound, ‘key’ refers throughout the text to a musical key: a tonal system derived from a diatonic (heptatonic) scale and named after its central note, the tonic, e.g. C major, D minor, etc.

xv Online materials

The work presented in this text is interdisciplinary. It includes a theoretical discourse, pre- recorded sound examples, software tools where a user by graphical means controls produced sound, and a dynamic computer system for musical interplay between a human and a machine. Due to the restrictions of the traditional written thesis format,16 some of those will be only verbally discussed in the text. The reader should, however, listen to the provided sound examples and experiment with the “pitch spiral”; both activities are most essential for appreciation of the discussed aspects of musical sound. The “pitch spiral” applet and sound examples are provided on web pages that accompany this text:

Main web page http://qvin.in/CC Interactive “pitch spiral” applet http://qvin.in/CC/PitchSpiral Sound experiments http://qvin.in/CC/Experiments The C53 Suite of musical etudes http://qvin.in/CC/C53

16A multimedia-friendly format, such as a set of dynamic, hyperlinked web pages, would be more suitable for this type of work.

xvi Chapter 1: Introduction and background

Principia essentialia rerum sunt nobis ignota.1

A violinist bows the strings of a violin. Their vibrations pass through the bridge onto the soundboard, excite acoustic pressure waves that propagate through air, eventually strike the eardrum and through the mechanical action of the ossicles in the middle ear enter the cochlea. Up to this point, the physical principles of sound creation and propagation are well understood and can be directly observed and measured. The processes that take place within and past the inner ear, processes that are essential in the sensation and perception of sound, are much less apparent and measurable, and can be directly investigated only with diﬃculty. We are therefore compelled to study this ﬁnal part of the sound creation–transmission–sensation– perception chain as a “black box”, by observing the causality between the character of sound waves and their subjective perception. This limitation, of course, by no means invalidates the conclusions that can be drawn from such observations.

Interdisciplinary text: This text is written for a mixed audience of readers: for computing scientists2 that are interested in music, and for musicians (theorists, composers, performers) that do realize and acknowledge the importance of physics and mathematics in arts. Many concepts that are well-known to one group of readers may be new to the other; we shall attempt to approach the material in such a way that it can be easily understood by both groups. It will therefore occasionally be necessary to review introductory terms and ideas as they are encountered. That itself is a useful exercise: reviewing foundational concepts in a course of study is certainly worth the time, since they are closely related to understanding of fundamental principles. We hope that the gentle reader ﬁnds even the familiar narrative interesting.

1Sed quia principia essentialia rerum sunt nobis ignota, ideo oportet quod utamur diﬀerentiis accidentalibus in designatione essentialium. “Since the essential principles of things are hidden from us we are compelled to make use of accidental diﬀerences as indications of what is essential.” Thomas Aquinas, Commentary on Aristotle’s De Anima, Book One, I/1 §15. http://dhspriory.org/thomas/DeAnima.htm 2The usual terms are “computer science” and “computer scientist”. Yet computer science studies computation, (the principles, methods, and applications of computing), and not computers. Computers are tools of computer science, but not necessarily objects of the study. For this reason it may be preferable to use the terms “computing science” and “computing scientists”, or even the terms (of European origin) “informatics” and “informatician”.

1 Music and machines Douglas Hofstadter in his seminal book G¨odel,Escher, Bach explored the potential and also the limits of formal symbolic systems that are the very foundation of computing, and also made many references to visual arts (M.C. Escher’s drawings) and music (J.S. Bach’s canons and fugues). As much as there is a strong connection between mathematics, computation, and music, Hofstadter did not suggest that computation would or could take on a creative role of an artist anytime soon:

Music is a language of emotions, and until programs have emotions as complex as ours, there is no way a program will write anything beautiful. There can be [...] shallow imitations of the syntax of earlier music, but [...] there is more to musical expression than can be captured in syntactical rules. There will be no new kinds of beauty turned up for a long time by computer music-composing programs.3

Hofstadter’s view has been, by his own account, shaken by compositions written by “Emmy” (EMI — Experiments in Musical Intelligence), a computer program written by David Cope.4 Emmy writes so-called recombinant music: it analyzes a given collection of musical works of a particular composer, breaks the works down to fragments, extracts semantic rules that supposedly guided how the fragments were originally combined, and reassembles them into new “templagiarised” compositions, that “sound like” Bach, Chopin, Beethoven, Gershwin, or Joplin. In Starring Emmy straight in the eye — and doing my best not to ﬂinch,5 Hofs- tadter recounts how, during his lectures on EMI, audience were often unable to distinguish between a “genuine Bach invention or a Chopin mazurka”, and Emmy-created compositions in the style of Bach or Chopin.

From his experience with Emmy, Hofstadter drew a pessimistic conclusion that:

• Either Chopin (for example) is shallower than Hofstadter had ever thought (since Emmy-Chopin piece had spoken to Hofstadter the same as the genuine Chopin had), and • music is shallower than he had ever thought (since what applies to Chopin, may apply to other composers as well), or perhaps even that

3Hofstadter, G¨odel,Escher, Bach: An Eternal Golden Braid, “Ten Questions and Speculations”. 4Cope, Experiments in musical intelligence. 5Hofstadter, “Creativity, Cognition, and Knowledge: An Interaction”.

2 • the human soul/mind is a lot shallower than he had ever thought (since music created by a relatively simple computer and about twenty thousand lines of LISP code, can be deeply moving).

[...] the day when music is ﬁnally and irrevocably reduced to syntactic pattern and pattern alone will be, to my old-fashioned way of looking at things, a very dark day indeed.6

While Hofstadter was deeply worried that the creation of beauty could be delegated to machines, the audience of his lectures apparently were not. We should not be worried, either: Hofstadter’s experience (that music created by a computer “can be deeply moving”) confirms that music does not ultimately exist on its own, but that it depends on its reflection within a human mind: beautiful minds find beauty in (certain) music. Minds that are not sensitive to this kind of beauty do not perceive it, but those that do find, for example, serenity in choral music, magnificence in symphonic music, stimulation and excitement in jazz music, visceral motion in rock music, eroticism in soul music, etc. Even though most music is nowadays marketed, distributed, and consumed as a commodity, its full potential can still be reached only through participation: by creating music as a music composer, playing music as a performing musician, by dancing to the sound of music, whistling while you work, humming a merry tune, and first and above all: by attentive listening.

Hofstadter’s “dark day” may surely come, but it will be not the day when music is reduced to a syntactical pattern and when machines therefore compose beautiful music, but the day when we no longer listen to it.

Computer-assisted music Having satisﬁed ourselves that making music with the help of machines is not necessarily a bad thing, we may now proceed to narrow the scope of our investigation: since participation is essential in making music, we shall not consider replacing human composers, performers, and listeners by machines; we shall use the computer only to assist with making music: as a laboratory for experiments with sound, as a tool for music composition (and real-time improvisation), and as a musical instrument.

6Cope later responded to Hofstadter’s and others’ comments in: Cope, Virtual Music: Computer Synthesis of Musical Style.

3 In this text we shall always keep in mind a distinction between ‘computer-assisted music’ and ‘algorithmic music’: in computer-assisted music, the machine helps a human composer and/or a human performer, who remains in charge of the creative process. That is in contrast with algorithmic music where once the machine is set up, the process takes its course with only a limited interactive control. In both cases, however, the algorithms are always set up by a human who thus participates in the music creation.

It may be tempting to approach computer-assisted or algorithmic music by constructing a program that simulates or imitates a human composer; for example, one that combines the twelve tones of the Western musical scale, according to a set of rules captured in some music theory, into a musical composition in a familiar style.7 Such approach can yield quite satisfactory results.

Another frequently used method is mapping (projecting) of some computed mathematical structure, a system of simulated physics, or spatial gestures and other input data made with human–computer interfaces, onto a domain of tones and other musical sounds. Success of a mapping method depends on whether a good mapping function is found. Some of these systems do produce impressive and interesting results. For example, in “an experiment in a new kind of music,” the output data of twelve-cells wide Wolfram “elementary cellular automata” are mapped to twelve tones of the chromatic scale.8

Mapping of abstract mathematical structures, dynamic physical systems, or spatial “gestures” works very well for visual arts. For example, simple iterative mathematical operations performed on complex numbers yield the well-known Mandelbrot9 or Julia10 fractals. That is, however, not a suﬃcient reason to assume that a similar approach should necessarily work for music. There is a fundamental diﬀerence between the two: music is based mostly on immediate sensation of audible sound, its precise change in time, and associated complex abstract mental concepts; while visual art is based more on real images, their shapes and

7Emmy’s recombinant music may be an example of such method. It is diﬃcult to say surely, because internal structure and function of EMI has not been published by its author, to my knowledge. An example of an “open source” system that does work by enforcing rules of counterpoint on a tone structure is COMPOzE (Henz, Lauer, and Zimmermann, “COMPOzE — Intention-based Music Composition through Constraint Programming”). 8WolframTones™ . 9“Mandelbrot Set”. Wolfram MathWorld™ . 10“Julia Set”. ibid.

4 colours.11

The problem of finding an appropriate mapping of various systems to music is quite elu- sive.12 In the area of gestural control of musical instruments, contemporary, easily accessible electronic technology allows intensive research and experimentation with “new interfaces for musical expression”.13 The innovative designs of various interfaces for human-computer interaction are stunning and offer many means of expression to electronic music composers and performers. The challenge is how to find and develop ways of using the new tools to their full potential, and how to construct different mappings for various styles of music, namely those that require musical sounds with a definite pitch.14 The evolution of technology and its possibilities takes place at a breathtaking speed. Musical genres, however, need time for their establishment.

Working definitions Before we further discuss computer-assisted music, we must address the question of what is music, even if we may be unable to answer it completely convincingly. It is also prudent that, in order to focus on the objectives of the work presented here, we quite narrow the working definitions, almost to the point of oversimplification.

To properly deﬁne ‘music’ (that which is a-musing)15 “would require discussion from many vantage points, including the linguistic, biological, psychological, philosophical, historical, anthropological, theological and even legal and medical, along with the musical in the widest sense.”16 Music has been described as “art concerned with combining vocal or instrumental sounds for beauty of form or emotional expression, usually according to cultural standards of

11“Music stands in a much closer connection with pure sensation than any of the other arts. The latter rather deal [...] with the images of outward objects [...] The plastic arts [...] their main purpose is to excite in us the image of an external object of determinate form and colour. [...] in music, the sensations of tone are the material of the art [...] we do not create out of them images of external objects or actions.” (Helmholtz, On the sensations of tone as a physiological basis for the theory of music, pp. 3–4) 12Interesting possibilities of mapping in the opposite direction, of music to topological structures, were explored, for example, in the work of Gueriono Mazzola (Mazzola, G¨oller,and M¨uller, The topos of music: geometric logic of concepts, theory, and performance) or Dmitri Tymoczko (Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice). 13NIME: International Conference on New Interfaces for Musical Expression. 14Using non-pitched sounds favours rhythm and sound “texture” over melody and harmony. 15From Greek mousik¯e(tekhn¯e), “(art) of the Muses” (“Music”. Oxford Dictionaries). 16“Music”. Oxford Music Online, (Nettl, Bruno).

5 rhythm, melody, and, in most Western music, harmony”,17 or “the art of arranging tones in aesthetically satisfying form, one after another (melody) and one against other (harmony), rhythmically subdivided, and put together in a completed work.”18

Musical sound

Music is an art form that uses sound as its material. Not all sound is music, but all music is realized through sound. To become music, sound must be somehow organized. Not all organized sound is music, but all music is made of organized sound.

Sound is the energy of mechanical vibrations that radiates as a wave from its source (a vibrating “sonorous” body) through an elastic medium (air) and reaches a listener’s ear. In the ear the mechanical energy of air pressure waves is captured by the eardrum and passed by the small bones of the inner ear into the cochlea, where it causes sympathetic vibration of the basilar membrane. The displacement of the basilar membrane is then converted by hair cells of the Corti organ into neural signals that ultimately lead to the perception of the sound.19 Sound generation and transmission are thus for a great part mechanical in nature and as such subject to laws of wave propagation and interference. Sound sensation is then a psychoacoustic phenomenon, i.e. the combined result of physical functioning of the ear and processing of the auditory stimuli by the mind.

Within this work, the smallest entities of musical sound are tones. A tone is a sound with a periodic waveform, sustained for the minimal length of time that is required for its perception as such by the sense of hearing. The sensation and perception of tones are the results of the periodic nature of the vibrations that generate tones. A periodic vibration, and the resulting wave, are characterized by their frequency, amplitude, shape (waveform), and phase.

The frequency of the wave is perceived as pitch, its amplitude as loudness, and its shape is the main contributing factor of the tone timbre. The phase of a sound wave is not detected by hearing, and plays a role only if two or more waves with comparable frequencies do interfere. 17“Music”. Encyclopædia Britannica Online. 18Die Kunst, Tönein ästhetisch befriedigender Form nacheinander (Melodie) und nebeneinander (Harmonie) zu ordnen, rhyth- misch zu gliedern, und zu einem geschlossenen Werk zusammenzufügen. Brockhaus-Wallring deutsches Wörterbuch, Wiesbaden, 1982. In: “Music”, Oxford Music Online. 19Some of this pathway is a “black box”, see p.1.

6 ‘Timbre’ is the characteristic perceived “colour” of the sound of various musical instruments (sources of musical sound). Timbre is the result of a frequency (pitch) spectrum, an amplitude envelope (loudness varying in time), and other factors, such as transient noises at the onset of the sound, or resonant frequencies of the instrument.

The terms ‘frequency’ and ‘amplitude’ are generally preferred in a scientiﬁc or engineering context, where quantitative measurements are important. In a casual or artistic context, with main focus on human perception, one would instead refer to pitch and loudness.

Organized sound

All music is made of “organized sound”. The term was introduced in the early twentieth century by Edgard Var`ese,and although it is now mostly used in the context of “music technology”,20 the concept applies to a variety of musical genres, beginning with traditional instrumental music. Var`esegave an account of how the term was coined, and also expressed some of his other views on art and music, including machine-assisted music, in a conversation with Alfred L. Copley:21

But how I envy you painters! You are in immediate communication with your audience. A canvas [is] a ﬁnished work. A score is only a blueprint at the mercy of performers and their ‘interpretations’. [...]

Musical freedom is of recent date. Thanks to electronics, composers are no longer deaf to the beauty of what once would have been called unpleasant noise. Music has at last decided to use its ears. [...] Electronics has freed music from the tempered system.

Electronics is an additive, and not a destructive element in the art and science of music. Western music has such a rich and varied patrimony, because new instruments have been added to the old ones. [...]

In the twenties [...] I got sick of the stupid phrase ‘Interesting, but is it music?’ After all, what is music but organised sound, all music! So, I said that my music was organised sound and that I was not a musician, but a worker in frequencies

20See Organised Sound, An International Journal of Music and Technology, ed. Leigh Landy, Cambridge University Press. Published since 1996. 21Varèseand Alcopley, “Edgard Varèse on Music and Art: A Conversation between Varèseand Alcopley”.

7 and intensities.

A machine can only give back what is put into it. A bad musician with instruments will be a bad musician with electronics. The composer should, in building his sonorous constructions, have thorough knowledge of the laws governing the vibratory system.

It was Helmholtz who ﬁrst started me thinking of music as masses of sound evolving in space, rather than, as I was being taught, notes in some prescribed order.22

Let us introduce a quasi-technical quasi-deﬁnition of ‘music’ that we shall keep in mind during the course of our argument:

Music: A combination of sound elements, organized in a structured form. Music has aspects of science, craft, and art. The science of music investigates how sound elements (in particular, tones of a deﬁnite pitch) may be combined successfully for the desired eﬀect; the discoveries of musical science make the basis of music theories. The purpose of music theories is to formulate rules for the craft of music composition, i.e. arranging tones in time into musically valid relationships. Finally, the art of music strives that the tone arrangements are meaningful structures with certain aesthetic qualities.

Sound element: A coherent (whole, unit) sound with a distinct beginning (onset) and ending, that itself is not an apparent combination of lesser (smaller, shorter etc.) sound elements.23 A sound element is constrained by an amplitude envelope.

Although the meaning of the term “sound element” is quite restricted in this context, it is related to the well-known concept of “sound object”. Curtis Roads defined nine levels of “time scales of music”, from “infinite” to “infinitesimal”. The fifth, middle level is the scale of “sound object”:24

A basic unit of musical structure, generalizing the traditional concept of note to include complex and mutating sound events [...] ranging from a fraction of a second to several seconds. 22Emphasis mine. 23The term ‘sound element’ is introduced to avoid confusion with ‘sound’ as a general physical entity. 24Roads, Microsound, p.3.

8 Pierre Schaeﬀer’s concept of “sound object” is:25,26

[...] every sound phenomenon and event perceived as a whole, a coherent entity, and heard by means of reduced listening27 [...] independently of its origin or its meaning.

In the remainder of this text, tones are the only type of sound elements that we shall discuss. Other types of sound elements (such as various percussive sounds) are less essential for the presented work.

Western music Taste in music varies from person to person. It depends on the current and past cultural standards, on a listener’s upbringing and schooling, acuteness of hearing and sound perception, exposure to various styles of music as a listener, participation in musical activities as a performer, personal preferences, etc.

We assume that the reader of this text has been exposed to and is reasonably familiar with common styles of Western music, i.e. the music that evolved from its roots in the music of ancient Greece and Rome, through the one-voice chant and two-voice organum of the Middle Ages and the Renaissance polyphony of many voices, through the Baroque sonata, concerto and fugue, classical symphony, and romantic chromaticism, and ﬁnally into the musical maelstrom of the twentieth and twenty-ﬁrst centuries.28

Western music relies on:

• Rhythm: The placement of sound elements in time.29 • Melody: A coherent sequence of tones, characterized by orderly change from pitch to pitch (by steps and leaps) and forming an orderly rhythmic structure.30,31

25Chion, Guide to Sound Objects, p.32. 26“Sound Object”. Landy et al., The ElectroAcoustic Resource Site (EARS). 27Listening to sound for its own sake. 28“Western music”. Encyclopædia Britannica Online. 29“Rhythm (music)”. ibid. 30“Melody (music)”. ibid. 31Melody is often referred to as the “horizontal dimension” of music, since it is related to the ﬂow of time that is notated horizontally, left-to-right, in musical scores.

9 • Harmony: Two or more tones with diﬀerent pitches that sound simultaneously.32,33 Harmony can be, namely, the result of combination of several melodies (voices) with diﬀerent rhythmical structures (polyphony), of several voices that move rhythmically together (homophony), or of simultaneous variations of the same melodic line (heterophony). Groups of (usually three or more) tones sounding together (vertically simultaneous tones) are called chords.

The set of pitches used in most of Western music is based on the division of the octave into twelve parts (called semitones). As we shall see, while the twelve-tone division of the octave is natural34 and also quite practical,35 it contains inherent contradictions that lead to tuning problems that cannot be solved to a perfect satisfaction; all we can do is to deal with them as well as we can.

Music theory rudiments There are several ways how the term music theory is understood: as “rudiments” of music, as theoretical writings about all possible aspects of music (the nature of sound, acoustics, aesthetics, instrument building, and so on), and as studies of the structure and fundamental principles of music.36 The latter vary from collections of practical rules that were acquired mostly empirically from analyses of music created by notable composers of the past, through various esoteric schemes, such as the “telluric” concepts of Levy,37 to systems that attempt to explain music from natural principles, such as the works of Rameau,38 Vogel,39 Parncutt40 and, most importantly for this text, the work of Helmholtz.41

Teaching of rudiments of music is a part of basic musicianship training. It includes the elements of musical notation, time and rhythm, intervals, scales, modes, keys and chords. We shall now brieﬂy review, as a necessary preparation for the argument that follows, some

32“Harmony (music)”. Encyclopædia Britannica Online. 33Also referred to as the “vertical dimension” of music, notated by vertical superposition of notes. 34“Existing in or derived from nature; not made or caused by humankind.”(“Natural” Oxford Dictionaries). 35The number twelve can be divided into two, three, four, or six equal, whole parts, which allows the construction of a number of symmetric (or not) musical pitch structures. 36“Theory”. Oxford Music Online, (Fallows, David). 37Levy, A theory of harmony. 38Rameau, Treatise on Harmony. 39Vogel, On the relations of tone. 40Parncutt, Harmony: A Psychoacoustical Approach. 41Helmholtz, On the sensations of tone as a physiological basis for the theory of music.

10 aspects of the rudiments; namely the basics of the system of intervals and scales.

The diatonic and chromatic scales

The usual approach to teaching music theory rudiments begins with the Western diatonic scale in the key of C major, as represented by the seven white keys on the piano keyboard (Fig. 1.1), and notated in the standard Western notation by the seven “natural” (not raised or lowered) notes (Fig. 1.2). The white keys and the corresponding notes are named using the ﬁrst seven letters of the Latin alphabet: CDEFGAB.42

The white keys are either immediately adjacent (E–F , B–C) and form the interval of a semitone, or have one black key in between them and form the interval of a tone (two semitones).

Figure 1.1: Piano keyboard

CDEFGAB

Figure 1.2: C major scale

The black keys do not have assigned their own letter names. The name of each black key is a combination of the letter assigned to either of the two neighbouring white keys and an “accidental” sign that indicates the “altering” — either raising (“sharpening”, by the “sharp” sign ]), or lowering (“ﬂattening”, by the “ﬂat” sign [) — of the pitch by the interval of a semitone.

Each black key can thus be named in two diﬀerent ways: for example, the key between C and D is both C] and D[. Each such two names are so-called enharmonically equivalent, i.e. “harmonically the same” (Fig. 1.3). On the piano keyboard, they correspond to the same black key and therefore their pitches are exactly the same, but as we shall see, that does not

42Which is the straight sequence ABCDEFG, rotated to constitute the “major” (Ionian) mode.

11 have to be necessarily so. If precise tuning is of importance, enharmonic notes do differ by a small interval (a comma).

C♯/D♭ D♯/E♭ F♯/G♭ G♯/A♭ A♯/B♭

Figure 1.3: Enharmonically equivalent altered notes.

The seven notes of the diatonic scale, together with the ﬁve altered notes, constitute a twelve- tone, symmetric chromatic scale with one size of the step (the semitone). This system of 7 + 5 = 12 tones in the chromatic scale is periodic and repeats every octave (the span of eight white keys).

Intervals

Within the octave there are eight possible intervals between two notes, named after the number of diatonic steps (white keys) they span. Table 1.1 lists the intervals, each with an example rooted (starting) on C (intervals can be rooted on any note).

key span interval name example 1 unison C–C 2 (the) second C–D 3 (the) third C–E 4 (the) fourth C–F 5 (the) ﬁfth C–G 6 (the) sixth C–A 7 (the) seventh C–B

8 octave C4–C5

Table 1.1: The eight musical intervals

There are two kinds of intervals: perfect and imperfect. The perfect intervals are the unison, the fourth, the fifth, and the octave. The perfect intervals can be rooted on any of the white keys and retain their size in semitones, as shown in Table 1.2, with one notable exception for the fourth and one for the fifth. The two exceptional intervals, the augmented fourth F –B and the diminished fifth B–F , are equal to the tritone (TT), the “devil in the music”, three tones or six semitones in size.

12 interval name abbrev. semitones on white keys (perfect) unison P1 0 C–C, D–D, etc. (perfect) fourth P4 5 C–F , D–G, etc., except F –B (perfect) ﬁfth P5 7 C–G, D–A, etc., except B–F

(perfect) octave P8 12 C4–C5, D4–D5, etc.

Table 1.2: Perfect intervals

The octave is an interval that serves as an equivalence relation (see p.47). That means that all Cs are “equivalent”, or “similar” to each other, so are the Ds, Es, etc. This is why the keyboard and the system of note names are periodic, and do repeat every octave.

An interval can be inverted by moving its lower note an octave higher or by moving its higher note an octave lower.

Since the octave is an equivalence relation, there are in reality only two kinds of perfect intervals with respect to the octave, the octave itself and the ﬁfth: the octave and the unison are in a way interchangeable (since the unison is an inversion of the octave), and so are the ﬁfth and the fourth. The singular interval, the tritone, is its own inversion.

The imperfect intervals are the second, the third, the sixth and the seventh. Each has two species: major and minor, that diﬀer in size by one semitone, as listed in Table 1.3.

interval name abbrev. semitones on white keys major second M2 2 C–D, D–E, F –G, G–A, A–B minor second m2 1 E–F , B–C major third M3 4 C–E, F –A, G–B minor third m3 3 D–F , E–G, A–C, B–D major sixth M6 9 C–A, D–B, F –D, G–E minor sixth m6 8 E–C, A–F , B–G major seventh M7 11 C–B, F –E minor seventh m7 10 D–C, E–D, G–F , A–G, B–A

Table 1.3: Imperfect intervals

Again, with respect to the octave, there are only two kinds of imperfect intervals: the third and the second, each in the major or minor variety. The sixths are inversions of the thirds

13 and the sevenths are inversions of the seconds; inverting an imperfect interval changes its character from major to minor and vice versa. Table 1.4 lists all intervals paired side-by-side with their inversions.

Note that the three major triads are rooted on C, F , and G, the three notes connected by two ﬁfths: F –C–G. After C (the tonic), F and G are the most important notes, the subdominant and the dominant, in the key of C major. This fact — that only the three thirds rooted on the tonic, subdominant, and dominant (C–E, F –A, and G–B in the key of C major) and no other are major thirds — is perhaps the reason why the major mode is the one most frequently used in Western tonal music.43

interval semitones interval semitones P1 0 P8 12 m2 1 M7 11 M2 2 m7 10 m3 3 M6 9 M3 4 m6 8 P4 5 P5 7 TT 6 TT 6

Table 1.4: Intervals and their inversions

Music notation Western music uses the “staﬀ notation” which is a result of centuries of evolution of Western music theory and practice based in the system of twelve tones in an octave.

Friedemann Sallis wrote about the relation of Western music and its system of musical notation:44

[...] change in the status and function and concept of Western music has usually been accompanied by changes in the way this music is notated. [...] in our musical

43The longest chain of alternating major and minor thirds in the diatonic scale is D–F –A–C–E–G–B–D. (The two adjacent minor thirds B–D–F do not form a perfect fifth.) Note that there are two possibilities for three interlinked triads (subdominant, tonic, dominant) within this chain: F –A–C, C–E–G, G–B–D (three major triads: the major mode), and D–F –A, A–C–E, E–G–B (three minor triads: the minor mode). 44Sallis and Burleigh, “Seizing the Ephemeral: Recording Luigi Nono’s “A Pierre dell’azzurro silenzio, inquietum. A piùcori” (1985) at the Banff Centre”.

14 culture notation and theory are two inseparable sides of the same coin. [...] it [tends] to be more than a system of signs indicating how to produce a given piece of music [...] our system of notation has always been a visual representation of the music-theoretical concepts [...] as well as a set of pragmatic instructions for the performance of a speciﬁc work.

The staﬀ is a grid of horizontal lines with spaces between them. Notes are graphical repre- sentations of tones; their vertical placement in the grid, on the lines or in the spaces between the lines, indicates the “height” of their pitch. The horizontal position of notes corresponds to the order of the tone sequence in time; the exact time of onsets of the tones and their duration are encoded through a variety of means: measure lines, various kinds of note heads that indicate the length of the tones, etc., but in principle the staﬀ approximates a two- dimensional pitch×time coordinate system: the vertical coordinate represents tone pitch, and the horizontal coordinate is related to the time dimension.

The staﬀ notation thus suggests the two-dimensional organization of sound elements (p.8) that form a musical piece. Notation systems that are more “machine-oriented” would indicate the pitch more precisely by the vertical position of the corresponding graphical symbol, and the onset and duration by its horizontal position and extent. A typical case of such machine-oriented notation is the “piano roll” that is ubiquitous in MIDI sequencers (historically preceded by player pianos45). Use of a discrete graphical system of notation was also advocated by Joseph Schillinger,46 and two-dimensional coordinate graphical systems are commonly used to display sonograms (see p.30).

The Western staff notation is widely accepted as a “common music language”47 and as such serves well its purpose. It is however, biased toward the diatonic tonal system. Notes on the staff directly represent the “natural” diatonic scale (as played on the white keys of the piano). The five “accidental” notes (black keys) that complete the chromatic scale are notated, using the accidental signs, as deviations from (alterations of) the “natural” notes. If pitches that are outside the twelve-tone system are to be included, they have to be notated, using some extended set of signs (such as quarter-tone accidental signs), as further deviations from either the diatonic or the chromatic scale. 45Pianos that were played by pneumatic actuators which action was controlled by a perforated paper roll. Player pianos were common in the late nineteenth and early twentieth centuries. 46Schillinger, The Schillinger System of Musical Composition. 47“Musical notation”. Encyclopædia Britannica Online.

15 Western musicians are trained almost exclusively with the use of the Western staﬀ notation, and its role is so ingrained in music practice that one is strongly compelled to employ it to convey all musical ideas, even those that are outside the bounds of the Western twelve-tone system. The usual solution in such cases is to use some extensions of the staﬀ notation to indicate deviations of pitches from those in the chromatic scale.

Fig. 1.4 shows the staﬀ notation of several opening measures of J.S. Bach’s Prelude in C. The chord symbols placed above the top staﬀ are a modern way to show the harmonic structure of the work.48 The Prelude is used throughout this text as a basis for musical examples that demonstrate the viability of the proposed tone arrangement method. Prelude in C, BWV 846a Johann Sebastian Bach C Dm7/C

G7/B C

Am/C D7/C

Figure 1.4: Prelude in C, opening measures

48For practical reasons, the harmonic structure is shown using the type of chord symbols used in contemporary jazz and popular music, rather than the Roman numeral system for classical music, or the ﬁgured bass system of the Baroque era.

16 Division of the octave Most of works of Western music have been composed, notated, and performed using a twelve- tone division of the octave. The choice of twelve distinct pitches is not arbitrary; it is a natural consequence of acoustical properties of tones (p.62).

The problem of how to divide the fourth and the octave into smaller steps and the consequences of the chosen division have been intensively studied throughout the history of music until about the end of the nineteenth century:49 ﬁrst in ancient Greece50 and mediaeval Persia,51 and then in the Western world primarily since the Renaissance in the works of musicians and mathematicians such as Vincenzo Galilei,52 Ren´eDescartes,53 Marin Mersenne,54 Jean-Philippe Rameau,55 Leonhard Euler,56 Jean Lerond d’Alembert,57 Giuseppe Tartini,58 and a number of others.

The problem of the division of musical intervals translates to the arithmetical problem of expressing a ratio of small whole numbers as a product of several such ratios that represent the intervals that the original interval is divided into. For example, the octave (2:1) can be 2 3 4 divided into a ﬁfth (3:2) and a fourth (4:3): 1 = 2 × 3 . The fourth (4:3) can be divided into 4 5 16 a major third (5:4) and a diatonic semitone (16:15): 3 = 4 × 15 , or a minor third (6:5) and a 4 6 10 minor tone (10:9): 3 = 5 × 9 . (An introduction to the intervals and their numerical ratios of frequencies starts on p.34; more in-depth discussion of this topic is beyond the scope of this text, and can be found in a large number of existing treatises.)

Music is number made audible.

Bo¨ethius,Anicius Manlius Severinus (480–524).59

49With the rise of chromaticism in music and with more frequent use of modulation to different keys, it became expedient to disregard the finer points of tuning. 50Barker, Greek Musical Writings. 51Shiloah, The Theory of Music in Arabic Writings c. 900–1900 . 52Galilei, Dialogue on Ancient and Modern Music (English translation 2004). 53Descartes, Compendium Musicae. 54Mersenne, Harmonie universelle. 55Rameau, Treatise on Harmony. 56Euler, Tentamen novae theoriae musicae. 57d’Alembert, Elémensde´ musique, théoriqueet pratique, suivant les principes de m. Rameau. 58Tartini, Trattato di musica secondo la vera scienza dell’armonia. 59In: Grattan-Guinness, The Fontana History of Mathematical Sciences.

17 Bo¨ethiusmight have been alluding to the fact that proportions of small numbers (ratios) lead to harmonious consonances.60 At ﬁrst, ratios of vibrating string lengths (provided that the strings — their material, thickness, and tension — were otherwise the same) were used to reason about musical intervals. Ratios of string lengths are inversely proportional to the frequencies of their vibrations; since about the middle of the nineteenth century when it became technically possible to measure frequencies (called then “vibration numbers”), it is preferable to use ratios of frequencies, keeping in mind that the ratios need to be inverted when reading older texts.

Although there is not much known about the actual music of ancient Greece, there are many accounts of the tuning systems that were used. Greek music was based on tetrachords, scales of four notes bounded by the perfect fourth.61,62 The original three Greek genera of tetrachords were:63

• Diatonic:64 Two descending whole tones, and the remaining semitone. • Chromatic:65 A descending minor third and two semitones; a “colourful” genus. • Enharmonic:66 A descending major third, with the remaining semitone divided into two halves; a “harmonious” genus.

The above descriptions of the three genera are simplified and use the modern terms ‘whole tone’, ‘semitone’, ‘minor third’, and ‘major third’ that are quite imprecise in the context of Greek tetrachords. The Greeks used many alternative divisions of the fourth in each genus, defined by many specific number ratios that offered many subtle audible shades of the intervals.

The modern use of the terms ‘diatonic’, ‘chromatic’, and ‘enharmonic’ is more or less related to the ancient Greek genera:

• Diatonic: The modern diatonic (seven-tone, heptatonic) scale is made of ﬁve whole- tone steps and two remaining semitone steps.

60And also to regular rhytmic patterns. 61“Tetrachord”. Encyclopædia Britannica Online. 62Greek scales were descending; a descending fourth is an inversion of an ascending ﬁfth. 63Barbera, The Euclidean Division of the Canon (Greek and Latin Sources). 64Diatonikos, ‘at intervals of a tone’, from dia- ‘through’ + tonos ‘tone’. (“Diatonic.” Oxford Dictionaries) 65Khr¯omatikos, from khr¯oma, ‘colour’. (“Chromatic.” ibid.) 66Enarmonikos, from en- ‘in’ + harmonia ‘harmony’. (“Enharmonic.” ibid.)

18 • Chromatic: The modern chromatic scale is a scale of twelve semitones. • Enharmonic: In modern music theory that assumes twelve-tone temperament (12- tet, see p.66), two tones that are named diﬀerently but fall on the same keys on the modern keyboard (such as F] and G[), are called “enharmonically equivalent” and are considered to be “the same”. However, in other systems of tuning (see p.70), such tones do differ by a small interval that is less than a semitone in size (a comma).

The twelve-tone division of the octave In Chapter 2 we shall brieﬂy discuss the harmonic series, a natural property of many vibratory systems, such as of stretched strings or columns of air that are used to generate sound in musical instruments.

We will now demonstrate how the division of the octave in Western music into twelve tones naturally follows from a sequence of (perfect) fifths; the choice of the sequence of fifths for division of the octave is itself a consequence naturally following from the harmonic series. Similarly as the fourth was differently divided in the Greek genera, the octave can also be divided into twelve parts in numerous ways.

Early music used so-called Pythagorean tuning (the “fifth-tuning”) that emphasizes the purity of the fifths. When the thirds became an integral part of harmony, it required adoption of the so-called just tuning where both the fifths and the thirds are properly tuned. Just tuning, however, does not allow easy modulation through keys and therefore various temperaments (systems of tuning that in a controlled manner compromise the purity of intervals) were introduced. A representative example is the “mean-tone” temperament, that in its several forms allows modulation, even though mostly only through several core keys. Later, various so-called well-tempered temperaments allowed modulation to all twelve keys, while the keys retained their distinct character.

Eventually, apparently around the beginning of the twentieth century, the twelve-tone equal temperament, where all the semitones have the same size67 and all keys have identical character (and where also all the subtle diﬀerences in interval and chord shades that the Greeks and others had are lost) was widely accepted in music practice and became “the only” standard. Or so many musicians think. In fact, the piano that is thought to have forced the √ 67The “ratio” of the two frequencies in the equally tempered semitone is 12 2, an irrational number.

19 acceptance of the equal temperament, cannot be easily tuned that way; it is unnatural and therefore diﬃcult for singers and players of string and wind instruments to sing or play in equal temperament; the pipe organ, if tuned in the equal division, does not sound very well. Perhaps the only instruments that can be tuned accurately in the equal temperament are electronic devices, be it digital hardware synthesizers or software instruments.

Just intonation Just intonation means tuning tones precisely, to the maximum of consonance. Harmony played in just intonation sounds well, and a critical ear recognizes that tones, intervals and chords are played “in tune”, rather than “out of tune”. As music gets more complicated, and begins to “change keys” (to modulate between keys), just intonation indeed gets in the way and on keyboard instruments needs to be replaced by some kind of a temperament. Nevertheless, just intonation does not have to be completely compromised.

Harry Partch said in his lecture on intonation:68

[...] just intonation [...] is an ancient truth [...] based on the relationship of small numbers, small numbered parts of a string [...] or [...] small frequency ratios [...] And this was true in the world prior to the advent of the piano, or let’s say the keyboard scale, which has been a monolith for about three hundred years, and in my opinion, a rather tyrannical monolith. Certainly some great things came out of it, yet it has its own truth. It’s just not the truth that I happen to like.

[...] The tempered triad is strangely uneasy, and no wonder. It wants nothing so much as to go away and sit down someplace.

[...] Equal temperament is a current habit as is also the scope for modulation which it allows. Composers can think only in equal temperament for just one reason: because it is all they have got to think in.

[...] I have said many times, and I am by no means the ﬁrst to say it [...] that 12-tone equal temperament not only slams doors against any further investigation of consonance, but it also slams doors in the entire balance of the temple against any further investigation of dissonance.69

68Partch, “Barbs and Broadsides”.

20 The practice of the equal division of the octave

Equal temperament is indeed a modern state of mind, rather than a tuning that is in real use. The twelve tones of the chromatic scale are often perhaps considered to be wider “pitch bands” without predetermined precise pitch values, and the adjustment of their intonation, for music played in mathematically precise equal temperament sounds poorly, is silently left to performing musicians and to listeners.

The musicians apply what can be called “corrective tuning”, that is, they micro-adjust the pitch of played tones “by ear”, if and when possible. The listeners intuitively employ “corrective listening”: they get used to the various imperfections of intonation and consider them to be “normal”. The timbres of instruments with ﬁxed pitch (such as the percussive character of the piano sound or the vibrato, phasing and other eﬀects that are integral parts of the “synthesizer sound”) do “mask” tuning faults.

Hence practically performed music depends on the sensibilities of musicians and the insen- sitivity of listeners to partially correct and partially mask the artefacts of the twelve-tone tuning compromise. Machines do not have a sensibility and if not programmed to do so, will not make such tuning adjustments as a sensitive human musician would, and therefore a sensitive listener’s ear might not be satisﬁed.

However, thinking in terms of a general twelve-tone division of the octave, as twelve (equal?) semitones which size is not precisely speciﬁed, is pragmatic and expedient for these main reasons:

• Music composition is a complex combinatorial task. Dealing with intonation in addition to the possible tone combinations would make the task more difficult to solve. At the least, it would force composers to create much less complex works. • Playing the pitches precisely on musical instruments with a continuous pitch range (e.g. the violin) would require much more refined skills; on instruments with fixed set of pitches (e.g. keyboard instruments) it would need more complex instrument design and also more complicated playing techniques.

69Like Varèse (p.8), Partch also credited Helmholtz for showing him the way: “Before I was twenty, I had [...] rejected the intonation system of modern Europe [...] When I was twenty-one I finally found [...] the key for which I had been searching, the Helmholtz-Ellis work, On the Sensations of Tone [...] and I began to take wing [...] I wrote a string quartet in Just Intonation [...]” (Partch, Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, Preface to the Second Edition)

21 Computer-assisted microtonal music Microtonal music (used here as a blanket term) is made from tone material other than the standard Western chromatic scale. That includes tones and semitones of diﬀerent sizes, and potentially more than twelve pitches in the octave.70

Systems with more than twelve pitches pose a diﬃculty: higher number of pitches make composition more complex, and intervals smaller than the semitone are diﬃcult to intone (or even impossible to play on keyboard instruments). However, modern computers:

• can process large amounts of data fast, and therefore handle the complexity of microtonal music (if a suitable algorithm exists), and • used as electronic instruments, can synthesize musical sounds with precisely set frequencies and intensities.

Computers offer a great opportunity not only to simulate traditional instruments and or- chestras, but also to be used for sound and music analysis, as an experimental tool (perhaps the greatest “laboratory” that ever existed) and, first and foremost, a new kind of musical instrument that is flexible, precise, and can be made “smart”.

Experiments and experimental tools The ultimate test of music is how it is perceived by the listener; music must be heard to exist. The scientiﬁc approach requires that assumptions, theories, and facts be tested by experiments. In the course of our argument we therefore must conduct several fundamental experiments in perception of musical sound.

Music perception is in part innate, but also quite subjective and individual, and is very much influenced by previous conditioning on the part of the listener.71 We recommend that each reader carries out the experiments individually, to confirm that the perception of the test sounds is indeed similar to what is described in this text. As Helmholtz wrote: 70“Microtonal music”. Encyclopædia Britannica Online. 71According to recent research, music perception — the perception of its “spectral and temporal organization” (“pitch and rhythmic aspects”) — requires specialized processing in the brain. The ability to discriminate tones that is sufficient for perception of pitch is already present in infants; appreciation of harmony seems to fully develop by early teenage years. There have been reports of structural brain differences between adult musicians and non-musicians, detected on MRI scans and in EEG activity. (Trainor and Unrau, “Human Auditory Development”)

22 It is of course impossible for any one to understand the investigations thoroughly, who does not take the trouble of becoming acquainted by personal observation with at least the fundamental phenomena mentioned. Fortunately with the assistance of common musical instruments it is easy for any one to become acquainted with harmonic upper partial tones, combinational tones, beats, and the like. Personal observation is better than the exactest description, especially when, as here, the subject of investigation is an analysis of sensations themselves, which are always extremely diﬃcult to describe to those who have not experienced them.72

The experiments and demonstrations within this text were prepared and carried out with the use of either freely available open-source programs, or programs that I have developed over the past number of years for research purposes.73 The used programs are available for all three major operating systems (MacOSX, Linux, and Windows):

• Audio files were edited with Audacity, “an open source program for recording and editing sound.”74 Audacity has a simple, intuitive interface, it is very easy to learn and therefore is widely used as a “Swiss Army knife” for work with digital audio files. • Sonograms were made with Sonic Visualiser,75 “an application for viewing and analysing the contents of music audio files.”76 Sonic Visualiser is a program that creates excellent sonograms that make it very easy to “look at music while listening to it.” • Time and frequency-reassigned (sharpened) sonogram-like scores were prepared with Loris,77 a library for sound morphing, and graphically presented with Qvorum,78 an experimental software for transcription of music.79 • The notation of all musical examples was typeset using ABC,80 a plain text format and a system for notating music. • Sound for examples used in tone perception and other experiments was synthesized 72Helmholtz, On the sensations of tone as a physiological basis for the theory of music, p.6. 73Development of the custom programs is ongoing, to satisfy the needs of continuing research. 74Mazzoni and Dannenberg, Audacity. 75British spelling. 76Cannam et al., Sonic Visualiser. 77Fitz and Haken, Loris. 78Burleigh, Qvorum. 79Qvorum is beeing developed as part of an ongoing research project in “mixed” music studies (Zattra, Burleigh, and Sallis, “Studying Luigi Nono’s A Pierre. Dell’azzurro silenzio, inquietum (1985) as a Per- formance Event”) and was presented at a 2012 IRCAM colloquium “Analyser la musique mixte.” (Sallis, Burleigh, and Zattra, “Seeking Virtual Voices in Luigi Nono’s A Pierre (1985) through a Study of a Per- formance and the Creative Process”) 80Walshaw et al., The ABC Music project.

23 with Csound,81,82 an audio synthesis and processing system that has a long genesis going back to the MUSIC-N programs developed by Max Mathews,83 beginning in the 1950s at the Bell Labs. • The tone arrangements in the 53-division of the octave were completed and sequenced in Qvinta,84 another experimental software that is under development. Their sound was synthesized using the SuperCollider 85 sound synthesis server. • The real-time interactive music system that is brieﬂy introduced in Chapter 5 has been implemented in Qvin.86 Qvin is a suite of software components that include a custom scripting language, a collection of various libraries, and a plug-in system through which the suite can be extended by new components. (Qvorum and Qvinta may be in the future integrated into Qvin as its plug-ins.)

Sound files The sound for experiments in musical tone perception was synthesized using the Csound software and stored as high-quality sound files (48 kHz sampling rate and 32-bit depth samples). Their compressed mp3 version is available for online listening, along with the source Csound files that were used to synthesize the sound. Whenever they are referred

to in this text, the names of the online mp3 ﬁles are labelled by an MP3 tag, and the names

of the corresponding Csound source ﬁles are labelled by CSD .

The reader is encouraged to re-synthesize the sound, possibly changing parameters of the experiments (frequencies, amplitudes, times, etc.), if so required. To play the synthesized sounds in real time, run:

csound -f -m6 -d -odac

The command line flag -f selects the floating-point (32-bit) representation of the samples (replace it with -3 to use 24-bit samples if the computer audio system does not accept 32-bit samples), -m6 and -d are used to suppress unimportant informational messages, and -odac directs the synthesized sound to play through the default audio interface. To store the synthesized sound as an audio file (WAVE format), replace -odac with -o.

81Vercoe and Ellis, “Real-Time CSOUND: Software Synthesis with Sensing and Control”. 82Vercoe et al., Csound. 83http://www.csounds.com/mathews/max_bios.html 84Burleigh, Qvinta. 85McCartney et al., SuperCollider. 86Burleigh, Qvin.

24 Audio equipment It is necessary to conduct the experiments in an environment and using audio equipment that allow attentive listening to high-quality sound, reasonably free from distortion, noise, hums, and other interferences. The built-in sound chipsets or sound cards of the modern laptop or desktop computers are satisfactory, but playing sound through a good quality external audio interface is still preferable.

It is most important to use a good quality pair of headphones. The computer built-in speakers or small external computer speakers are unacceptable: they do not radiate frequencies below about 160 Hz and their sound is often distorted. Various “earphones” or “earbuds” and “in-ear monitors” are not sufficient, either: they are designed to fit into ear canals, tightly seal them and thus to isolate the listener from outside noise. In doing that, they change the natural air pressure conditions within the ear and also exclude the effects of the ear canals and earlobes that are essential for natural sound perception.

The best results should be achieved with a pair of over-the-ear headphones that have a ﬂat frequency response, such as are used for audio monitoring and mixing in a recording studio. Listening to the synthesized sound over a pair of high-quality loudspeakers would be also satisfactory, albeit less than when using headphones: many of the experiments include pure sine waves (tones without higher harmonics) and unless one listens in an anechoic room, natural room resonances would also interfere with the experience.

When listening to computer-generated sounds, especially using headphones, always be careful and protect your hearing! To avoid discomfort and potential hearing damage, begin with the volume of the sound lowered, and keep the headphones slightly oﬀ your ears. If all sounds well, then put the headphones over your ears and listen more closely.

The outline of the work The outline of this work is as follows:

1. First, we will introduce and discuss the sensation and perception of tone pitch, and the phenomenon of the harmonic series as a natural origin and also a constraint of the Western music system. 2. Next, we will show why Western music is based on the division of the octave into twelve steps of the chromatic scale, as a practical compromise between the precision of intonation and the complexity of the tone system. We will point out the fundamental

25 problem of tuning that exists due to incompatibility of intervals, and briefly review and test the psychoacoustic theory of tone consonance developed by Herman Helmholtz. 3. Next, we will demonstrate the traditional method of resolving the tuning problem: tempering the twelve-tone system. We will compare the Pythagorean tuning, just intonation, a “quarter-comma” mean-tone temperament, and the equal twelve-tone temperament. We will show on a brief example how the character of the same piece of music changes when the music is played in different temperaments. 4. Next, we will make a case for a fifty-three tone system as the domain for machine- assisted microtonal tone arrangement, and propose how to notate the fifty-three pitches in the Western staff notation. 5. Finally, we will show how the method of machine-assisted tone arrangement has been used to compose a suite of short musical etudes, and to create a dynamic system for musical interplay between a man and a machine.

26 Chapter 2: Tone sensations

The basic building elements of music are tones. Tones are sound elements with periodic waveforms, deﬁnite frequency and therefore distinct pitch. Even though the perception of pitch has a subjective component, it is in principle logarithmically related to tone frequency, at least in the middle range of audible frequencies.

The sensitivity and precision of hearing and therefore the ability to perceive and analyze musical sound varies signiﬁcantly from person to person. It also depends on other factors: on how rested the person is, on the listening environment, on the time of the day, etc.

It is beneﬁcial to complement listening analysis of musical sound by its analysis with the help of machine-based (computer) analytical tools, using visual representation of the sound in question; this allows the listener to “see the sound while listening to it”. Seeing a visual image associated with a sound helps to focus hearing to its aspects that otherwise might be missed. Visual images, however, should be used only for guidance and ought not to replace careful listening to the sound. Listening tests and “judgement of the ear” are most important.

In this work, two such tools are used: the sonograms (of the usual kind) obtained by Fourier analysis that provide a graphical representation of sound intensity in the two-dimensional time×pitch domain, and the “pitch spiral”, a novel tool for visual representation and control of synthesized tones in pitch perception experiments.

The pitch spiral The “pitch spiral” is a novel tool that has been developed for experiments that investigate perception of tones and their combinations. The tool integrates sound synthesis of tones with a graphical display that indicates the presence and relations of the tone frequencies and, optionally, their differences (that are perceived as difference tones). The visual representation of sounding pitches thus supports and clarifies the perception of the sound. A number of music theory concepts, such as the character of various possible musical intervals and chords, their voicings and inversions, or the nature of scales, tuning systems and temperaments, etc., can be also demonstrated on the spiral.1

1A detailed discussion of those is beyond the scope of this text, and is reseved for another publication.

27 The current version of the pitch spiral2 is a web-based application, written in JavaScript using the D3.js3 and Vinci 4 libraries. The graphical part is realized as an SVG5 code embedded in an HTML6 page with standard HTML user interface elements. The pitch spiral works in any modern web browser that supports the HTML5 standard. A running instance of the spiral is available online. Since web technologies are ready for sound ﬁle playback, but not yet for eﬃcient sound synthesis, it requires locally installed Csound and a running WebSocket7 bridge (that passes commands from JavaScript running inside the HTML page to Csound) to synthesize sound. Appendix B describes in more detail how to use the pitch spiral.

Figure 2.1: The pitch spiral

The curve of the spiral is a continuous scale of a uniformly stretched continuum of pitches,

2There were two previous versions of the pitch spiral, prototyped in SuperCollider/Tk and Java/Swing programming environments. 3Bostock, Heer, and Ogievetsky, D3.js. 4Burleigh, Vinci. 5Scalable Vector Graphics, a language to describe two-dimensional vector graphics. http://www.w3.org/ Graphics/SVG/ 6Hypertext Markup Language, a standard language for creating web pages. http://www.w3.org/html/ 7A protocol for communication between web pages and network servers. http://dev.w3.org/html5/ websockets/

28 from C1 to C8. Each turn of the spiral represents one octave. C4 (the “middle C”) serves as an origin of the pitch scale and is indicated by a red circle (Fig. 2.1).

Figure 2.2: Source and diﬀerence tones on the pitch spiral

In the centre of the spiral there is a “repository” of five coloured discs (blue, green, orange, yellow, and white). When the discs are dragged from the centre onto the spiral, they represent respective pitches (fundamental harmonics) of up to five tones. Each tone can have a harmonic spectrum of between one to six harmonics,8 and its synthesized sound re- flects the displayed spectrum. The difference tones (see p.52) are displayed as circles with perimeters drawn by a combination of the colours of their two resepective source harmonics (Fig. 2.2).

8The range of six harmonics (the “senario”, see p.46) was chosen for the purpose of investigation of 5-limit (see p.35) sonorities. The range of harmonics can be extended beyond six for experiments with higher-limit tuning systems (that are among the planned topics of future work).

29 Just tuning 12-tet 53-tet Figure 2.3: Various grids on the pitch spiral

The spiral can be overlaid by various radial grids that indicate the positions of pitches in respective tuning systems, namely just tuning and 12-tone and 53-tone equal divisions of the octave (Fig. 2.3), but also a number of other equal divisions, meantone temperaments, tuning systems generated by other intervals than the ﬁfths, etc.

Sonograms The theoretical foundation for sonograms (“images of sound”), also known as spectrograms (“images of [sound] spectrum”) is the Fourier theorem that is based on these principles:

• Complex periodic waves are composed from a series of simple (sinusoidal) waves, • the frequencies of the sinusoidal component waves are whole multiplies of the fundamental frequency (the frequency of the complex wave), and • each component wave is characterized by its frequency, phase, and amplitude.9

Fourier theorem is the basis for a Fourier transform, a mathematical method that breaks down a complex wave into a series of its component simple waves.

Helmholtz hypothesized that the inner ear performs a type of Fourier transform by mechanical means: the basilar membrane within the cochlea resonates (sympathetically vibrates) with the sound; the resonant frequency of the membrane changes along its length, and therefore sounds of diﬀering frequencies (that is, the component waves of a complex wave) induce vibrations at diﬀerent places along the membrane where they are detected by the hair cells in the organ of Corti. Although this “place theory” is an explanation that is no longer fully accepted and the mechanism of frequency separation within the ear is apparently much

9“Fourier theorem”. Encyclopædia Britannica Online.

30 more complicated,10,11 it is a useful simpliﬁed analogy that helps to explain how we perceive complex sounds.

Sonograms are computed by application of Short-time Fourier transform (STFT) that uses a “sliding window” to isolate short intervals of the sound signal that is being analyzed. For digital signals, STFT is realized as Fast Fourier transform (FFT), an eﬃcient algorithm that allows real-time processing of sound.12,13 The sound signal is converted from the “time- domain” (the instantaneous amplitude of the sound waveform as a function of time) into the “frequency domain” (the sound amplitude as a function of frequency). A sonogram shows the frequency content (the so-called frequency spectrum) of sound in time, and the image is strongly related to what the listener hears.14

The typical sonogram is a two-dimensional graph: time is plotted on the horizontal axis in the left-to-right direction, frequency (or pitch) on the vertical axis in the bottom-up direction, and the sound amplitude is indicated by colour. A sonogram, read left-to-right, shows the “evolution of sound” in time.

FFT operates on digital audio data. Digital audio is sampled, i.e. a continuous waveform is represented by discrete samples (measurements of its instantaneous amplitude) taken at regular time intervals (the samling rate). The width of the STFT window, as a number of samples,15 determines the number of so-called “frequency bins”: frequency ranges of cumulative frequency domain data; there is not individual information for every possible component frequency in FFT data.

The width of the STFT window (related to its length in time) also determines the compromise between the time and frequency resolution of the sonogram. This is not unlike listening to sound by ear: a longer window (longer lasting sound) leads to better frequency accuracy at the expense of time resolution. A shorter window gives better time resolution but not enough time to determine the frequency accurately. In addition to this unavoidable compromise

10“Human ear: Transmission of sound within the inner ear”. Encyclopædia Britannica Online. 11Wightman and Green, “The Perception of Pitch”. 12Benson, Music: a mathematical oﬀering. 13Loy, Musimathics: The Mathematical Foundations of Music, Volume 1 and 2 . 14Fourier transform yields two values for each component frequency: its amplitude and phase. Since the phase of vibrations makes no diﬀerence for sound sensation, in sonograms the phase information is usually omitted. 15In FFT, the width of the window is a power of two.

31 between the accuracy of time and frequency information, the windowing principle itself introduces additional artefacts into a sonogram. Sonograms therefore must be interpreted with caution.

This phenomenon can be demonstrated by a simple experiment (Example 2.1):

Example 2.1: MP3 2.1 CSD 2_1_tones.csd This sound ﬁle contains a synthesized sequence of seven one second long tones with

frequencies that correspond to a whole-tone scale from C4 to C5 in twelve-tone equal temperament. The tones have a sinusoidal waveform and a constant amplitude envelope with short linear rise and release segments. The frequencies of the tones were calculated as f = 261.63 × 2n/6, n = 0..6.

(a) w=1024 (21ms); lin (b) w=4096 (85ms); lin (c) w=16384 (340ms); lin

(d) w=1024 (21ms); log (e) w=4096 (85ms); log (f) w=16384 (340ms); log Figure 2.4: Sonograms of a whole-tone scale

Fig. 2.4 shows six diﬀerent sonograms of the synthesized whole-tone scale, created with three diﬀerent widths of the STFT window and displayed on either a linear or logarithmic frequency scale. The sonograms in the left column (a, d) were made with the window width of 1024 (210) samples, which corresponds to about 21 milliseconds of the sampled signal. The sonograms in the centre column (b, e) were made with the window width of 4096 (212) samples, about

32 85ms, and in the right column (c, f) with the width of 16384 (216) samples, about 340ms. It can be clearly seen how shorter lengths of the window discriminate frequencies less precisely (the frequency bins are wider) but do show the onset and the duration of the tones very well. Longer window lenghts, on the other hand, discriminate frequencies well but the onset times and tone lengths are “blurred”.

The sonograms in the top row of Fig. 2.4 (a, b, c) display the frequency on a linear scale. Since the computed tone frequencies rise exponentially, the series of displayed tone images curves slightly upward. In the bottom row (d, e, f), the frequency is displayed on a logarithmic scale, and the tone images lie on a straight line. Since the frequency bins of FFT are of the same size throughout the frequency range, on a logarithmic scale the lower bins are wider and the higher frequency bins are narrower.

This observation relates to how we hear musical sound: the perceived pitch of tones is roughly logarithmic in relation to their frequency,16 the pitch of tones in the lower frequency range is perceived less acutely than the pitch of higher tones, the pitch of long sustained tones can be determined more precisely, while the pitch of short tones in fast musical passages is perceived as being less distinct.

Reassigned sonograms The technique of time-frequency reassignment can be used to “sharpen” the sonogram image. Since musical sound for the most part consists of tones (distinct frequencies separated by frequency gaps) and silences (time gaps), it is possible by using the phase information (that is otherwise ignored in sonograms) to move the frequency and time points within the frequency bins and window lengths to more speciﬁc positions.17,18

An implementation of the time-frequency reassignment technique for sound analysis is available in the open-source library Loris.19 The Loris library performs time-frequency reassigned analysis of a sound ﬁle, and then detects “partials”: higher intensity “ridges” in the

16This is a simpliﬁed relation that holds well for the middle range of audible frequencies and for complex tones that contain a number of harmonics. For simple tones of lower frequencies (below ca. 500Hz) the relationship is roughly linear. Individual results vary and are subject to perceptual biases (Stevens, Volkman, and Newman, “A Scale for the Measurement of the Psychological Magnitude Pitch”). 17Fitz and Haken, “On the Use of Time: Frequency Reassignment in Additive Sound Modeling”. 18Fitz and Fulop, “A Uniﬁed Theory of Time-Frequency Reassignment”. 19Fitz and Haken, Loris.

33 time×frequency space of the reassigned sonogram that likely represent continuous pitched sounds (typically the harmonics of complex tones). Data that describe the detected partials are then available to a calling program.

Figure 2.5: The whole-tone scale in Qvorum

The Qvorum20 software is a graphical editor that displays time-frequency reassigned data provided by Loris. Each “partial” is shown as a single glyph.21 Fig. 2.5 shows the sound ﬁle from Example 2.1 analyzed by the Loris library and displayed in Qvorum. Notice that the displayed glyphs indicate very closely the pitch and time span of the synthesized sound, and align well with the background 12-tet grid. Small artefacts on each end of each glyph are the result of STFT windows “sliding in and out” and are related to the “blurring” seen on the respective sonogram in Fig. 2.4.

This simple experiment shows that for sparse musical textures reassigned sonograms ade- quately represent musical sound, and can be therefore used for analysis of its microtonal pitch content.

The harmonic series Many physical bodies can vibrate in a simple harmonic motion, repeatedly to-and-fro through a point of equilibrium.22 Simple harmonic motion produces waves of a sinusoidal character and therefore the sound of simple tones. Many “sonorous” physical bodies can vibrate simultaneously in several modes of vibration with diﬀerent frequencies. In particular, stretched strings or enclosed columns of air, such as those that are the basis of string and wind musical

20Burleigh, Qvorum. 21A glyph is a small graphical symbol, such as a short line or curve. 22“Harmonic motion”. Encyclopædia Britannica Online.

34 instruments, vibrate as their whole lengths, in two halves, three thirds, four fourths, etc. The frequencies of their vibrations are thus related by ratios 1:2:3:4:. . . , and form the so-called harmonic series. The lowest frequency in the series is called the fundamental frequency, or the ﬁrst (the lowest) harmonic. The higher frequencies within the series are its higher harmonics.

Vibrations related by the harmonic series, in most cases produced by the many modes of vibration of the same sonorous body, occur so frequently that they are perceived by hearing as a single complex tone, rather than as a set of individual simple tones; this eﬀect might be, at least in part, the result of our conditioning by a lifetime experience of listening to such sounds. It is certainly often possible to hear individual harmonics and distinguish them by ear, but in most cases the harmonics of a complex tone do perceptually “fuse”. Most of the time we recognize changes in relative strength of the harmonics only as changes in the timbre of the musical sound.23

Since the frequencies of the harmonic series are equidistant (separated by the same frequency diﬀerence) and pitch is logarithmically related to frequency, the musical intervals formed by subsequent pairs of frequencies in the series progressively decrease in size.24

Table 2.1 lists the ﬁrst twenty-four intervals found in the harmonic series and indicates which of them are part of the twelve-tone system of Western music. Note that those intervals are formed by frequency ratios of numbers which prime number factorization contains only prime numbers 2, 3, and 5. Such intervals and tone systems made with their use are called to be 5- limit. The intervals formed with higher prime numbers, most notably the number 7 (the ﬁrst prime number after 5) that occurs in the 7:6 ratio (the seventh harmonic), sound as if they

23Reportedly, there are detectable differences in neural activities between non-musicians and musicians, and even non-musicians develop brain networks that are influenced by the pitch structure of the music in their environments. There is also reported evidence suggesting that pitch (as a fusion of individual harmonics) is extracted from the complex tone prior to its processing in the auditory cortex. The detection of a harmonic series might be peripheral in the neural system, which would explain the emergence of sensitivity to consonance and dissonance early in human development, and the apparently innate preference for consonance over dissonance (a mistuned fifth) that has been found in six months old infants. (Trainor and Unrau, “Human Auditory Development”; Folland et al., “Processing simultaneous auditory objects: infants’ ability to detect mistuning in harmonic complexes”) 24The size of a musical interval between two tones is related to the ratio of the two frequencies. Adding intervals requires multiplication of frequency ratios; an inversion of an interval requires inversion of the 2 1 frequency ratio. For example: 1 is the ratio of an ascending octave, 2 of a descending octave. Adding a 4 3 3 4 2 6 3 fourth ( 3 ) to a fifth ( 2 ) gives an octave ( 2 × 3 = 1 ). Subtracting a minor third ( 5 ) from a fifth ( 2 ) gives 3 5 5 a major third ( 2 × 6 = 4 ), etc.

35 were “outside” of the twelve-tone system. “Higher-limit” tone systems (systems that contain higher-limit intervals) are of progressively higher complexity than the 5-limit twelve-tones system of Western music.

frequency ratio interval frequency ratio interval 2:1 octave (P8) 14:13 3:2 ﬁfth (P5) 15:14 4:3 fourth (P4) 16:15 diatonic semitone (m2) 5:4 major third (M3) 17:16 6:5 minor third (m3) 18:17 7:6 19:18 8:7 20:19 9:8 major tone (M2) 21:20 10:9 minor tone (M2) 22:21 11:10 23:22 12:11 24:23 13:12 25:24 chromatic semitone (m2)

Table 2.1: Intervals of the harmonic series

Tuning by the harmonic series, i.e. using frequency ratios of small numbers, is the so-called natural tuning or just tuning. Any 5-limit interval can be tuned by a sequence of octaves (the prime number 2 in 2:1), of fifths (3 in 3:2), and of major thirds (5 in 5:4). For example, a diatonic semitone can be tuned by this sequence: one octave up, one fifth down, one third 2 2 4 16 down, 1 × 3 × 5 = 15 (the difference between a perfect fourth and a major third). A chromatic 5 5 2 25 semitone can be tuned as two thirds up and a fifth down, 4 × 4 × 3 = 24 (the difference between an augmented and perfect fifths), etc.

Note (and this is a clear indication that the twelve tone system is a compromised simpliﬁ- cation of a much more complex system) that in just tuning there are two kinds of the major second (the major and minor whole tones) and two kinds of the minor second (the diatonic and chromatic semitones).

Relations among the harmonics of a complex tone (that is, among the simple tones that form its harmonic series) can be nicely illustrated on the pitch spiral. Fig. 2.6 shows a sequence of six images of the spiral that display one to six harmonics of C4. The spiral is

36 overlaid by two grids of radial lines: the thicker lines show the angles that correspond to justly tuned intervals, while the thinner lines divide the octave into fifty-three equal small intervals. The coloured discs represent the harmonics of the tones; the position of each disc indicates the pitch of the corresponding harmonic and the size of each disc is related to the strength (loudness) of the harmonic. There is one full clockwise turn of the spiral between the first and second harmonics (an octave), slightly over half a turn between the second and third harmonics (a fifth), less than half a turn between the third and the fourth harmonics (a fourth; added to the fifth between the second and third harmonics they make an octave between the second and the fourth harmonics). The fourth and the fifth harmonics, and the fifth and the sixth harmonics form a major third and a minor third, respectively.

Note that the thicker radial lines that indicate the justly tuned intervals, relatively to C4, do quite nicely align with the thin lines of the 53-division. That is a fortuitous coincidence that is exploited further in this work.

The harmonic series is inherent in most musical sounds (made by vibrating strings and pipes) and therefore strongly inﬂuences, or even determines the character of what is music. Even if all musical instruments were made to avoid the harmonic series (for example by allowing only percussion instruments, or instruments with so-called non-harmonic timbres, such as bells, xylophones, steel drums etc.) and thus prevent its impact on what is music, the harmonic series and our conditioning to it cannot be escaped for the one and very reason that it is present in the human voice! Therefore anyone who sings or is sung to is inevitably conditioned to perceive the harmonic series of simple tones as a “fused” unity of a complex tone.

It is possible to learn, quite easily for an attentive listener, to hear separately a few of the harmonics sounded by, for example, a vibrating string. Helmholtz claimed to be able to hear up to sixteen to twenty harmonics of thin strings, with the help of his custom-made ear-ﬁtted and sealed resonators.25

Harmonics of the singing voice Let us now analyze a short recording of a singing voice (Example 2.2). The singer is Iva Bittov´a,a Moravian natural-born musician.26 Her trademark is solo singing while accompa-

25Helmholtz, On the sensations of tone as a physiological basis for the theory of music, p.47. 26http://www.bittova.com/

37 nd (1) The fundamental of C4. (2) Add 2 harmonic (P8).

(3) Add 3rd harmonic (P5). (4) 4th harmonic (P4).

(5) 5th harmonic (M3). (5) 6th harmonic (m3). Figure 2.6: Harmonics of a complex tone, shown on the pitch spiral

38 nying herself on a violin; the combination of the voice and the sound of violin open strings strongly emphasizes the role of the harmonic series in her music. In the example she sings an opening of a Moravian folk song.27 Her intonation is ﬂawless and, except for the folk-style voice modulation at the onsets of sung tones, there is not any prominent vibrato that would mask the tuning precision. Music notation of the example is shown in Fig. 2.7.

Example 2.2: MP3 2.2

rit. Figure 2.7: Tuˇzba(Desire)

Figure 2.8: A sonogram of the singing voice, linear frequency scale

Fig. 2.8 shows a sonogram of the recording on a linear frequency scale. The displayed frequency range (on the left) is from about 100 Hz to about 15 kHz. The horizontal dark lines show the detected harmonics of the voice. The lines are mostly straight, showing steady, sustained pitch. The ocassional, limited vibrato can be seen as the orderly undulation of the lines. The vertical darker bars show the fricative and plosive consonants (in the sung lyrics) that are characterized by broadband frequency content.

27Jan´aˇcek, Tuˇzba (Desire).

39 On the sonogram image it is clearly possible to identify up to 32 harmonics, from the fundamental frequency in the range of about 300–500 Hz, to the highest harmonic frequency in the range of 9–16 kHz. Note that all harmonics are vertically equally spaced, since they are integer multiples of the fundamental frequency (the harmonic frequencies are equidistant).

Figure 2.9: A sonogram of the singing voice, logarithmic frequency scale

Fig. 2.9 shows the same sonogram on a logarithmic frequency (linear pitch) scale. The harmonics of the voice now clearly form a series of intervals of decreasing size (the octave, the ﬁfth, the fourth, etc.).

The recording has been converted to the time-frequency reassigned sonogram. Fig. 2.10 shows the data directly as produced by the Loris library and displayed by the Qvorum software. The thickness of the glyphs indicates the amplitude of corresponding harmonics. 28 The horizontal grid lines labelled 8–14 indicate the octaves from C4 to C10.

The weak harmonics were then eliminated from the sonogram. Fig. 2.11 shows a sonogram that contains only the stronger harmonics.

28The ordinal number scale for octaves follows the convention used, for example, in Csound. If the number 8 indicates C4 (the “middle C”) that has a frequency of about 262 Hz, then 0 indicates the C eight octaves below with a frequency of about 1.02 Hz, and the full range of notes can be comfortably represented by non-negative numbers.

40 Fig. 2.12 shows a closer view of the sonogram overlaid by a 12-tet grid. The coincidence of the partials with the grid lines clearly suggests that during the recording session the 29 musicians tuned to the A4 = 440 Hz pitch standard. Note that the first five harmonics are approximately 12, 7, 5, 4, and 3 equally tempered semitones (an octave, fifth, fourth, major third, and a minor third) apart.

In Fig. 2.13, only the fundamental harmonics (that correspond to the perceived pitch of the complex tones) remain. The sonogram shows only the melodic range G4–D5. Finally, Fig. 2.14 shows the opening phrase in close view. Note that the pitches of the sustained tones, especially the opening ascending ﬁfth G4–D5 and the C5 in the ascending step-wise motif very well coincide with the 12-tet grid.

Figure 2.10: Time-frequency reassigned sonogram of Example 2.2

30 31 In the second example of a singing voice (Example 2.3a), the violin plays a G4 drone (apparently as the second harmonic of the open violin G string — a “ﬂageolet” tone) and the voice sings a melody that is very well tuned relatively to the drone pitch. Music notation

29The pitch standard historically varied from about 370 to 560 Hz. The current standard of 440 Hz, known as the Stuttgart pitch, was recommended by Johann Heinrich Scheibler and accepted by the German Association of Natural Philosophers (die deutsche Naturforscherversammlung) in 1834. (Ellis, “The History of Musical Pitch”) 30Jan´aˇcek, Jabluˇnka(Apple Tree). 31Played by a member of the Skampaˇ String Quartet

41 Figure 2.11: Example 2.2, strong harmonics only

Figure 2.12: Example 2.2, 12-tet grid of the example is shown in Fig. 2.15, and the time-frequency reassigned sonogram in Fig. 2.16. Note that in this case not all the glyphs on the sonogram align completely with the 12-tet grid (although the tones apparently do sound to be very well in tune). Speciﬁcally:

42 Figure 2.13: Example 2.2, fundamental harmonics

Figure 2.14: Example 2.2, the ﬁrst phrase

• The G4s played by the violin and sung by the voice are perfectly in alignment with the horizontal grid line. That makes them a good reference point.

• The two D5s are slightly above the grid line. A justly tuned ﬁfth (G–D) is apparently sharper than the equally tempered one.

• The ﬁrst two sung B4s are below the grid line. This demonstrates that the equally tempered major third (G–B) is sharp compared to natural tuning. The second two

43 sung B4s are closer to the grid line (they are intoned slightly higher, since they “lead” to

the C5 tones that follow them), which clearly demonstrates that tones notated by notes of the same name can have slightly (“microtonally”) diﬀerent pitches, and therefore diﬀerent character or “shade”.

Example 2.3A: MP3 2.3A

Figure 2.15: Jabluˇnka (Apple Tree)

Figure 2.16: A sonogram of a violin/voice duet.

The different tuning and the consequent difference in the character of the two pairs of B4s can be heard in the next example and seen in Fig. 2.17. The four tones were cut from the original sound file and spliced together.32

32 Note that in the spliced excerpt one can hear the pitches G4 – G4 - D5 - A4, tones that were not apparent in the original, as if they were played by plucking the violin strings. This is an artefact caused by reverberation that is present in the recording. What we hear are the reverberation tails of the tones sung just before the selected part. The sung tones to which the reverberation would be normally perceptually attributed are absent from the spliced sound ﬁle. The cut parts have a very short fade-ins that are reminiscent of percussive sounds and therefore create an illusion of the sound of a plucked violin string.

44 Example 2.3B: MP3 2.3B

Figure 2.17: Two shades of B4

Microtonal notation

Microtonal variations in tone pitch can be, with some difficulties, notated using the Western staff notation. For example, the standard natural and accidental signs (\, ], [) can be adorned by small arrows pointing up or down that indicate that the pitch of the respective tone should be slightly sharpened or flattened relative to its normal value. What is a normal value of the pitch and what should be the size of the notated differences from it is a good question, of course.

Fig. 2.18 illustrates how the last example, as shown on the sonogram in Fig. 2.16, could be notated microtonally. It assumes that the microtonal accidental signs indicate a deviation from the twelve-tone equal temperament. Note that in the microtonal notation as used within this text, the meaning of the natural and accidental signs is not carried all the way to the end of the measure, but alters only the immediately following note.

Figure 2.18: Jabluˇnka — microtonal notation

45 The intervals of the harmonic series The harmonic series is the foundation of just (natural) tuning that we have seen in the two examples of a singing voice. The series and the intervals found in it have been studied since Antiquity. Until the early Middle Ages, only the unison, the octave, the ﬁfth, and the fourth were considered to be consonant intervals. The thirds and the sixth were deemed dissonant since, in the Pythagorean view, in their justly tuned form they were “incompatible” with the consonant intervals. The polyphony of early European Renaissance gradually introduced the thirds and the sixths as consonant intervals.33

The first six integer numbers (1 to 6, the “senario”) and the first five intervals (or six, if the unison is included) of the harmonic series formed by them were, and indeed still are, the basis of Western twelve-tone music. At first, numerology was usually used to explain the importance of the number six in music: God created the world in six days, there were six known planets at the time, six faces on the cube, etc. Since the seventeenth century there were numerous advances in music theory based in physics and mathematics, presented in the works of music theorists and scientists: Descartes, Mersenne, Rameau, Euler, d’Alembert and others,34 and the work culminated in the middle of the nineteenth century in Helmholtz’s Die Lehre von den Tonempfindungen.

The modern view of the function of the intervals found in the harmonic series is based on the psychoacoustic theory that Helmholtz proposed and which, at least in my view and for our purposes, still ﬁrmly stands. The role of the ﬁrst several intervals of the harmonic series is summarized below. Keep in mind that:

• In an interval sounded by two complex tones (that may or may not be related by the harmonic series), each of the complex tones is itself a sum of a harmonic series of simple tones. • In a complex tone that is naturally created, for example by a vibrating string, the amplitude of higher harmonics progressively decreases; the lower harmonics (the lower components of the harmonic series) are stronger and therefore more audible and perceptually more important. • By the principle of parsimony of the Ockham’s razor, simpler explanations are to be 33Cohen, Quantifying music: the science of music at the ﬁrst stage of the Scientiﬁc Revolution, 1580–1650 . 34Barbour, Tuning and Temperament: A Historical Survey.

46 preferred.

The unison, 1:1

In the domain of musical intervals, the unison is an identity relation with respect to pitch. Two complex tones of the same timbre (of the same harmonic content) are (do sound) the same, since all their respective harmonics coincide in frequency:

1 2 3 4 5 6 ... (harmonics of the one tone) 1 2 3 4 5 6 ... (harmonics of the other tone)

The octave, 2:1

The octave is an equivalence relation35 with respect to pitch. A tone that is transposed by an octave (or several octaves) up or down is “equivalent” to (i.e. it can reasonably well replace) the original tone. The (numbered) harmonics of two tones that are separated by an octave align this way:

1 2 3 4 5 6 7 8 9 10 ... (the lower tone) 1 2 3 4 5 ... (the higher tone)

The higher tone does not contribute any new harmonic content to the aggregate result of the two tones, it merely strengthens every other harmonic of the lower tone. The sensations of the two tones therefore fuse very well.

The octave is the most “simple” (most consonant) interval of two tones with diﬀerent pitches.

The ﬁfth, 3:2

Between two tones that form a ﬁfth, one half of the harmonics of the higher tone strengthen the harmonics of the lower tone and the other half introduce a new harmonic content: harmonics that were not present in the lower tone.

1 2 3 4 5 6 7 8 9 10 11 12 ... (the lower tone) 1 2 3 4 5 6 7 8 ... (the higher tone)

34“Entities are not to be multiplied beyond necessity.” (“Ockham’s razor”. Encyclopædia Britannica Online) 35A relation that is symmetric, reﬂexive, and transitive. (“Equivalence relation”. ibid.)

47 Transposing a tone by a fifth thus introduces a new pitch that is not “equivalent” to the first one; at the same time, two tones at the distance of a fifth still “fuse” well enough that the fifth can be used as a simple harmonic “colouration” (organum36 ).

The fourth, 4:3

The fourth is an inversion of the fifth with respect to the octave: transposing the lower tone of a fifth or a fourth an octave up, or transposing the higher tone an octave down turns 3 4 the fourth into a fifth and vice versa; a fourth added to a fifth make an octave ( 2 × 3 = 2 1 ). Therefore the fourth is inherently present in a system constructed from octaves and fifths and is not an interval that would introduce new complexity (a new dimension) to the system.

1 2 3 4 5 6 7 8 9 ... (the lower tone) 1 2 3 4 5 6 7 ... (the higher tone)

The (major) third, 5:4

The third does introduce a new dimension into the system. (We shall see how the essentially one-dimensional tuning system of octaves and fifths becomes two-dimensional with the introduction of the third.) Every fourth harmonic of the higher tone coincides with every fifth harmonic of the lower tone. The two tones are thus reasonably strongly “anchored” to each other,37 yet each has a quite distinctly different harmonic content.

1 2 3 4 5 6 7 8 9 10 ... (the lower tone) 1 2 3 4 5 6 7 8 ... (the higher tone)

The minor third, 6:5

The minor third is simply an inversion of the major third with respect to the fifth, analogous to the fourth being an inversion of the fifth with respect to the octave. The minor third is thus inherently present in a system that contains fifths and major thirds.

36In mediaeval chant, a second voice doubling the principal voice a fourth or a fifth below. (“Organum”. Encyclopædia Britannica Online) 37Provided that their timbres have at least five harmonics. However, if difference tones (p.52) are taken into account, two harmonics suffice to strongly connect the two tones of the major third. E.g. two tones with frequencies of 400 and 500 Hz also contain harmonics of 800 and 1000 Hz. The difference tones of 500−400 = 100, 1000−800 = 200, 800−500 = 300, and 1000−400 = 600 Hz nicely complete the “senario” of 100–200–300–400–500–600 Hz.

48 Furthermore, the inversion of the major or minor third with respect to the octave makes the minor or major sixth, respectively; the inversion of the major or minor third with respect to the fourth makes the minor or major second, and the inversion of the major or minor second with respect to the octave makes the minor or major seventh. All intervals of the Western chromatic scale are present in a system generated by the octave, the ﬁfth, and the third.

The subminor third, 7:6

The next interval of the harmonic series is, in Helmholtz’s terminology, the subminor third. It cannot be derived from the first six intervals and therefore introduces another level of complexity (a third dimension) to the tuning system. The twelve-tone system generated by the three of the first five intervals in the harmonic series (that is, the octave, the fifth, and the third) is already sufficiently complex to support a multitude of musical works created over the course of a number of centuries. The complexity of a system that includes the subminor third among the generating intervals would, conceivably, be overwhelming, and therefore this interval and also all other higher-limit intervals (intervals which frequency ratios contain prime numbers 7 and higher) are not used in Western music.

Consonance and dissonance, redeﬁned In the usual context of a music discourse, consonance is “a harmonious sounding together of two or more tones [...] with an ‘absence of roughness’.” Dissonance is “then the antonym to consonance with corresponding criterion of ‘roughness’ [...] [that] implies a psychoacoustic judgement.”38 This meaning of the word ‘consonance’ applies to “stable” intervals and chords (the unisons and octaves, ﬁfths and fourths, thirds and sixths, major and minor triads, and such) and ‘dissonance’ to intervals and chords with more “tension” (the seconds and sevenths, augmented and diminished chords, and such). The perception of certain intervals and chords as consonant or dissonant has changed throughout music history and is, to a point, graded subjectively.39 There is, of course, the implied assumption that the intervals and chords in question are played well “in tune” and that they are judged to be consonant or dissonant solely based on the intervallic relations among their tones.

For the remainder of the text we redeﬁne the meaning of ‘consonance’ and ‘dissonance’ in

38“Consonance”. Oxford Music Online, (Claude V. Palisca). 39“Consonance and dissonance”. Encyclopædia Britannica Online.

49 the following way:

Consonance: Two or more tones sounding together are consonant if the perceived “roughness” or “tension” is locally minimized and cannot be improved by microtonal adjustment of their pitches. That is, if any of the consonant pitches is slightly sharpened or ﬂattened (by signiﬁcantly less than a semitone), the “roughness” or “tension” increases. By “locally” we mean that the roughness cannot be lowered by adjusting the tone pitches within their “neighbourhood”,40 but could be lowered “globally”, i.e. by changing the pitches by larger amounts (more than a fraction of the semitone). Such adjustment would, of course, change the character of the interval or the chord.

Dissonance: The opposite of ‘consonance’.

“Consonant” resp. “dissonant” thus means “sounding well in tune” resp. “out of tune”. A sensible musician that sings or plays a string or wind instrument in a choir or an orchestra, given a note to sing or play, naturally does what we have just described: makes the tone sound “locally consonant”, i.e. microtonally adjusts its pitch so that the tone sounds as much in tune as possible, however strong the cumulative tension among all tones sung or played by the musical group may be.

Our goal is to do the same thing computationally: ﬁnd the points of local consonance and use them in computer-assisted microtonal music.

Helmholtz’s Tonempfindungen Hermann von Helmholtz (1821—1894) was a truly versatile scientist and “the last scholar whose work [...] embraced all the sciences, [...] philosophy and the fine arts,” who made a number of significant contributions to the fields of anatomy, physiology, neurology, op- tics, thermodynamics, hydrodynamics, electrodynamics, acoustics, geometry, and philosophy.41

Helmholtz carried out rigorous investigations into the nature of musical sound, supported by an extensive set of experiments. In 1863 he published his ﬁndings in a seminal work Die Lehre von den Tonempﬁndungen, translated by Alexander John Ellis as On the Sensations of Tone

40“Neighborhood”. Wolfram MathWorld™ . 41Turner, “Dictionary of Scientiﬁc Biography”.

50 and ﬁrst printed in English in 1875. The most consequential result of Helmholtz’s work is his psychoacoustic theory of tone consonance that explains harmonious relationships of musical tones, intervals and chords, that also allowed him to make connections between physiological acoustics and the theory and aesthetics of tonal music. Ellis writes in his Translator’s Notice to the second English edition that the book is “a work which all candidates for musical degrees are expected to study.”

In his translation of Helmholtz’s book, Ellis promoted the ‘cent’, a unit for measuring musical intervals that Ellis introduced.42 Cent measurements are clearly marked additions to Helmholtz’s text (Helmholtz himself did not use cents). The cent is widely adopted today, however in this text we will use it sparingly and only as a reference unit.43

Helmholtz’s theory of tone consonance is beautiful in its simplicity. Recall that complex tones are sums of harmonic series of simple tones with sinusoidal waveforms. Two simultaneously sounding simple tones with sufficiently close frequencies produce audible “beats”. The beats are periodic fluctuations in the amplitude of the combined sound, caused by the interference of the two sound waves. The frequency of the beats is equal to the difference of the two frequencies. Beats that are faster than a few cycles per second cause a rough, unpleasant sensation that leads to a perception of tone dissonance. Two or more tones are consonant when the rough sensation is minimized. That is certainly so for the intervals of the harmonic series (see p.46), which harmonics are either coincident or sufficiently separated.

Helmholtz also explained the known phenomenon of combination tones that can often be heard in addition to simultaneously sounding simple tones. When two simple tones of a

42“cent”. Oxford Music Online. 43In my view, the wide acceptance of the cent as a measurement unit, without realizing its full impact, is unfortunate. The cent is 1/100 of the equally tempered semitone. The equally tempered intervals have nice round cent values, and thus may seem somehow more correct or natural than justly tuned intervals which cent values are fractional numbers. Using the cent unit therefore reinforces the notion of certain dominance of the equally tempered tuning over just intonation. Paradoxically, Ellis himself was much in favour of just tuning, as evidenced in his report to the Musical Association of London (1875): “At any rate just intonation, even upon a large scale, is immediately possible. And if I long for the time of its adoption, in the interests of the listener, still more do I long for it in the interests of the composer. What he has done of late years with the rough-and-ready tool of equal temperament is a glorious presage of what he will do in the future with the delicate instrument which acoustical science puts into his hands. The temporary necessity for equal temperament is passing away. Its defects have been proved to be ineradicable, because inherent in the nature of sound. An intonation possessing none of these defects has been scientiﬁcally demonstrated. It is feasible now on the three noblest sources of musical sound — the quartet of voices, the quartet of bowed instruments, and the quartet of trombones. The issue is in the hands of the composer.” (Vogel, On the relations of tone, p.9)

51 sufficient intensity are sounded simultaneously, one can hear a third difference tone with a frequency equal to the difference of the two source frequencies and, under certain conditions, possibly several summation tones with frequencies equal to various sums of the two source frequencies. Demonstrably, combination tones do not physically exist outside the ear, but are created by the nonlinear mechanical response of the eardrum and the ossicles in the middle ear. Once created in the ear, the combination tones are indistinguishable from the source tones, contribute to the composite sensation of tone, and are also an additional source of beats.

Several ﬁner points of Helmholtz’s theories have been challenged in relation to various psychoacoustic phenomena, such as the explanation of the “missing fundamental”.44 Neverthe- less, it can be veriﬁed that in reasonable situations and within reasonable ranges of sound frequencies and intensities Helmholtz’s model correlates very well with the results of experiments in perception of musical sound.

Helmholtz performed his experiments using mechanical devices. For tone generation he used mechanical sirens: rotating metal discs with concentric rings of equidistant holes that modulated jet streams of air. Two simultaneous tones were sounded by a double siren, in which mechanical gears set the ratio of tone frequencies. To analyze complex tones (to detect the presence of their harmonics), Helmholtz used sets of tuned resonators with protruding necks that could be ﬁtted directly into the ear canal and sealed with wax.45

Experiments Today, most of Helmholtz’s experiments can easily be reproduced with a personal computer, digital audio software, and a pair of quality headphones. The reader is encouraged to conduct the following experiments or, at the least, listen closely to the sound ﬁles that are provided on the accompanying web site.

Example 2.4A (audible range): MP3 2.4A CSD 2_4_range_A.csd

44Wightman and Green, “The Perception of Pitch”. 45The mechanical nature (and therefore imperfections) of sound generation by the siren may have been the cause of a rare, interesting mistake in Helmholtz’s text: he thought the upper limit of hearing to be around 40 kHz, twice the actual limit of about 20 kHz. My conjecture is that Helmholtz might have heard a subharmonic frequency of the siren tone.

52 The purpose of this experiment is to test the frequency range of the used audio system (mainly the headphones) combined with the frequency range of the listeners hearing. The example also demonstrates that the perception of the pitch of simple tones is not precise.

The audio ﬁle contains series of simple tones, synthesized as sine waves with plain amplitude envelopes. The series begins with the frequency at the low end of the audible range, and progresses by octaves (by doubling the frequency) toward the high limit of the audible range. Each frequency is played as a sequence of three two seconds long tones: two seconds for the left ear only, two seconds for the right ear, and two seconds for both ears. The frequencies are: 31.25 Hz, 62.5 Hz, 125 Hz, 250 Hz, 500 Hz, 1 kHz, 2 kHz, 4 kHz, 8 kHz, and 16 kHz.

With the possible exception of the lowest (31.25 Hz) and the highest (16 kHz) frequencies, you should clearly hear all tones. The lowest frequency might be less audible due to the naturally lower sensitivity of hearing in the low range, and also due to a possible roll-oﬀ in the frequency characteristic of the used audio interface and headphones. With a pair of quality studio headphones, however, even the lowest frequency should be heard or at least “sensed”. If you can hear a tone in one ear but not the other, try to turn the headphones around; that should show whether it is due to a diﬀerence in sensitivity between your ears or the headphones.

Whether you are able to hear the 16 kHz tones depends very likely less on the headphones and more on your hearing. Young, healthy ears hear tones of frequencies up to 20 or 21 kHz; with age (and possible hearing damage mostly due to exposure to high intensity sound) the high limit of the hearing range decreases.46 It is possible that one hears another sound when the 16 kHz tones are being played: sound that apparently is of a lower frequency, or a mix of frequencies. That would be most likely some kind of a distortion or cross-modulation in the audio system. If such sound is present and its intensity is not negligible, it is recommended to use a better quality audio system.

You should be able to hear all the frequencies between 62.5 Hz and 8 kHz clearly and with no apparent distortion. It is a necessary condition for success of the experiments that follow. Note that the lower frequencies and the higher frequencies sound less loud than the middle- range frequencies. This is a normal eﬀect that is due to the varied sensitivity of the ear,

46As of the time of writing this text my own hearing range seems to end somewhere between 15 and 16 kHz.

53 known as the Fletcher-Munson curves of equal loudness.47

Observe that in each group of three tones of the same frequency (left ear, right ear, both ears) the perceived pitch is not necessarily the same. The pitch perceived in one ear may be higher than in the other ear, and the perceived pitch of the tone that is played to both ears is then somewhere between the two individual pitches. Listen closely and note which ear hears each frequency as a higher pitch than the other ear. Turn the headphones around and repeat the experiment: does the difference in perceived pitches at all depend on which ear hears the tone first? Return the headphones to their original position and listen again: has the experience changed? I myself perceive the pitch of tones in the low range (below 500 Hz) to differ by about a semitone, consistently higher in one ear (which one, depends on the frequency). From 500 Hz up, the perceived pitch difference is very small but still noticeable, and which ear perceives the pitches higher changes back and forth with repeated listening and changed loudness of the tones.

Example 2.4B (low range): MP3 2.4B CSD 2_4_range_B.csd

This experiment tests the low limit of the audible range in more detail. A series of simple (sinusoidal) tones with frequencies starting from 250 Hz and descending by one third of an octave (the equally tempered major third) is again played for each ear separately and then for both ears together; starting with the right ear for a change. The frequencies are: 250, 198, 157, 125, 99.2, 78.7, 62.5, 49.6, 39.4, 31.3, 24.8, 19.9, and 15.6 Hz.

This time, perhaps because of the melodious relationship among the tones (descending thirds in a regular time pattern) I tend to perceive the ﬁrst tone (in the right ear) as having the higher pitch in each pair of tones.

Note that there are two competing sensations in the low frequency range: one is that of a pitched tone, the other of “growling”: separate, distinct sound pulses. Helmholtz calls such sensation to be “intermittent”, and a source of unpleasant “sensory roughness”. Below about 60 Hz the intermittent “growling” sensation becomes more dominant than the sensation of the pitch. Below about 30 Hz the sensation of the pitch vanishes, and only the feeling of separate pulses remains.

47“Sound: Dynamic range of the ear”. Encyclopædia Britannica Online.

54 Example 2.4C: MP3 2.4C CSD 2_4_range_C.csd

The high limit of the audible range is tested in this experiment. The series of simple sinusoidal tones starts from 4 kHz and ascends by equally tempered major thirds. The frequencies are: 4.00, 5.04, 6.35, 8.00, 10.1, 12.7, 16.0, and 20.2 kHz. There is no intermittent sensation of any sensory roughness in this range.

Example 2.4D: MP3 2.4D CSD 2_4_range_D.csd

The high limit of hearing is tested even in more detail. A series of simple tones with frequencies ascending from 8 kHz to 18 kHz by 500 Hz steps is played for both ears. Since the hearing sensitivity decreases in high frequency range, you should notice a progressive drop in loudness from about 12 kHz up to the limit of your hearing.

Example 2.5A: MP3 2.5A CSD 2_5_pitch_A.csd

In Example 2.4B we have encountered a case of what might be called a “corrective perception of pitch” (see p.21). That is, in situations where the pitch perception is uncertain or where it is in conflict in what the listener expects (such as a familiar musical context), the perceived pitch differs from the pitch that would be otherwise perceived as a logarithmic function of the given frequency. As a real-life example: the sound of a musical group that plays poorly in tune feels insufferable at first, but it may become to be perceived tolerable as the concert proceeds, since the ear attempts to “correct” the out-of-tune pitches.

In this experiment, a sequence of tones of the same pitch (C4) that alternate between the left and right ears is played. At first, the sequence is played in a slow tempo, with longer silences between the tones. The pitch of the tones may be perceived by each ear as being slightly different. As the speed of the sequence increases and the silences shorten, the pitches get perceptually closer. If the sequence is played again, it may seem that the pitches, as perceived by each ear separately, are the same from the very beginning. This effect could be explained by short-term musical memory: the listener remembers the pitches from the first time, and therefore is more likely to identify them as being identical when the sequence is repeated.

55 On the other hand, if one concentrates on hearing the tones as diﬀerent in pitch, it is more likely that they may be perceived as such.

Example 2.5B: MP3 2.5B CSD 2_5_pitch_B.csd

In the previous experiment, I perceived the pitch of the tones to be slightly higher in the left ear than in the right ear. I therefore changed the frequency of one of the tones, to make them perceptually of equal pitch. In this example the frequency of the tones played on the

left is 261.6 Hz (C4), and on the right 263.2 Hz (the diﬀerence is 9/1000 of an octave).

The reader is encouraged to repeat this experiment individually with the help of the provided Csound source ﬁle, and compare the results with mine.

Conclusion: The perception of pitch of simple tones is inaccurate; it depends on the frequency range, the loudness of the sound, and other factors. Each ear may perceive the pitch of a tone of the same frequency as being slightly diﬀerent. Perception of pitch is also inﬂuenced by the musical context, the memory of a past experience, or a conscious expectation.

Example 2.6A: MP3 2.6A CSD 2_6_series_A.csd

There are some musical instruments with timbres that are similar to that of a simple tone, with a strong fundamental harmonic and weaker higher harmonics, such as the ﬂute, glass harmonica, or the Fender Rhodes piano (with its ampliﬁed sound of metal tines), but most musical instruments produce complex tones that consist of a number of harmonics.

In this example we introduce complex tones with rich harmonic content. The sound example has several parts:

1. Sixteen simple tones which frequencies are related by the harmonic series are played.

The frequency of the lowest tone is 263.2 Hz (the middle C4). 2. A complex tone is built from the same set of simple tones that now function as its harmonics. Note that as the harmonic content changes (new high harmonics are added to it), one can always hear the highest harmonic distinctly (since it is the cause of the most recent diﬀerence in the character of the sound), but the lower harmonics do

56 seamlessly blend into the composite sensation of a complex tone. As the number of harmonics increases, the sensation of the complex tone becomes richer. 3. A series of tones that at ﬁrst are becoming gradually more complex (by more harmonics added to them) and then, in reverse order, less complex (the harmonics being removed) is played. The perceived pitch of all tones remains the same and corresponds to the frequency of the fundamental harmonic, but their timbre changes. The highest harmonics do perceptually stand out because they are the diﬀerence between each two consecutive tones, but the rest of harmonics do blend well together.

Figure 2.19: Example 2.6A

Fig. 2.19 shows the sound ﬁle analyzed by the Loris library and displayed by the Qvorum software. The horizontal grid lines indicate the pitches of the 12-tet system. Note that the “5-limit” harmonics (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16) are aligned reasonably closely with the 12-tet grid lines. The higher-limit harmonics (7, 11, 13, 14) do clearly lie outside of the twelve-tone system.

Example 2.6B: MP3 2.6B CSD 2_6_series_B.csd

In this example, a complex tone with the same range of harmonics as the one in Example 2.6a is built, but only from the “5-limit” harmonics. Harmonics 7, 11, and 13 are omitted.

57 This complex tone sounds more harmonious. The included 5-limit harmonics emphasize the intervals of the octave, the ﬁfth, and the third: the intervals that are the building blocks of Western harmony. Including higher-limit harmonics would lead to “extended just intonation”, more complex and unusual tonal systems, as the one employed by Harry Partch in his music.

Figure 2.20: Example 2.6B

Fig. 2.20 (rotated right to save space) shows in detail how the 5-limit harmonics align with the 12-tet grid.

Example 2.7: MP3 2.7 CSD 2_7_complex.csd

The musical etudes that accompany this text (the “C53 Suite”, see p.94) were rendered using several synthesized timbres that are arguably quite harsh in sound. Most of the timbres were constructed by additive synthesis from a required number of harmonics with simple amplitude envelopes and their frequency spectrum was limited by a low-pass filter to avoid aliasing of high frequencies. There is no vibrato, chorus effect, detuning of harmonics, phasing, or other effects that are usually used to make synthesized sound sound smoother and more interesting. The effects were omitted on purpose: our attention ought to be focused on issues of tuning, and for that reason there should be a minimum of sound masking and other distractions.

In this example, ﬁve instruments play the same well-known melodic theme. Note that although the pitches of the tones (their frequencies) are exactly the same in each version of the theme, the perceptual results diﬀer depending on the timbre of the instrument:

58 1. Simple sinusoidal tones. The melody is recognizable, but the exact tuning of the pitches seems uncertain. 2. The “flute” timbre with only two harmonics. The pitch of the tones is more determined. 3. The “harmonium” timbre with four harmonics. The pitch and the contour of the melody are now quite certain. 4. The “organ” timbre contains all 3-limit harmonics up to the 16th (1, 2, 3, 4, 6, 8, 9, 12, 16) that emphasize the octaves and fifths. The pitch is well-defined, but also very sensitive to tuning, and some tones in the melody sound slightly out of place. That is because the excerpt is in just 5-limit tuning while the timbre favours Pythagorean tuning (see p.70). 5. The “full” timbre contains all 5-limit harmonics up to the 16th (1, 2, 3, 5, 4, 6, 8, 9, 10, 12, 15, 16). The sound of the instrument is less clear (it has more inner tension) than the “organ” timbre. However, since the timbre contains thirds in addition to octaves and fifths, there is no sensation hinting that any of the tones of the melody would be mistuned as it was in the case of the “organ” timbre.

Example 2.8A - beats of simple tones: MP3 2.8A CSD 2_8_beats_A.csd

The reader is encouraged to conduct this and the following experiment using the pitch spiral tool. A sound ﬁle is provided for illustration, but to appreciate the real nature and extent of this sensation requires interactive control of the tones. Becoming familiar with the sensation of beats is most important for apprehension of tone consonance.

Listen to two simple sinusoidal tones. Begin with both tones at the same pitch (for example,

C4), sounding in unison. Keep the pitch of one tone fixed while slowly changing the other. The interference of the waves with close, but unequal frequencies creates beats with frequency that is equal to the difference in frequencies of the two tones. Notice how the beats are slow at first, then increase in speed and eventually morph into an intermittent sensation of sensory roughness that is the strongest around 20–40 Hz. Up to this point of the maximum roughness, the two tones were perceived as a single, perceptually fused pitch. If the frequency difference is increased further, the two pitches become perceptually separated and the strength of the rough sensation begins to decrease. Eventually one begins to hear the difference tone, a tone with frequency equal to the difference of frequencies of the two tones, created by a nonlinear distortion within the middle ear.

59 Notice that, apart from the unison and the octave, no interval can be clearly perceptually determined. A perfectly tuned unison does not beat at all, and therefore even a slight mistuning is easily recognizable. In a mistuned octave one can perceive beats between the lower pitch and the difference tone. The fifths, the fourths, the thirds and so on, whether equally tempered or justly tuned, do not stand out in any distinct way and can be recognized only on the basis of a listener’s long-term pitch memory and therefore with a certain difficulty and within a margin of error.

Example 2.8B - beats of complex tones: MP3 2.8B CSD 2_8_beats_B.csd

Repeat the experiment 2.8A, now with two complex tones of a timbre with richer harmonic content of four to six harmonics. Even though the simple tones of the harmonic series in each complex tones are perceived together as one “fused” sensation of a complex tone, each pair of harmonics does create their own beats.

Since the fundamental harmonics are the strongest, the sensation caused by the beats between them prevails over the sensations caused by beats between other pairs of harmonics and the overall sensory roughness is similar to that of the two simple tones in the previous example. However, the beats between higher harmonics contribute to the composite sensation and do further increase the overall roughness. Whenever any two harmonics coincide in frequency, the beats between them cease. As a result, the overall roughness has distinct “local minima” that clearly mark justly tuned intervals: the octave (for 2-limit timbres with at least two harmonics), the ﬁfth and the fourth (for 3-limit timbres), the thirds and the sixths (5-limit timbres) and, with attentive listening also the seconds, the sevenths, etc.

60 Chapter 3: Octave divisions

We are now familiar with the harmonic series and the sensations of simple and complex tones, and can proceed to investigate how a tone system (a musical scale) can be constructed on the basis of these principles (and as always, the principle of Ockham’s razor).

Octave equivalence

Figure 3.1: Octave equivalence: C2, C3, C4, and C5

The construction of a tonal system begins with an arbitrary choice of a single pitch that serves as the pitch standard and the central point of the system. We will start with C4 (the “middle” C), set to 261.6 Hz (the “Stuttgart”1 pitch).

The octave is an equivalence relation (see p.47). Every given pitch can be replicated by octave transpositions (multiplying and dividing its frequency by two) across the full range of the system. From the original C4 one thus obtains all Cs. Fig. 3.1 shows the pitch spiral

1See the footnote on p.41.

61 Figure 3.2: Octave equivalence with diﬀerence tones

with complex tones C2, C3, C4, and C5. Each tone has six harmonics (the “senario”). Note that all harmonics of the tones are perfectly lined up; there are no beats and no sensory roughness caused by them, that would not be already inherently present among the six harmonics of any one of the complex tones. Even when diﬀerence tones are considered, as shown in Fig. 3.2, the image is one of calmness.

Division of the octave by cycles of ﬁfths Using the principle of octave equivalence, all pitches with the same name can be generated from a single one from among them. Next, the second most simple and most consonant interval of the harmonic series, the ﬁfth, is used to generate other pitches. The new pitches will then also be replicated by the octave equivalence through the whole pitch range.

Since the fifth is the simplest interval that is not an equivalence, pitches that are related by the interval of the fifth, although distinctly different, are harmonically very closely related (half of the harmonics of the higher tone strengthen those already present in the lower tone, see p.47). The sequence of pitches linked by the interval of the fifth is known as the cycle

62 C C–G C–G–D

C–G–D–A–E C–G–D–A C...E–B MOS: pentatonic scale

C...B–F] C...F]–C] C...C]–G] MOS: heptatonic scale

C . . . A]–E] C . . . G]–D] C...D]–A] MOS: chromatic scale Figure 3.3: The cycle of ﬁfths with moments of symmetry

63 of ﬁfths.2 Fig. 3.3 shows the sequence of screenshots of the pitch spiral that illustrate the construction of the cycle of ﬁfths and its “moments of symmetry” (MOS).

Moments of Symmetry

A moment of symmetry occurs when a tone system has only one or two sizes of steps (intervals between adjacent notes).3,4,5

Among chains of ﬁfths of increasing length, moments of symmetry occur for (see Fig. 3.3):

• The two fifths long chain (three tones, C–G–D): this chain is too short and does not make a usable musical scale. • Four fifths (five tones, C–G–D–A–E): a usable minimal scale; one of the possible pentatonic scales. • Six fifths (seven tones, C–G–D–A–E–B–F]): the most common scale in Western music, the heptatonic (diatonic) scale. The chain of seven fifths generates the modern Lydian mode, and not the Ionian (major) mode. This will be corrected shortly. • Nine fifths: there is no strong significance of this system, since it is practically the twelve-tone system (below), only with two tones missing. • Eleven fifths (twelve tones, C...E]): The Western chromatic scale is completely symmetric with only one size of step, from which various combinations of seven tones, i.e. various heptatonic scales, can be drawn.

Just intervals are not compatible: The Pythagorean comma

The remaining, twelfth fifth that would be expected to “close” the cycle (E]–B], B] being enharmonically equivalent to C) does, in fact, generate a new pitch (Fig. 3.4). The small interval between C and B] (B] is higher in pitch than C) is known as the Pythagorean comma; it is the difference between twelve fifths and seven octaves, or the interval that

2In the traditional musical context, the ‘cycle of ﬁfths’ is more often referred to as the ‘circle of ﬁfths’. In this text we prefer the term ‘cycle’ as a “recurring series”, rather than ‘circle’ that might wrongly suggest a geometric shape. 3Schulter, Why are there 12 notes per octave on typical keyboards? 4Wilson, Moments of Symmetry. 5“Moment of symmetry” in this sense does not imply that the scale itself is necessarily symmetric. A musical scale needs at least two sizes of steps to establish a tonal centre. Completely symmetric scales, such as the “augmented” scale of six whole tones (also known as the whole-tone scale) or the “diminished” scale of four semitones alternating with four whole tones (also known as an octatonic scale) are ambivalent in that respect: there is no clear point of origin. On the other hand, a scale with three sizes of steps can be symmetric, but violates the Ockham’s razor principle.

64 remains after tuning five fifths up (5×7 semitones) and seven fourths down (7×5 semitones): 3 5 3 7 312 531441 ( 2 ) × ( 4 ) = 219 = 524288 ≈ 1.14. If the cycle is limited to twelve tones (the original C substituted for the derived B]), the twelfth fifth E]–C is by the Pythagorean comma narrower and is severely mistuned; such out-of-tune interval is called a wolf interval, because it “howls” when played.

Figure 3.4: The Pythagorean comma

Similar situations (other kinds of “commas”) arise when octaves are compared with chains of major thirds (the diﬀerence is the “lesser diesis”) or minor thirds (the “greater diesis”), or 81 6 ﬁfths compared with thirds (the syntonic comma, 80 ). The intervals are, in a Pythagorean view, “incompatible” (cf. p.46).

Possible solutions

The problem of incompatible intervals can be alleviated, if not solved, by these means:

1. Avoid the use of wolf intervals; this choice requires that the structure of the music is

6This is related to the fact that every natural number has a unique prime factorization; see the “fundamental theorem of arithmetic”. (“Fundamental theorem of arithmetic”. Wolfram MathWorld™ )

65 kept simple, without the use of chromaticism and modulation to diﬀerent keys. 2. Make a tuning compromise: intentionally mistune (temper) the justly tuned intervals in order to make the wolf intervals less mistuned. 3. Dynamically adjust the pitch of the twelve tones as required. 4. Use more than twelve pitches in an octave.

We shall brieﬂy demonstrate the principle of the solution 2) and then discuss the solution 4) that is at the core of our method.

Tempering out the Pythagorean comma

The first eleven fifths in the Pythagorean tuning are justly tuned; the remaining, twelfth fifth is narrower by the Pythagorean comma. If each of the eleven fifths is flattened by one twelfth of the Pythagorean comma, then the wolf fifth that is flat by the full comma will 1 1 become also flat only by 1 − 11 × 12 = 12 of the comma, and all twelve fifths will be equally mistuned; but not enough to be considered “wolf” intervals. This principle is the foundation of the twelve-tone equal temperament. Fig. 3.5 shows equal tempering on the pitch spiral (left to right).

Just ﬁfths ⇒ Equally tempered ﬁfths Figure 3.5: Tempering out the Pythagorean comma

Fig. 3.6 shows a justly tuned fifth on the left with its harmonics and difference tones nicely lined up. On the right it shows an equally tempered fifth. Its harmonics are very slightly misaligned; the sensory roughness due to beats is quite acceptable.

The sound of the third is a more serious problem in equal temperament. In the Pythagorean tuning there are really no thirds; rather, a chain of four ﬁfths (e.g. C–G–D–A–E) generates

66 Just fifth Equally tempered fifth Figure 3.6: Just and equally tempered fifths

a ditone C–D–E, an interval that is sharper than the justly tuned third by the syntonic 3 4 1 2 4 81 comma: ( 2 ) × ( 2 ) × 5 = 80 . The equally tempered third is narrower than the Pythagorean ditone, but it is still quite sharp. Fig. 3.7 shows the ditone (left), the just third (centre), and the equally tempered third (right).

Pythagorean ditone Just third Equally tempered third Figure 3.7: Pythagorean, just and equally tempered thirds

Tempering out the syntonic comma

To temper out the syntonic comma of the ditone, each ﬁfth must be ﬂattened by one quarter of the syntonic comma (SC). Fig. 3.8 illustrates the process (left to right).

67 Equally tempered ﬁfths ⇒ Fifths tempered by SC/4 Figure 3.8: Tempering out the syntonic comma

The result is the quarter-comma meantone temperament (Fig. 3.9). Its name is derived from the incidental fact that its major second (whole tone) is the mean between the justly tuned minor and major tones.

The meantone temperament has semitones of diﬀerent sizes, and it also contains wolf intervals. As a result, some keys sound very well in meantone temperament, mainly because their important major triads (tonic, subdominant, dominant) contain properly tuned major thirds and therefore have a “sweet” sound. Other keys sound more rough, and some keys are not usable at all.

Figure 3.9: The meantone temperament (unequal semitones)

68 Temperaments — how do they sound? The following set of examples demonstrates the impact of the necessary tuning compromises in the four discussed tuning systems:

• The Pythagorean tuning: eleven pure fifths and one wolf fifth; the quality of the rest of the intervals is incidental. Pythagorean tuning is not adequate for Western tertian harmony, since it contains ditones rather than thirds.7 • Just tuning: all steps in a scale are tuned to frequency ratios of small numbers. In particular, the tonic, subdominant, and dominant major triads are tuned perfectly. Just tuning contains the best tuned intervals and chords, but only in one key. • The meantone temperament: the eleven fifths are flattened, each by a quarter of the syntonic comma, and provide nine justly tuned major thirds. The twelfth fifth and the three remaining major thirds are too wide. The meantone temperament is very well usable in several, but not all keys. Each key has some tuning imperfections in different places; the keys are thus not equal and each has a distinct character. • The equal temperament: all twelve fifths are equally narrowed by one twelfth of the Pythagorean comma and all twelve thirds are equally sharp. The equal temperament does not contain any wolf intervals, but besides the octaves there are also no perfectly tuned intervals. All keys are equally usable in equal temperament, and each key has the identical, slightly mistuned character.

Listen to the following sound examples and compare the tunings of the musical excerpts:

Example 3.1 (chromatic scale): MP3 3.1P/J/M/E CSD 3_1_P/J/M/E.csd

The twelve-tone chromatic scale as tuned in (P)ythagorean tuning, (J)ust intonation, (M)eantone temperament, and (E)qual temperament.

Example 3.2 (major scale and triad): MP3 3.2P/J/M/E CSD 3_2_P/J/M/E.csd

The ascending and descending heptatonic major scale, followed by the tonic major triad. The most calmly sounding is the perfectly tuned triad in just tuning, next is the triad of

7To be precise, Pythagorean tuning does contain four good thirds: those generated by the chains of four ﬁfths among which one is the wolf ﬁfth.

69 the meantone temperament (with its just third and a tempered fifth), and then the equal temperament (both the third and the fifth are tempered). The Pythagorean tuning (a just fifth, but a ditone instead of a third) is the least adequate.

Example 3.3 (Prelude in C opening): MP3 3.3XP/J/M/E CSD 3_2_XP/J/M/E.csd

In this example there are five groups of four sound files. The opening four measures of the Prelude (the harmonic progression C–Dm7–G7–C) are played in the original arpeggiated form that is followed by sustained chords, in the four different tunings. The Prelude is first presented in the original key of C, and then transposed to keys F , G, E[, and E (while keeping the tunings set for the key of C).

The Pythagorean tuning is, as could have been expected, not suitable for the Prelude in any of the keys. The Prelude sounds the purest in just tuning, but only in its original key of C (for which this particular instance of just tuning has been adapted). In all other keys to which the Prelude has been transposed the just tuning is unacceptable. In the equal temperament, the Prelude sounds reasonably well (or equally badly, one could say) in all keys.

In the meantone temperament the sound of the Prelude is, perhaps, the most interesting. None of the triads are tuned perfectly, but since the purity of major thirds is more vital than the purity of the fifths, the major thirds give most of the triads a sweet, full character. The scale obtained by meantone temperament is irregular and therefore different triads are mistuned differently, which gives them individual character. However, the five wolf thirds and the wolf fifth are badly out of tune and therefore the keys in which they would play a fundamental role are unusable. In this example, the key of C is the best in tune, the keys F , G, and Eb are quite acceptable with an “interesting” sound, and the key of E cannot be used at all.

Pythagorean (the fifth) tuning We have shown how the Pythagorean tuning generates a twelve-tone scale, by tuning a chain of eleven fifths (p.62).The first six fifths starting from C generated a heptatonic scale:

C - G - D - A - E - B - F#

70 Sorted with the help of octave equivalence and assuming that the starting tone C is the tonic, that created a diatonic scale in the Lydian mode: CDEF] GAB.

The most common major (Ionian) mode can be tuned, with C as the tonic, by tuning one fifth down (the subdominant F ) and five fifths up (the dominant G and the sequence of secondary dominants):

F- C-G-D-A-E-B

The C major scale (CDEFGAB) that is thus obtained corresponds to the seven white keys on the piano keyboard. To tune the notes on the five black keys, the chain is to be extended by five more fifths, both down and up. That gives two sets of altered, enharmonic notes:

Gb-Db-Ab-Eb-Bb -F- C-G-D-A-E-B- F#-C#-G#-D#-A#

In Pythagorean tuning, the “enharmonically equivalent” pitches are not the same; they differ by the Pythagorean comma. Each such two pitches, for example, G[ and F], are separated by a chain of twelve fifths. To reach F] from G[ requires tuning by twelve fifths up and 3 12 1 7 312 531441 seven octaves down: ( 2 ) × ( 2 ) = 219 = 524288 > 1. In Pythagorean tuning, the “sharp” notes are higher in pitch than the enharmonic “flat” notes. To make the available fifths all pure, each black key on the keyboard would have to be split in two and the system would have seventeen, and not just twelve, tones in an octave.

Just tuning A justly tuned (5-limit) system is created by tuning both ﬁfths and thirds to their just frequency ratios, 3:2 and 5:4. The C major scale is generated, starting from C, by tuning the tonic, subdominant, and dominant major triads, C–E–G, F –A–C, and G–B–D:

F-A- C-E- G-B-D 4:3 5:3 1:1 5:4 3:2 15:8 9:8

It is more intuitive to reason about just tuning using the tuning lattice. The tuning lattice is a tool commonly used by microtonal music theorists. It was introduced by Leonhard Euler.8

8Euler, Tentamen novae theoriae musicae, “De genere diatonico-chromatico,” p.147.

71 Fig. 3.10 shows a modern version of the lattice with justly tuned C major scale, the five “sharp” and five “flat” accidentals, to a total of seventeen pitches.

The horizontal lines indicate tuning by fifths; each horizontal leg corresponds to tuning up by a fifth, in the left-to-right direction. Each leg of the vertical lines that are slanted to the right corresponds to tuning up by a major third, in the bottom-left to top-right direction, and the remaining lines correspond to tuning up by minor thirds, top-left to bottom-right. The lines of the fifths and major thirds are the two coordinate axes of the two-dimensional tuning system.

The equilateral upward and downward pointing triangles on the lattice correspond to major and minor triads, respectively. Fig. 3.10 clearly shows that the C major diatonic scale was constructed by first tuning the tonic, subdominant, and dominant triad, and then the accidental pitches were added by tuning additional fifths and/or major thirds. Enharmonic pairs of notes are connected by chains of three major thirds and differ in pitch by the lesser 5 3 1 125 diesis (( 4 ) × 2 = 128 < 1). In just tuning, the “sharp” notes are lower in pitch than the “flat” notes (the opposite of their relation in the Pythagorean tuning).

♯ ♯ ♯ ♯

♯

♭ ♭ ♭ ♭ ♭

Figure 3.10: Just C chromatic scale on a tuning lattice

♯ ♯ ♯ ♯ ♯

♭ ♭ ♯ ♯

♭ ♭ ♭ ♭ ♭

♭ ♭ ♭ ♭ ♭ Figure 3.11: Continued tuning lattice

72 ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♭ ♭ ♯ ♯ ♭ ♭ ♯ ♯

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ C Dm7 ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♭ ♭ ♯ ♯ ♭ ♭ ♯ ♯

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ G7 C Figure 3.12: Comma drift

♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♭ ♭ ♯ ♯ ♭ ♭ ♯ ♯

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ C Dm7 ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯ ♯

♭ ♭ ♯ ♯ ♭ ♭ ♯ ♯

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭

♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ ♭ G7 C Figure 3.13: Comma shift

Tuning of intervals on the tuning lattice by fifths and thirds can be continued indefinitely in both dimensions. Depending on the path taken, any note can have many different pitches. Two notes of the same name are connected by four fifths and one third (transposed by 3 4 4 1 81 two octaves) and differ in pitch by 2 × 5 × 4 = 80 , the syntonic comma. Fig. 3.11 shows this phenomenon of just tuning. It has a practical consequence of which vocal groups that perform music in just tuning (a cappella) must be aware: the comma drift. If the common notes within a harmonic progression are kept at the same pitch, the overall tuning is likely to drift through the lattice. As an example, the harmonic progression through a sequence of fifths that occurs frequently in Western music, causes intonation to fall in pitch. Fig. 3.12 shows the comma drift caused by the opening cadence C–Dm7–G7–C of the Prelude.

73 Comma drift can be countered by properly executed comma shift: a microtonal adjustment by a comma in the pitch of the common note between two chords in a harmonic progression. Fig. 3.13 illustrates how a comma shift in the pitch of D prevents the comma drift that would otherwise happen in the opening of the Prelude.

74 The 53-tone microtonal division of the octave Although the division of the octave into twelve steps is a good practical compromise that follows from the 5-limit intervals of the harmonic series, the octave can obviously be divided in many other ways. The 43-tone scale of Harry Partch,9 for example, is constructed as a “tonality diamond” of higher-limit frequency ratios. Ezra Sims’ 18 or 24-tone scales are selected from a 72-tone asymmetric division of the octave that was constructed in such a way to make it possible for musicians to play the microtonal pitches correctly.10

1 1 1 On the other hand, it could be argued that the 4 , 5 , or 6 -tone systems (48, 60, or 72-tone equal division of the octave) used, for example, by Alois Hába,11 contain an inherent flaw: they are constructed as subdivisions of the twelve-tone system. But any twelve-tone system is already a compromise that is subject to tuning imperfections! One of the main reasons for a finer division of the octave is better tuning of its intervals. Tuning precision is the very quality of microtonal music which already is (necessarily) lost in a twelve-tone system, and that defies the purpose of further subdividing it.

Equal divisions of the octave

Dividing the octave equally has a strong practical advantage: having only one size of a semitone step dramatically reduces the complexity of the system. No system of equal division can contain the just intervals of the harmonic series except the octave precisely, but some may satisfactorily approximate them.

Table 3.1 lists the equal divisions of the octave from the 12-tone division up to the 53-tone division. For each division, the table shows the number of steps that approximate the just ﬁfth, the third and the diatonic and chromatic semitones, the corresponding tuning error (as a proportion of the octave), and the sum of their absolute values that serves as a measure of the aggregate tuning error of the division. Only the divisions with their aggregate tuning error less or equal to that of the 12-tone division are listed in the table. The tuning errors

9Partch, Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments. 10“Since 1961 [...] all my major pieces that haven’t been tape music have been microtonal. At first, it was ¼-tone, but from 1964 it has been 72-note, using at any moment one transposition or another of an asymmetrical mode of 18 or 24 pitches drawn from a 72-note division of the octave, much as tonal music used a 7-note mode drawn from a 12-note chromatic. The mode consist of 8 pitches of superior importance, the 8-15th harmonics of the tonic of the moment, plus 10 (or 16) chromatic pitches also drawn from its harmonic series. [...] I trust the performers naturally to play as near Just [sic] perfection as their musicality permits, but sometimes it helps to remind them.” http://www.ezrasims.com/ 11“Hába,Alois”. Oxford Music Online.

75 that are small or practically negligible are underlined, and the divisions with low aggregate errors are shown in bold typeface.

The good equal divisions between 12 and 53 are:12

• 12: Good fifths, out-of-tune thirds and semitones. • 31: Great thirds, acceptable fifths and semitones. • 34: Great thirds and chromatic semitones, acceptable fifths and diatonic semitones. • 43: Great diatonic semitones, acceptable fifths and thirds, out-of-tune chromatic semitones. • 53: Nearly perfect fifths, great thirds and semitones.

The cycle of fifty-three fifths almost closes with respect to the octave! (53 fifths are almost 3 53 1 31 353 equal to 31 octaves, ( 2 ) × ( 2 ) = 284 ≈ 1.002). The system of equal division (Fig. 3.14a) of the octave into fifty-three steps has these advantages:

353 1. It requires very little tempering to divide the reminder 284 ≈ 1.0021 among the 53 ﬁfths. 2. The just 5-limit intervals of the harmonic series can be very well approximated (Fig. 3.14b). 1 81 3. The size of the step, 53 ≈ 0.0189, is close to log2 80 ≈ 0.0179, the syntonic comma. 4. Also the intervals of the equally tempered 12-tone systems can be reasonably well approximated (Fig. 3.14c).

Table 3.2 shows in a comprehensive form how just intervals and twelve-tone equal temperament are approximated by the 53-tone system. The table is, for convenience and practicality, extended to ﬁve octaves. For example, the twelfth (an octave and a ﬁfth, 1:3) is approximately equal to 84 (53+31) steps in 53-tet.

New unit: the ﬁtt

Here we introduce a new unit for measuring the size of intervals: the fitt (from fifty three). One fitt is equal to one step in the 53-division, i.e. 1/53 of the octave, similarly as one

12Many properties of equal divisions of the octave were noted by a number theorists both in the past and the present times; reportedly, the 53-tone scale was known in ancient China. Today, with the help of simple computation and with basic knowledge of the theory of just intervals and their frequency ratios, any serious student of the topic can certainly draw the conclusions. The author of this text ﬁrst made this small discovery himself that way, and only then found references to it in Helmholtz’s text and elsewhere, a few years before articles about microtonality appeared on Wikipedia^ ¨

76 a b c Figure 3.14: The 53-tone system

cent is equal to 1/100 of the equally tempered semitone, or 1/1200 of the octave. (One ﬁtt is approximately equal to 22.6 cents.) Table 3.3 lists the steps of the 53-tone division, the approximation of just and 12-tet intervals, and their sizes as parts of the octave and in cents.

Let us now compare how the intervals of the ﬁfth and the third, and the major triad are approximated by the 12-tone and 53-tone systems:

just 12-tet 53-tet Figure 3.15: Just, 12-tet, and 53-tet ﬁfths

The ﬁfth is slightly mistuned in the 12-tone division, and nearly perfect in the 53-tone division (Fig. 3.15). The third is very sharp in the 12-tet, and also nearly perfect in 53-tet (Fig. 3.16).

The major triad is mistuned in the 12-tone system, mainly due to its sharp major third.13

77 just 12-tet 53-tet Figure 3.16: Just, 12-tet, and 53-tet thirds

12-tet 12-tet, ﬁrst inversion 53-tet Figure 3.17: 12-tet and 53-tet major triads

Note that in its ﬁrst inversion (the root C transposed an octave higher), some of the beats between the harmonics of the root C and of the third E are eliminated, and therefore the triad apparently sounds more stable. In the 53-division, the triad can be tuned almost perfectly.

Microtonal notation of the 53-tet system The twelve-tone system is so ingrained in Western music practice that a notation of any other tone system needs to be related to the traditional staﬀ notation, if it should be accepted with no great reluctance.

13It really “wants nothing so much as to go away and sit down someplace” (p.20).

78 In a similar way that the usual accidental signs (the sharp and ﬂat signs ] and [) indicate the deviation of chromatic twelve-tone pitches from the seven “natural” pitches of the diatonic scale, the established microtonal notations use various kinds of microtonal accidental signs to indicate the deviation of the microtonal pitches from the twelve pitches of some twelve- tone division of the octave. For example, the quarter-tone accidental signs ( [ , ], etc.) can be used to notate the 24-tone division of the octave as a subdivision of the 12-tet system. As another example, the quite complex “Sagittal” system of musical notation contains a number of accidental signs that indicate the relations of pitches in terms of intervals of the harmonic series.14

For the purpose of this text, to notate the musical examples that demonstrate our work in the system of equal 53-tone division, we adopt a practical version of such microtonal staﬀ- notation. The notation is not intended for performance, but rather to provide a “listening score” which purpose is to communicate rhythmic, melodic, and harmonic aspects of the examples to listeners trained in reading staﬀ notation, by relating the music to a familiar frame of reference.

The notation is based on the following assumptions and rules:

• The notation is relative to the diatonic scale in the key of C; there is no key signature. • The seven reference notes — C, D, E, F , G, A, B — indicate the pitches of the 53-tet system that are the closest approximations of the corresponding pitches in the 12-tet system: 0, 9, 18, 22, 31, 40, 49 fitts or approximately 0, 204, 400, 498, 702, 906, 1109 cents, respectively (see Table 3.3). • The two basic accidental signs (] and [) divide each interval of a tone (C–D, D–E, F –G, G–A, and A–B) into two halves. Similarly as in just tuning, the two notes of each enharmonic pair differ in pitch by a small comma-sized interval of 1 fitt. For example, the pitch of C] is lower than D[ by one fitt. • The intervals of the twelve “semitones” (C–C], D[–D, etc.) are further subdivided by half-sharps and half-flats (] and [ ). The two “natural” intervals of a semitone (E–F and B–C) are subdivided by F [ and C ,[ respectively. It would be equally possible to use E] and B]; the choice of F [ and C [ was made to suggest that these two pitches might be related to the subdominant F and the tonic C, respectively. • Finally, the last round of subdivision indicated by arrows pointing up or down that 14Keenan, Secor, and Smith, The Sagittal notation system.

79 indicate sharpening or ﬂattening of the pitch by one ﬁtt, completes the system of 53-tone scale notation.

Note that the semitones C–C], D[–D, D–D], E[–E, F –F] etc., and E–F and B–C happen all to be “neutral” semitones of 4 fitts. Chromatic semitones (of 3 fitts) would be notated as C–C] ↓, D[ ↑ –D, E ↑ –F , E–F ↓, etc.and diatonic semitones (of 5 fitts) as C–D[, C]–D, E ↓ –F , E–F ↑, etc.

Table 3.4 summarizes the complete system of 53-tet notation, and Fig. 3.18 shows the relation between the 53-tet and 12-tet notations. Remember that unlike in the traditional notation, in the microtonal notation the natural and accidental signs modify only the immediately following note and do not carry their meaning all the way to the end of the measure.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49 50 51 52 53

Figure 3.18: The 53-tet and 12-tet notations compared

80 the ﬁfth the third diat. semit. chrom. semit. edo steps ∆oct steps ∆oct steps ∆oct steps ∆oct Σ|∆oct| 12 7 -0.002 4 0.011 1 -0.010 1 0.024 0.047 19 11 -0.006 6 -0.006 2 0.012 1 -0.006 0.031 21 12 -0.014 7 0.011 2 0.002 1 -0.011 0.038 22 13 0.006 7 -0.004 2 -0.002 1 -0.013 0.025 24 14 -0.002 8 0.011 2 -0.010 1 -0.017 0.040 28 16 -0.014 9 0.000 3 0.014 2 0.013 0.041 29 17 0.001 9 -0.012 3 0.010 2 0.010 0.033 30 18 0.015 10 0.011 3 0.007 2 0.008 0.041 31 18 -0.004 10 0.001 3 0.004 2 0.006 0.014 32 19 0.009 10 -0.009 3 0.001 2 0.004 0.022 33 19 -0.009 11 0.011 3 -0.002 2 0.002 0.025 34 20 0.003 11 0.002 3 -0.005 2 0.000 0.010 35 20 -0.014 11 -0.008 3 -0.007 2 -0.002 0.030 36 21 -0.002 12 0.011 3 -0.010 2 -0.003 0.026 37 22 0.010 12 0.002 3 -0.012 2 -0.005 0.029 38 22 -0.006 12 -0.006 4 0.012 2 -0.006 0.031 39 23 0.005 13 0.011 4 0.009 2 -0.008 0.033 40 23 -0.010 13 0.003 4 0.007 2 -0.009 0.029 41 24 0.000 13 -0.005 4 0.004 2 -0.010 0.020 42 25 0.010 14 0.011 4 0.002 2 -0.011 0.035 43 25 -0.004 14 0.004 4 0.000 3 0.011 0.018 44 26 0.006 14 -0.004 4 -0.002 3 0.009 0.021 45 26 -0.007 14 -0.011 4 -0.004 3 0.008 0.030 46 27 0.002 15 0.004 4 -0.006 3 0.006 0.019 47 27 -0.010 15 -0.003 4 -0.008 3 0.005 0.026 48 28 -0.002 15 -0.009 4 -0.010 3 0.004 0.024 49 29 0.007 16 0.005 5 0.009 3 0.002 0.023 50 29 -0.005 16 -0.002 5 0.007 3 0.001 0.015 51 30 0.003 16 -0.008 5 0.005 3 0.000 0.016 52 30 -0.008 17 0.005 5 0.003 3 -0.001 0.017 53 31 0.000 17 -0.001 5 0.001 3 -0.002 0.005

Table 3.1: Equal divisions of the octave

81 ratio 12-tet 0 1 2 3 4 5 ratio 12-tet 0 1 2 3 4 5 0 53 106 159 212 265 1 54 107 160 213 266 27 80 133 186 239 292 2 55 108 161 214 267 28 81 134 187 240 293 25:24 3 56 109 162 215 268 29 82 135 188 241 294 m2 4 57 110 163 216 269 30 83 136 189 242 295 16:15 5 58 111 164 217 270 3:2 P5 31 84 137 190 243 296 6 59 112 165 218 271 32 85 138 191 244 297 7 60 113 166 219 272 33 86 139 192 245 298 10:9 8 61 114 167 220 273 34 87 140 193 246 299 9:8 M2 9 62 115 168 221 274 m6 35 88 141 194 247 300 10 63 116 169 222 275 8:5 36 89 142 195 248 301 11 64 117 170 223 276 37 90 143 196 249 302 12 65 118 171 224 277 38 91 144 197 250 303 m3 13 66 119 172 225 278 5:3 M6 39 92 145 198 251 304 6:5 14 67 120 173 226 279 40 93 146 199 252 305 15 68 121 174 227 280 41 94 147 200 253 306 16 69 122 175 228 281 42 95 148 201 254 307 5:4 17 70 123 176 229 282 43 96 149 202 255 308 M3 18 71 124 177 230 283 16:9 m7 44 97 150 203 256 309 19 72 125 178 231 284 9:5 45 98 151 204 257 310 20 73 126 179 232 285 46 99 152 205 258 311 21 74 127 180 233 286 47 100 153 206 259 312 4:3 P4 22 75 128 181 234 287 15:8 48 101 154 207 260 313 23 76 129 182 235 288 48:25 M7 49 102 155 208 261 314 24 77 130 183 236 289 50 103 156 209 262 315 25 78 131 184 237 290 51 104 157 210 263 316 26 79 132 185 238 291 52 105 158 211 264 317

Table 3.2: Just tuning and 12-tet approximated in 53-tet

82 just int. 12-tet 53-tet ∆oct cents just int. 12-tet 53-tet ∆oct cents 1 0.019 23 27 0.509 611 2 0.038 45 28 0.528 634 chrom. sem. 3 0.057 68 29 0.547 657 m2 4 0.075 91 30 0.566 679 diat. sem. 5 0.094 113 ﬁfth P5 31 0.585 702 6 0.113 136 32 0.604 725 7 0.132 158 33 0.623 747 minor tone 8 0.151 181 34 0.642 770 major tone M2 9 0.170 204 m6 35 0.660 792 10 0.189 226 36 0.679 815 11 0.208 249 37 0.698 838 12 0.226 272 38 0.717 860 m3 13 0.245 294 M6 39 0.736 883 minor third 14 0.264 317 40 0.755 906 15 0.283 340 41 0.774 928 16 0.302 362 42 0.792 951 major third 17 0.321 385 43 0.811 974 M3 18 0.340 408 m7 44 0.830 996 19 0.358 430 45 0.849 1019 20 0.377 453 46 0.868 1042 21 0.396 475 47 0.887 1064 fourth P4 22 0.415 498 48 0.906 1087 23 0.434 521 M7 49 0.925 1109 24 0.453 543 50 0.943 1132 25 0.472 566 51 0.962 1155 26 0.491 589 52 0.981 1177 octave 53 1 1200

Table 3.3: The 53-tone equal division of the octave

83 note steps ∆oct note steps ∆oct C\ 0 0.000 C\ ↑ 1 0.019 G[ 27 0.509 C] 2 0.038 G[ ↑ 28 0.528 C] ↓ 3 0.057 G [ 29 0.547 C] 4 0.075 G\ ↓ 30 0.566 D[ 5 0.094 G\ 31 0.585 D[ ↑ 6 0.113 G\ ↑ 32 0.604 D [ 7 0.132 G] 33 0.623 D\ ↓ 8 0.151 G] ↓ 34 0.642 D\ 9 0.170 G] 35 0.660 D\ ↑ 10 0.189 A[ 36 0.679 D] 11 0.208 A[ ↑ 37 0.698 D] ↓ 12 0.226 A [ 38 0.717 D] 13 0.245 A\ ↓ 39 0.736 E[ 14 0.264 A\ 40 0.755 E[ ↑ 15 0.283 A\ ↑ 41 0.774 E [ 16 0.302 A] 42 0.792 E\ ↓ 17 0.321 A] ↓ 43 0.811 E\ 18 0.340 A] 44 0.830 E\ ↑ 19 0.358 B[ 45 0.849 F [ 20 0.377 B[ ↑ 46 0.868 F\ ↓ 21 0.396 B [ 47 0.887 F\ 22 0.415 B\ ↓ 48 0.906 F\ ↑ 23 0.434 B\ 49 0.925 F] 24 0.453 B\ ↑ 50 0.943 F] ↓ 25 0.472 C [ 51 0.962 F] 26 0.491 C\ ↓ 52 0.981 C\ 53 1

Table 3.4: The 53-tone notation

84 Chapter 4: Calculated consonance and its applications

Dissonance curves and landscapes Recall how the beats created by interference of simple tones are the source of an unpleasant sensation of sensory roughness, and how the aggregate sensation of all beats created by all possible pairs of harmonics in complex tones has distinct local minima of sensory roughness that distinguish justly tuned intervals (Example 2.8, p.59).

The relative sensation of roughness (dissonance) between two simple tones can be plotted as an idealized dissonance curve. Dissonance curves were discussed in detail by Plomp and Levelt,1 Haluˇska,2 or Sethares;3 their origin is in the work of Helmholtz. Helmholtz writes:

I have been forced to assume a somewhat arbitrary law for the dependence of roughness upon the number of beats. I chose for this purpose the simplest mathematical formula which would shew that the roughness vanishes when there are no beats, increases to a maximum for 33 beats, and then diminishes as the number of beats increases.4

The dissonance curve shown in Fig. 4.1 has two segments which shape is calculated as y = (ln(0.763 × x + 1) × e−0.763×x+5) ÷ 39.2. The curve has a maximum of y ≈ 1 for x ≈ 1, and can then be simply scaled to match the observed frequency value of the maximum roughness, for example by dividing x by 33, should we follow Helmholtz’s assumption.

To calculate the dissonance of a group of complex tones, the individual dissonance curves of each pair of harmonics, weighted by their amplitude, are simply added together5 and the result, with its height normalized, is the dissonance landscape.

Helmholtz himself constructed such a diagram for complex violin tones (Fig. 4.2):

1Plomp and Levelt, “Tonal Consonance and Critical Bandwidth”. 2Haluˇska, The Mathematical Theory of Tone Systems, “Fuzziness and sonance”. 3Sethares, Tuning, Timbre, Spectrum, Scale. 4Helmholtz, On the sensations of tone as a physiological basis for the theory of music, p.192. 5Assuming that the aggregate sensory dissonance is a linear combination of individual dissonances. That is, without any question, a substantial simpliﬁcation.

85 Figure 4.1: The dissonance curve

Fig. [...] shews the roughness of all intervals which are less than an Octave [...] Above the baseline there are prominences [...] The height of these prominences at every point of their bits is made proportional to the roughness produced by the two partial tones denoted by the numbers, when a note of corresponding pitch is sounded at the same time with the note c0. The roughnesses produced by the diﬀerent pairs of upper partials are erected one over the other. [...] the various roughnesses arising from the diﬀerent intervals encroach on each other’s regions, and only a few narrow valleys remain, corresponding to the position of the best consonances.6

Figure 4.2: Helmholtz’s construction of the dissonance landscape

Fig. 4.3 and Fig. 4.4 show two examples of dissonance landscapes created by three ﬁxed tones. The fourth (a “probe”) tone, that in both examples has four harmonics, “slides” across the whole pitch range. The height of the landscape contour at any point is related to the aggregate roughness of the four tones. The rounded peaks of the landscape represent areas of dissonance (mistuning), while the clearly delineated, narrow valleys are the points of local consonance where one or more pairs of partials coincide and the overall sensory

6Helmholtz, On the sensations of tone as a physiological basis for the theory of music, p.193.

86 roughness is locally minimized. The length of the short vertical annotated red lines above each valley is related to how “steep” the valey is. The vertical grid lines indicate 12-tet pitches.

Figure 4.3: The dissonance landscape of simple tones

Fig. 4.3 shows a simpler dissonance landscape created by three justly tuned simple tones C4,

G4, and C5. Note that, since the three fixed tones have no higher harmonics, a tone of any pitch above about G5 would cause no significant rougness. Below C4, a tone (that has four harmonics) would “fit in” at the indicated C, F , and G pitches.

Figure 4.4: The dissonance landscape of complex tones

The second example, shown on Fig. 4.4, is a dissonance landscape created by complex tones that form a major triad in the second inversion. In this case, due to much richer harmonic content of the triad, there are signiﬁcantly more pitch positions where dissonance is locally minimized (the resulting four-tone locally consonant chords would also have higher aggregate “tension”).

Dissonance landscape displayed on the pitch spiral The computed dissonance landscape can be displayed on the pitch spiral by following these steps:

• Move any of the four blue, green, orange, and yellow discs into chosen pitch positions on the spiral. Choose their requested harmonic content, from 1 to 6 harmonics, by adjusting the respective sliders. • Turn on the tones that are to be included in forming the landscape, by depressing the coloured buttons. • The “white” tone is the “probe”. Choose its requested harmonic content by adjusting the respective slider.

87 • Push the button labelled “cons” (consonance). The dissonance landscape will be calculated by a JavaScript worker thread and promptly displayed as a yellow band with varied thickness, with the position of consonance “valleys” shown by small red triangles.

Fig. 4.5 shows dissonance landscapes of a single ﬁxed tone (C4) and a probe tone, where

• (a) both tones have only one harmonic, • (b) the ﬁxed tone has six harmonics and the probe tone one harmonic, and • (c) both tones have six harmonics, and dissonance landscapes of groups of complex tones, all with six harmonics, where

• (d) the ﬁxed tones form a major triad in the root position, • (e) a minor triad in the root position, and • (f) a minor seventh chord in the third inversion (Dm7/C) such as occurs in the second measure of the Prelude.

88 a d

b e

c f Figure 4.5: The dissonance landscapes on the pitch spiral

89 Sequencing music in 53-tone division

[...] musical esthetics has made [...] advances [...] by introducing the conception of movement in the examination of musical works of art [...] the simple elements of melodic movement. [...] In the inorganic world the kind of motion we see, reveals the kind of moving force in action, and in the last resort the only method of recognizing and measuring the elementary powers of nature consists in deter- mining the motions they generate, and this is also the case for the motions of bodies or of voices which take place under the inﬂuence of human feelings. Hence the properties of musical movements which poses a graceful, dallying, or a heavy, forced, a dull, or a powerful, a quiet, or excited character, and so on, evidently chieﬂy depend on psychological action.7

Helmholtz suggests that there is a close connection between the motions of physical bodies and musical motions of melodic voices. It is true that the character of music that is considered to be calming or exciting, sad or cheerful, etc. is related to the motions that a human would do in a like mood: be still or restless, moving slowly or be dancing, a notion alluded to in these verses:

‘Twas brillig, and the slithy toves Did gyre and gimble in the wabe8

In this section we will present a practical application of the two principles we have been discussing, and the one more additional principle that has just been introduced. The three principles can be summarized as:

1. That relative consonance of groups of tones can be calculated as a dissonance landscape on the basis of Helmholtz’s theory of sensory roughness, that the choice of pitches of additional tones can be guided by the shape of the landscape, and that intervals and chords that were constructed from the choices taken from the landscape tend to be justly tuned, 2. that the 53-tone equal division of the octave is a discrete tone system that very well approximates just tuning (required for eﬃcient work with the dissonance landscape) and that at the same time can closely represent the tonal structure of twelve-tone

7Helmholtz, On the sensations of tone as a physiological basis for the theory of music, p.2. 8Lewis Carroll, “Jabberwocky”. In: Through the Looking-Glass and What Alice Found There.

90 Western music, and 3. that melodic contours are related to motions of physical bodies and therefore can be modelled as such.

In the presented applications, the calculated dissonance landscapes are used in various ways to assist both in arranging tones in harmony and to shape melodic lines.

The process of tone arrangement, in general terms, follows this simple strategy:

1. Begin with a harmonic plan in the form of chords in appropriate voicing (ordering and spacing of pitches) to create the initial dissonance landscape. The tones of the chords can either be included among the tones that are part of the arrangement, or they can play the role of “ghost tones”: tones that form the shape of the landscape but are excluded from the audible music (i.e. are muted). 2. Check and correct the tuning of the tones in the harmonic plan if necessary, especially if the plan was originally created in 12-tone equal temperament; for example, convert the pitches into the 53-tone system and microtonally adjust their values to approximate just tuning. 3. (Re)compute the shape of the dissonance landscape. 4. Devise various musical motifs in the form of geometric shapes (curves) in the two- dimensional time×pitch space. Their shapes can be drawn by hand or can be, for example, computed using some kind of simulated physics. 5. Divide (split) the geometric shapes into short segments that represent separate tones in the melodic motives, tones that have distinct pitch, onset, and duration. 6. Adjust the pitch of each tone to that of a nearby “consonance valley” of a required depth on the dissonance landscape. 7. Repeat steps 3 to 6 as long as required.

Note that since each tone added to the landscape may change the landscape contour, it may be necessary to perform the process of pitch adjustment incrementally and repeatedly, working to-and-fro among the voices. One can also order the voices by their importance or by their pitch range, work through the voices only once and adjust only the newer voices to match those that are already ﬁxed.

91 Qvinta

A software application (a kind of a microtonal music sequencer) has been developed to test and practically employ the described strategy for microtonal tone arrangement. The original version was written in Java and used a custom written OSC9 link to the SuperCollider10 server to synthesize the sound. The current version of the software, called Qvinta11 is under development, written in C++ using the Qt512 development framework and Csound for sound synthesis. The goal is to eventually develop Qvinta into a professional quality software.

Qvinta is a sequencer that allows eﬃcient arrangement of tones on a time×pitch grid. The tones are represented by simple horizontal line segments which position can be adjusted horizontally (in time) and vertically (in pitch). Various vertical resolutions of the grid can be chosen to allow work in diﬀerent systems of equal division of the octave, particularly in 12-tet and 53-tet.

A built-in module that calculates dissonance landscapes assists with a choice of tone pitches. The user can select either a single time instant on the grid, or a time range, and a number of applicable “layers” of tones (that represent separate voices, instruments, or “staves” of notation). From the selection, the software calculates a dissonance landscape that is made available to the user in two possible ways:

1. The pitches that are the local minima in the dissonance landscape are shown as small elliptic marks on a vertical cross-hair. The sizes of the marks indicate the relative depth of the respective consonance valleys (Fig. 4.6 and Fig. 4.7a). The user can then, using the marks as visual guidance, either draw new line segments representing tones or move existing ones into the consonant positions. 2. The software can also, at the user’s request, enter a complete sequence of tones of all locally consonant pitches. The duration of the tones is proportional to their relative consonance (Fig. 4.6 and Fig. 4.7c). The user then chooses the most suitable tones from the oﬀered sequence, adjusts their duration and position in time, and thus creates a melodic contour. More consonant tones can be, for example, used on accented beats or sustained for longer times, while the less consonant tones are choices more suitable

9Open Sound Control, a network protocol for communication among musical software. http:// opensoundcontrol.org/ 10McCartney et al., SuperCollider. 11Burleigh, Qvinta. 12http://qt-project.org/qt5

92 for passing notes.

Figure 4.6: The original sequencer Qvinta written in Java

a b c Figure 4.7: The current C++/Qt version of Qvinta

The pitch grid of Qvinta is made to suit work in 12-tet and 53-tet divisions of the octave the best. Fig. 4.7a shows the 12-tet variant of the grid. The horizontal lines indicate the position of the twelve pitches. The thickness and colour of the lines, and the colour of the horizontal bands between the lines help the user to ﬁnd a given pitch faster, without having to count the lines: six out of the twelve bands are coloured and thus show the (two possible) whole-tone scales; the thick gray lines and the red bands mark the positions of Cs (the middle C4 is marked by a red line); the blue lines and the blue bands mark F s and Gs.

Fig. 4.7b shows the 53-tet grid of Qvinta. The horizontal lines indicate the positions of

93 all ﬁfty-three pitches, with the thicker lines showing the 53-tet approximation of Cs, F s, and Gs. The coloured bands in the background show the twelve-tone equal division, for reference.

The C53 Suite: Variations on Prelude in C, BWV 846 A suite of several short musical examples or etudes, the “C53 Suite” demonstrates the feasibility and potential of a proposed method for microtonal tone arrangement.13 The Suite is based on J.S. Bach’s Prelude in C with some inspiration taken from Arvo P¨art’s Spiegel im Spiegel and motivic fragments from several Czech folk songs. The Suite was arranged and the sound ﬁles that are provided on the accompanying website were synthesized using the original Java version of the Qvinta software.

Since the foundation of the Prelude enforces both the strong tonal character of the suite and the key of major C, the etudes can be quite successfully notated using the microtonal notation described on p.78.

The microtonal notation of the Suite is provided in Appendix A, and synthesized sound ﬁles are available online at http://qvin.in/CC/C53.

The Suite has six movements. The ﬁrst movement is the original Prelude, verbatim as composed by J.S. Bach, but re-tuned to the 53-tet approximation of just tuning. The ﬁve movements that follow are a sequence of experiments that test and explore various possibilities of the 53-tet system and of the tone arrangement method. Each movement, in the order as they are presented in the Suite, is a stepping-stone that leads to the next level, a more daring adventure in machine-assisted tone arrangement.

The six movements/etudes are:

1. Prelude in C, re-tuned. 2. Rusty Reﬂections, an exercise in tuning of diﬃcult-to-tune timbres. 3. Farmer’s Fuguetta, a three-voice polyphonic texture created mainly by the way of weaving melodious strands into a braid, laid out in the graphical time×pitch space. 4. Martian Mazurek, a test of stability of a spiralling melody and an accompaniment that

13Proper arrangement of tones (or notes, in its written form) is only a necessary, but not a suﬃcient condition for success in music composition.

94 are deliberately upset by microtonal nutations. 5. Genaues Glockenspiel, a four-voice texture of bell-like voices, in part generated by a simulated physics algorithm. 6. Turing Toccata, a combination of a manual tone arrangement with an algorithmically generated virtuoso voice.

The Prelude, with its chord progression and its arpeggiated character, is the basis for all movements of the Suite. The complete harmonic progression of the Prelude as used in movements 1 and 6 is:

[1] C | Dm7/C | G7/B | C | [5] Am/C | D7/C | G/B | Cmaj7/B | [9] C6/A | D7 | G | Gdim7 | [13] Dm/F | Ddim7/F | C/E | Fmaj7/E | [17] Dm7 | G7 | C | C7 | [21] Fmaj7 | F#dim7 | Abdim7 | G7 | [25] C/G | G7sus | G7 | Adim7/G | [29] C/G | G7sus | G7 | C7 | [33] Fmaj7/C F6/C | G7/C C ||

For movements 2 to 5, the harmonic progression has been shortened by skipping from measure 22 directly to the ﬁnal cadence:

[1] C | Dm7/C | G7/B | C | [5] Am/C | D7/C | G/B | Cmaj7/B | [9] C6/A | D7 | G | Gdim7 | [13] Dm/F | Ddim7/F | C/E | Fmaj7/E | [17] Dm7 | G7 | C | C7 | [21] Fmaj7 | F#dim7 | Dm7/C | G7/C | [25] C ||

1. Prelude in C, BWV 846

The Prelude was entered directly from a published score, in 12-tone equal temperament. All pitches were then converted (rounded) to their nearest equivalent in the 53-tone system. The result was inspected with the help of the calculated dissonance landscape technique,

95 and some pitches were accordingly adjusted as described below. All given fitt measurements are relative to C4. (Refer to Table 3.2 and Table 3.3 for the fitt values of just and equally tempered intervals.) m.1 The pitches of the C major arpeggio were entered as 0-18-31-53-71 fitts. The sequence contained two wide major thirds C-E (18 and 53 + 18 = 71), due to the fact that major thirds are notoriously sharp in 12-tet. The Es were therefore flattened by one fitt each, to 0-17-31-53-70, which makes a nicely tuned major triad 0-17-31.

m.2 The pitches of the inverted Dm7 chord C4-D4-A4-D5-F5 were entered as 0-9-40-62-75

fitts. The interval between C4 and D4 (9 fitts, a major tone) is correct, but the size of D5-F5 was 13 fitts, one fitt less than the 14 fitts of a just minor third, again due to the badly tuned thirds of the twelve-tone equal temperament. The sequence was adjusted to 0-9-40-62-76 by sharpening F5 by 1 fitt, which makes a nicely tuned minor triad that in the root position would be 9-76-40 ⇒ 9-23-40 ⇒ 9 + [0-14-31].

m.3 Notice that in the inverted G7 chord B3-D4-G4-D5-F5,(−5)-9-31-62-75 ﬁtts, the F5

(75 fitts) is flatter by one fitt than the F5 (76) in measure 2 (this is an example of the comma shift). In m.2, the F5 was tuned as a minor third (14 fitts) up from D5, while in m.3 the F5 is tuned as a minor seventh of 44 fitts (the 3-limit 16:9 ratio) rather than 45 fitts (the 5-limit

9:5 ratio). The B3 has been flattened by one fitt to justly tune the G-B-D-F dominant seventh chord, (−5)-9-31-75 ⇒ 31 + [0-17-31-44] fitts. m.4 Measure 4 is the same as m.1. m.5 As and Es were flattened, 0-18-40-71-93 ⇒ 0-17-39-70-92 to tune the minor triad 39-53-70 ⇒ 39 + [0-14-31]. m.6 The D7 chord did not need any adjusting.

m.7 The B3 was flattened as in m.3. m.8 The Es were flattened as in m.1 and m.4, and Bs flattened to the (15:8) major seventh of −5 ⇒ 48 fitts.

96 m.9, 10, and 11 Same as m.5, 6, and 7, respectively.

m.12 The diminished seventh chord is a harmonically ambivalent, symmetric division of the octave into four parts; one of the few situations where equal temperament is most suitable. It poses a certain problem for the 53-division. One might be tempted to justly tune some of the minor thirds (major sixths) to relieve the tension of equally tempered intervals, but any such attempt would make the remaining parts of the chord sound worse, and also defy the purpose of diminished (and augmented) chords as intentional dissonances. The pitches in this measure were therefore left as close to their 12-tet values as 53-tet allows.

m.13 The minor third F was sharpened by one ﬁtt.

m.14 This diminished seventh chord Ddim7 (notated enharmonically) has a strong harmonic relation to the Dm triad in m.13. This fact weakens its harmonically ambivalent character, and strengthens its role in voice-leading between m.13 and m.15. Therefore the minor third D-F was kept justly tuned as in m.13, the Bs were flattened to tune the B-D minor third, and the G]s were kept close to equal tempering as a passing note, since . . . (see m.15) m.15 . . . the C major triad (with the flattened E) releases the built-up tension here. m.16 In the F maj7 chord, the As were tuned (flattened) to the just F -A third, and the Es were flattened to the 48 fitt (15:8) major seventh F -E and also to retain their pitch value from m.15 (to avoid an unnecessary comma shift). m.17 The F -A-C pitches were retained from m.16, and the Ds were flattened to tune the D-F minor third.

Notice that from this point to the end of the Prelude, the two sustained half and double- dotted quarter notes that are repeated twice in each measure keep descending toward lower pitches. While until now they always formed a second or a third, from this point on they tend to be separated by larger intervals. This is in accordance with the observation that the beats between two simple tones that are separated by the same interval cause stronger sensory roughness in lower ranges than in high ranges. For example, C4 and E4 diﬀer by

97 approximately 65 Hz, while C3 and E3 diﬀer by approximately 33 Hz which is near the maximum of sensory roughness.

We will discuss the tuning of the remaining part of the Prelude only brieﬂy, since it follows the same principles that were shown so far. We point out only the more notable facts as they are encountered:

m.18 A comma shift between D3s in m.17 and m.18.

m.19 A release of the built-up tension, C major.

m.20 and 21 A temporary modulation to the key of F major (tonicization). The ﬂattened E (the 48 ﬁtt major seventh F -E) acts as a suspension from the key of C.

m.22 Another diminished seventh chord, and therefore a tuning puzzle. Tuned again as equally tempered, except with As ﬂattened to avoid a comma shift between m.21 and m.22. m.23 A diminished seventh chord, descending by half a step from m.22, with added passing Cs. Tuned to avoid comma shifts between m.23 and m.24.

m.32 The base line has now reached the lowest pitch C2, with the frequency of only 65.4 Hz. m.33 and 34 The progression in these two measures is Am-F -Dm7-G7 over a pedal C and is tuned accordingly.

The re-tuned version of the Prelude, now very close to just intonation (within the small margin of error of 53-tet), served as a foundation for the remaining movements of the Suite.

2. Rusty Reﬂections

Rusty Reflections (inspired by Arvo Pärt’s Spiegel im Spiegel) was the very first test of tuning by consonance calculated as the dissonance landscape. On its success depended the very result of this work. In order to obtain a demonstrable proof of the applicability of the

98 method, the conditions of the experiment were set quite strictly: the synthesized timbre was to be harmonically rich and shrill in sound, to the point of being unpleasant. Any mistuning would be thus revealed with no mercy.

The timbre of the top voices was synthesized as a bank of resonators excited by a “gray” noise signal; its sound resembles a harshly (but precisely) played violin, if not the sound of a rusty hinge. The accompaniment has a simple additive timbre with four harmonics that provides a reasonably strong constraint for tuning of the upper voices.

The chord progression of the Prelude, as a foundation for this movement of the Suite, has been shortened (see p.95). There were two major reasons for doing so: to keep the extent of the experiments more manageable and, also, to avoid the more chromatic part of the harmonic progression with its diminished and suspended chords that would, presumably, make the tone arrangement more diﬃcult and keep attention from the goal of the experiment.

The more complex arpeggiated ﬁgures of the Prelude were reduced to simple arpeggiated triads in open voicing (a sparse texture characteristic of P¨art’s“tintinnabuli”). The inversions of the chords were chosen in such a way that the respective tones of each triad make three voices, each moving mostly in a step-wise downward motion. A particular attention was paid to make the lowest voice (the “bass” line) to move by steps that are as small as possible. These are deliberate constraints which purpose is to create a dissonance landscape which shape does not change dramatically by leaps and bounds as the music progresses, but more slowly so it resembles “rolling hills”, shifting with time.

A line of sustained tones, one for each chord in the harmonic progression, was then constructed over the arpeggiated accompaniment, following these few general rules: it was to be about two octaves or higher above the accompaniment, also moving only by small steps, and be alternatively ascending in a contrary motion to the accompaniment, and descending to stay at a more or less same distance from the accompaniment. Within the constraints of these rules, the pitches where chosen exclusively from the available local minima of the dissonance landscape formed by the arpeggiated triads.

Finally, a voice that begins in a flageolet-like register and complements the first two voices, first by filling in the “gaps” and then by contrasting the motion of the arpeggiated line, was added. The pitches were, again, selected from among the dissonance minima of the aggregate

99 dissonance landscape of the previous two voices.

For example, in m.1 the accompanying triad of 0, 31, and 70 ﬁtts (measurements relative to

C4) created a landscape for the “rusty” timbre with minima at 0, 9, 17, 31, 39, 48, 53, 62, 70, 84, 101, 106, 115, 123, 137, 140, 154, 168, 176, 193, and 207 ﬁtts. The three “deepest” minima in the range about two octaves above the accompaniment were 137, 154, and 176 ﬁtts. The three pitches were arranged as the built-up triad in the upper voices.

The process was repeated for several measures. It is logical that the best tuned pitches (the deepest minima) turned up to be mostly octave-shifts of the pitches in the accompaniment, with added major sevenths and ninths over major triads, minor sevenths over minor triads, minor ninths over dominant seventh chords, etc. Therefore after m.14 the process was modiﬁed to include also the less deep minima, and to form stepwise motions from other available pitches.

Note the comma shifts between measures 4 and 5, 19 and 20, and 23 and 24. The 1-ﬁtt microtonal adjustments of the pitches are perceived as subtle changes in the tone colour.

Fig. 4.8 shows the opening of Rusty Reﬂections as sequenced in Qvinta: all three voices are displayed over the 12-tet grid (a), and a magniﬁed detail of the top two voices is shown over the 53-tet grid (b).

(a) 12-tet grid (b) 53-tet grid Figure 4.8: Rusty Reﬂections, the opening

100 Note that the broad pitch range of the voices lead to the choice of pitches in the upper voice that in a number of places enriched the character of the triads in the accompaniment, by adding minor and major sevenths, ninths and elevenths, such as jazz musicians might routinely use: B6 in m.1, A7 in m.3, G7 in m.5, F7] in m.7, G6 in m.9, E6 in m.10, etc.

Note also that the voices often move only by a comma-sized (1 ﬁtt) steps, and that the proximity of certain pitch choices (local minima on the dissonance landscape) allowed for interesting “clashes” of voices that form intervals that are well-tuned, yet have high tension.

3. Farmer’s Fuguetta

The next experiment set out to ﬁnd if a simple imitative polyphonic texture could be created using such quasi-mechanical process. This movement was constructed as a texture of three voices, and also served as an exercise in using the chromatic semitone (25:24, 3 ﬁtts).

The “rules of counterpoint” used in this etude were set as follows:

• The voices were to imitate each other by copying sequences of tones and shifting them visually on the 53-grid, both in pitch and in time. • The voices were to stay out of each other’s way (also visually on the grid). • The voices were to avoid frequent duplication of pitches, including octave shifts. • The constraints of calculated consonance were to be satisﬁed by working back and forth among the voices, since all voices are of equal importance.

The voices were synthesized as organ-like timbres of four harmonics with slightly different amplitude envelopes. To help to differentiate the voices easier, the lowest voice is panned to the centre, the middle voice to the right, and the top voice to the left of the stereophonic field.

The melodic theme that is passed around among the voices is a concatenation of four short phrases that were borrowed (and modiﬁed) from several Czech popular songs.14 Six such distinct phrases can be identiﬁed, for example, in measures 1–4, 5–6, 7–8, 9–12, and 25–26

14The folk songs “Kdyˇzjsem j´aslouˇzil.. . ”15 , “Sk´akal pes pˇresoves” and “Koˇcka leze d´ırou”16 , and a popular Czech song “Pˇredvejtoˇnskou hospodou”17 .

101 of the lowest voice, and measures 25–26 of the middle voice.

Figure 4.9: Farmer’s Fuguetta, the opening

The “interchanging melodies” that make the polyphonic texture of the fuguetta were “woven in harmony”18 on the microtonal grid of Qvinta. The graphical patterns of the melodic phrases were visually replicated among the voices and their pitches adjusted with the guidance of the dissonance landscape (in this case, working back and forth among the three voices). Certain (limited) eﬀort was made to (visually) intertwine the three voices into a three-strand braid (Fig. 4.9 and Fig. 4.10).

4. Martian Mazurek

The adequate success of the Fuguetta exercise, speciﬁcally its more microtonal second part (mm. 17–28) encouraged the next experiment:

The Martian Mazurek explores how far one can push harmonic and melodic textures into farther regions of microtonality and still keep it within the tonal harmonic progression of the Prelude. The accompaniment of the two lower voices was adapted from Rusty Reflections, mainly by a number of deliberately introduced microtonal shifts in pitch. For example, the added two-measure introduction contains four repetitions of a three-tone arpeggio. The first and the third are two copies of the original C major triad (with the flattened E) and the second and the fourth are its modifications with pitches shifted by chromatic semitones

17A story of a young man who works as a farm hand for nine years before he can marry his sweetheart. 17A melody that traces back to 17th century Italian song La Mantovana, which in its minor mode version was adapted as the main theme in Vltava (Moldau) by Bedˇrich Smetana and in Hatikvah by Samuel Cohen (http://www.aish.com/j/as/The_History_of_Hatikvah.html). 17Composed and published by Karel Haˇslerin 1922. 18See Ainulindal¨e, p.iii.

102 Figure 4.10: Farmer’s Fuguetta, the ending and 1-ﬁtt commas. Note also the microtonal step-wise motion of the bass line, namely in measures 2, 11–16, and 23–25.

The twisty, spiralling melody was made from sets of tones that were automatically derived from respective dissonance landscapes, such as shown in Fig. 4.7c. The tones were arranged into shapes that resembled a dance-like motion that “gyred and gimbled.”

In mm. 13–16, a turn-like19 motif of six sixteenth notes was simply replicated eight times and each of its repetitions shifted in pitch by a comma, mostly in the downward direction. This is the only part of the Mazurek where pitches of the melody are drawn from all available

19Turn: a musical ornamentation by a group of notes that are close in pitch (“grupetto”).

103 steps of the 53-tone division, without preference for diatonic or chromatic (justly tuned) scales. By about m.16 the tonal stability is becoming lost and is therefore reinstated by re-introducing the main theme in m.17 (in which also the chord progression returns to C major).

5. Genaues Glockenspiel

By this point, it seemed that the pitch choice method works suﬃciently well, and therefore it became possible to attempt an algorithmic musical piece:

Genaues Glockenspiel is a justly tuned carillon. First, a sequence of sustained chords of the shortened harmonic progression of the Prelude was entered as background chords. The chord tones are “ghost” tones: their role is to be the source for computing the dissonance landscape, but their sound is muted.

The four bell-like voices (in one version synthesized as a bank of four slightly detuned resonators excited by a “pink” noise signal, and using FM synthesis in another version) were constructed within the constraints of the harmonic progression by an algorithmic process that was guided by certain aspects of simulated physics: each voice tends to be a sequence of three-tone arpeggios that once in a while slow down in speed as if running out of energy and needing a breath of air. There is also some degree of randomness that introduces variations (indeterminacy20) added to each voice.

The bottom voice was created first with its pitches constrained only by the dissonance landscape of the ghost tones. The second voice was then generated as if it were a physical body connected by a spring (by attractive and repulsive forces similar to those used in swarming “boid” model of animal groups21) to a physical body associated with the first voice. The pitches of the second voice were constrained to the dissonance landscape computed from the ghost tones and the tones of the first voice. The process was then repeated to generate the third and the fourth voices.

A more detailed description of the used algorithm is as follows: Both the Genaues Glock- enspiel and the Turing Toccata (below) etudes use a “pitch walker” simulation to generate melodic lines. Pitch walker is an application of the “random walk”22 concept combined 20Xenakis, “Determinacy and indeterminacy”. 21Reynolds, Boids. Background and update. 22A path that is a sequence of steps of random length taken in random directions. For example, the Max/MSP

104 with a simple simulation of a physical body23 motion. The pitch walker “walks” in one direction only: up and down the 53-tet pitch scale. It takes small steps, and occasionally larger leaps, which size is the combined result of the walker’s “velocity” and pseudo-random values with Gaussian24 distribution. A “path shape” that simulates a shape of a musical gesture can, optionally, guide the (drunken) walker motion: for example, the walker could be impelled to make a sequence of small steps in one direction, followed by a leap in the opposite direction.

Genaues Glockenspiel has four voices, simulated by four walkers: the “bass”, “tenor”, “alto”, and “soprano”. The bass is the most independent voice and simply follows its path shape that consists from two smaller leaps down and one bigger leap up: a series of descending three-tone arpeggios. The motion of the upper three voices is aﬀected by simulated forces. There are two “barriers” that limit the range of all the voices to ﬁve octaves. A walker is repulsed by each barrier the more the closer they are. By itself, a walker would end up walking around the centre point between the barriers. At the same time, however, there are repulsive forces between each pair of the walkers that keep the soprano above the alto, tenor, and bass; the alto below the soprano and above the tenor and bass; and the tenor below the soprano and alto and above the bass. All the forces combined with the inertia and randomness of the walkers form a dynamic system of four bodies that move in the pitch space. Their paths, further constrained by dynamically calculated dissonance landscape, create the four-voice texture.

6. Turing Toccata

The ﬁnal movement of the suite uses the harmonic progression of the complete Prelude. It consists from a background accompaniment made as a combination of the original Prelude and a high voice that resembles the violin sound of Rusty Reﬂections, arranged manually from pitch choices given by the dissonance landscape.

The foreground virtuoso melody was derived from a path taken by a single pitch walker. Its range of motion was constrained, similarly as in Genaues Glockenspiel, by two pitch barriers. This time, the low barrier was ﬁxed, but the pitch of the high barrier changed with the current pitch of the top, violin-like voice. The centre point of the two repulsive

object drunk is a random walker in the domain of MIDI note numbers (0–127). 23A body with mass (and therefore inertia) and velocity (that can be aﬀected by forces), located in space. 24A “bell-shaped” curve.

105 forces, and therefore the overall rising and falling of the generated voice thus loosely follow the shape of the top voice.

The walker is also guided by a path shape: it is more or less impelled to first take a series of up to seventeen downward microtonal steps, each between 1 and 2 fitts in size, and then five upward steps of twice the size. Whenever the walker completes one full cycle of the path shape, it pauses for a moment (there is a rest in the generated melody) and then resumes its motion with a new iteration of the cycle.

106 Chapter 5: Related work: interactive music

The principles of computing the dissonance landscape from beats caused by near-coincident harmonics, and of tuning of complex tones by ﬁnding local minima of dissonance in the dissonance landscape can be applied to interactive, machine-assisted music, provided that the required calculations can be performed fast enough and with minimal latency that would not adversely impact the timing of musical pulse. An experimental software solution has been developed and incorporated as part of the general software environment Qvin.1 Qvin is a cross-platform system written in C++ using the Qt5 framework.

Qvin is an “environment” that contains:

• its own object-oriented scripting language, • a compiler that translates the scripting language into a “bytecode”, the Q-code, • a stack-based virtual machine that executes the Q-code, • a detailed interface that allows the development of classes (written in C++/Qt) that can be incorporated into the class hierarchy of the scripting language, • a plug-in mechanism of dynamically loaded libraries through which the new classes can be added to the system, • a library that provides a set of core built-in, general use classes that are the foundation of the class hierarchy, and • an integrated development environment (IDE).

Instances of the available classes can be created and manipulated at runtime from within the scripting language. This combination of compiled and interpreted code provides the speed, robustness, and access to operating system services of C++, and the brevity and ﬂexibility of a scripting language.

The grammar of the built-in Qvin scripting language is inspired by the language introduced in SuperCollider.2 It is an imperative (procedural) language with some strict functional aspects, including tail recursion. The only data type is a ‘class’; every function call (a class method invocation) returns a value and the ‘dot’ operator makes it easy and handy to string

1Burleigh, Qvin. 2McCartney, “Rethinking the Commputer Music Language: SuperCollider”.

107 function calls one after another. Unlike in SuperCollider, the Qvin language is strictly typed and the type information is checked and enforced by the compiler at compile time.

The interface that is a part of the plug-in mechanism for implementing classes allows deﬁni- tion of class hierarchy and class methods including their type signatures, and provides the means for callbacks from C++ code back to the Qvin script.

The Qvin language uses reference counting of objects, and not garbage collection as many other scripting languages do. A reference-counted object gets destroyed automatically as soon as all references to it are lost and the object is thus “thrown away”. Reference-counting is compatible with the philosophy of C++ and Qt: it allows the use of destructors with strictly predictable timing of their invocation.

The scripting code is compiled into Q-code with added meta information; both are stored in a SqLite3 database. Storing information in the database allows incremental compilation of classes and their methods; as long as their signatures (their interface and type information) does not change, only the modified code needs to be recompiled. For execution, the Q-code and the accompanying meta-information are loaded into an instance of a virtual machine. After that, the source code can be again modified and recompiled. It is thus possible to run different versions of a program in several distinct instances of the virtual machine, side by side.

The Qvin language was designed to be compact and brief. Most of its keywords and names of the built-in classes are three letters long: cls (class), mth (method), Obj (Object: the ultimate base class), Int (integer number), Flt (floating point number), Chr (character), Str (string), Vec (vector), etc. Three-letter abbreviations seem to be sufficiently mnemonic, and in most cases there is no need to use longer identifiers.

Fig. 5.1 is a screenshot of the Qvin IDE showing an example program with a recursive factorial algorithm and a tail-recursive Euler’s greatest common divisor algorithm.

Qvin Musical Processes Fig. 5.2 shows the IDE with the ‘qmp’ (Qvin Musical Processes) plug-in that implements a collection of classes for real-time sequencing of data that represent time events, notes,

3http://www.sqlite.org/

108 Figure 5.1: The Qvin IDE instrument timbres and other information required for music, and for computation and use of dissonance landscapes.

Figure 5.2: The Qvin Musical Processes plugin

At the root of the musical processes class hierarchy is a class qmp::Proc. Each instance of the class has its own operating system-level thread for data processing and an associated

109 widget for its user interface. Instances of the class communicate by passing messages. The thread that processes data received by a particular instance of the class can post messages into the receiving “bag” of another instance of the class. Each message has a precise time stamp; when a message is posted into a bag, it can be either processed immediately by the receiving thread, or put into a timestamp-ordered queue and have its processing postponed until the designated time.

Multiple instances of “musical processes” can be connected into a network of “posters” and “receivers”. Each data-processing thread of execution can be dynamically started and stopped in real time. The processes can be disconnected and reconnected, also in real time, and the topology of the network and be thus changed at will.

The current version of the ‘qmp’ plugin provides two kinds of classes: classes to represent musical event messages, and classes that implement various specialized musical data-processing processes.

The basic musical event classes are:

• qmp::Evt: This class is the base of each “event” message passed among the musical processes. It contains the time stamp, which is precise to a millisecond, and other data and methods that are required for its handling by the data processing threads. • qmp::EvtTone is a direct subclass of qmp::Evt. It represents timed events for handling tones and contains additional information of the tone pitch, amplitude, type of a synthesized instrument, spatial information, etc. It has several subclasses: – qmp::EvtToneOn for tone onsets, – qmp::EvtToneOff for endings of tones, and – qmp::EvtToneChg for changing the parameters of an existing (sounding) tone. • qmp::EvtTick is also a subclass of qmp::Evt. Its time-stamped instances represent special moments in time, such as ticks of a metronome.

The musical data-processing classes are all subclasses of qmp::Proc. Each instance of a data-processing class contains its own thread of execution for processing qmp::Evt-based data. Fig. 5.3 shows the user interfaces of several interconnected instances of data-processing classes.

110 Figure 5.3: ‘qmp’ user interface example

Among the basic musical data-processing classes are:

• qmp::Ticker: Instances of this class generate qmp::EvtTick events. • qmp::Synth: In response to received qmp::EvtTone messages, an instance of this class controls a running instance of Csound. • qmp::Ewi: This class interfaces with the Akai EWI wind controller through its MIDI connection, and in response to a player’s actions generates qmp::EvtTone messages. • qmp::Fitter: This class rounds the pitch values of passing tone events to whole fitts (steps in 53-tet). • qmp::Imitator: Monitors passing sequences of tones, remembers and later imitates melodic motifs. • qmp::Drunk: A “random pitch walker” with a pitch bias (the centre of the walk). The walker first accumulates “energy” from passing tone events, and when its level exceeds a given threshold, the energy is expended for steps in the walk until its level falls again below the threshold (with certain hysteresis). • qmp::Score: Generates streams of tone events from a predefined score. • qmp::Tuner: Using the data of currently sounding tones, the Tuner tunes passing tones by adjusting their pitch values to local minima in a dynamically computed dissonance landscape.

The classes listed above were used in several experiments with real-time musical interplay of a machine and a human musician. At the core are, again, dynamically calculated dissonance landscapes that are used to adjust or alter pitches of simultaneously sounding tones. A number of instances of the above classes are connected into a network and communicate by exchanging (posting and receiving) events. The network may emulate the actions of

111 co-operating musicians that listen and complement each other.

The work is in its initial stages, and although there have been some demonstrably promising results, it would be premature at this time to speculate about the potential of the system. Let us, nontheless, present one example. It consists of:

• an instance of qmp::Ticker that sets the tempo and metre, • an instance of qmp::Score that plays the prepared accompaniment (acts as a “rhythm section”), • an instance of qmp::Ewi as an interface to a human musician that plays the Akai EWI wind controller (the main melodic line), • an instance of qmp::Drunk that generates the other melodic line, • two instances of qmp::Tuner that constrain pitches to dissonance landscapes, and • an instance of qmp::Synth that sends data to Csound for real-time sound synthesis.

For full appreciation of the dynamic result, the example requires a live demonstration; in this text we can only list excerpts of the code as follows, to illustrate the use of the Qvin language.

The code that creates an initializes the instances of ‘qmp’ objects is: cls Band { var ti = Ticker(), sc = Score(), ew = Ewi(), dr = Drunk(), tu = Tuner(), tt = Tuner(), sy = Synth(); } mth init { ti.stop; // stop previous run, if any

// reset visibility of controls ti.show; sc.hide; ew.hide; dr.hide; tu.hide; tt.hide; sy.show;

// disconnect the network

112 ti.disc; sc.disc; ew.disc; dr.disc; tu.disc; tt.disc; sy.disc;

// reset instrument assignment; detach tuners sc.inum(4).tuner(Tuner.nil); ew.inum(5).tuner(Tuner.nil); dr.inum(6).tuner(Tuner.nil);

// reset and reconnect the ticker ti.tempo(60.0); ti.sendTo(sc).sendTo(ew).sendTo(dr); }

// ... more methods here }

The network is then connected, and the music started by: mth ex1 { self.init; // (1) ew.show.amp(0.3).tuner(tu).tune(1,1).sendTo(dr).sendTo(tt).sendTo(sy); // (2) dr.show.amp(0.2).tuner(tt).tune(2,12).sendTo(sy); // (3) sc.amp(0.35).sendTo(tu).sendTo(tt).sendTo(sy); // (4) ti.tempo(78.0); sc.line(Prelude.in53).play; // (5) }

The above lines of code perform these functions:

1. Initialize, or reset the network. 2. The ew interface is shown; the amplitude of generated tone events is set to 0.3. Pitches converted from MIDI events (in 12-tet) are to be tuned (constrained by a dissonance landscape) by the tuner tu, using strategy 1 (the closest consonance valley) within the maximum distance of 1 ﬁtt, and then sent to dr for following, to tt for computation of its dissonance landscape, and to sy for sound sythesis. 3. The dr interface is shown; the amplitude of generated tone events is set to 0.2. Their pitches are to be tuned by the tuner tt, using strategy 2 (the deepest consonance valley) within the maximum distance of 12 ﬁtts, and then sent to sy for sound sythesis. 4. The amplitude of tone events from the score player sc is set to 0.35, and the events will be sent to both tuners tu and tt, and to sy for sound synthesis.

113 5. The ticker ti tempo is set, and the score sc begins to emit tone events.

Note that the two tuners are used to

• tune ew to sc, and • tune dr both to sc and ew.

The accompaniment is, again, the Prelude, encoded in the 53-tet just intonation approximation: cls Prelude { use Line = qmp::Line;

* in53 Line { Line(53) .m.ts(00,8).ts(17,7).g(31,0,6).m.t(31).t(53).t(70).d.d .m.ts(00,8).ts(09,7).g(40,0,6).m.t(40).t(62).t(75).d.d .m.ts(-5,8).ts(09,7).g(31,0,6).m.t(31).t(62).t(75).d.d .m.ts(00,8).ts(17,7).g(31,0,6).m.t(31).t(53).t(70).d.d // ... etc. } }

The above code captures the sequence of tones in the Prelude (first four measures are shown): each sequence between m (a repetition marker) and d (duplicate)4 is repeated twice; ts defines a tone with a pitch specified in fitts and sustained for a given number of beats, and t is a shortcut for a tone sustained for one beat; g represents a “ghost” tone (that shapes the dissonance landscape but is not to be played) with a specified pitch, a number of beats by which the score time is advanced (0 in this case) and a number of beats to sustain the tone.

However simple this particular system is, it has the ability to act as a partner to a human perfomer in two-voice polyphonic improvisation. The character of the interaction is aﬀected by the choice of dr attributes (controlled in its graphical interface): the pitch bias (as a ﬁtt

4 A feature obviously borrowed from Forth.

114 distance from the pitch intercepted in ew data), the thresholds and factors that inﬂuence how the “energy” is handled, and the maximum allowable ﬁtt distance for tuning. A proper combination of attribute values needs to be set to elicit the most interesting behaviour.5

The dynamically generated melody line is created as intermittent musical gestures of the pitch walker (shaped only by its pitch bias and “energy” handling, constrained by the un- derlying calculated dissonance landscape) and therefore most of the interaction must be done by the human performer. Despite of this limitation, the example system has been a source of an interesting musical entertainment.

5Seek and ye shall ﬁnd the edge of chaos.

115 Chapter 6: Conclusions and future work

The narrative of this text presented an argument for the use of just intonation in music, founded in physical properties of musical sound and in psychoacoustics of tone sensations, as proposed in the work of Hermann Helmholtz.

At the core is the concept of a “dissonance landscape”, a principle that allows the calculation of relative dissonance among a group of tones. Local minima in the landscape correspond to pitches of tones that are well in tune and can form harmonious sonorities.

Such harmonious sonorities naturally lead to a formation of the Western diatonic and chromatic scales, and the associated harmony. If the choice of pitches is ﬁxed and limited to twelve in an octave and if harmonic mobility is needed, just tuning must necessarily be compromised. Precise just tuning requires theoretically an inﬁnite, practically a large number of pitches in an octave.

A system of dividing the octave into fifty three equal steps (fitts) well approximates just intonation, and can be used for its practical approximation in computer-assisted music; calculated dissonance landscapes assist in choosing harmonious pitches from among the fifty three steps. Notation of music in 53-division of the octave is possible with the proposed microtonal extension of the common staff notation system.

A number of software tools (for visualization and investigation of harmonious tone combinations, for calculation of dissonance landscapes and their use in tone sequencing, and for construction of melodic shapes derived from simulated physics) were used in sound experiments and in construction of musical etudes that support the presented argument and propositions. Some of the tools were developed in the ﬁrst place for the purpose of the presented work.

A short self-contained suite of microtonal etudes, based on J.S. Bach’s Prelude in C, was constructed to verify some of the assumptions and demonstrate some of the potential of the proposed method in computer-assisted and algorithmic music. There is also demonstrable potential of using the presented method in real-time, dynamic system for a musical human- computer interaction.

116 Future work To plan and write about a future follow-up work is often a speculative exercise. In an eﬀort not to surmise what cannot be substantiated at this point, I will limit myself here to listing only the most immediate planned steps that can be directly related to the work that has been already completed and presented in this text, and omit the more ambitious ideas, lest the eventual results fall short of expectations. As in the presented work, every future step ought to be supported by experiments and practically implemented in a way that is relatively sound and leaves little ground for uncertainty and therefore the possibility of its deconstruction.

Even with the above stated limits, the planned future work shall occupy the author, if God permits, for many years to come, and supply more than enough subject matter to support the research component of university teaching. These are (some of) the planned steps:

• Improve and validate the used psychoacoustic model of consonance, and further investigate its possible variations. Namely: a) better determine the character of contribution of (pair-wise) beats to the aggregate sensation of dissonance, b) better assess the contribution of combination tones to the sensation, and c) investigate the role of secondary combination tones. These and similar investigations, supported by experiments, should extend to involve groups of musicians/listeners, to establish how d) the sound sensation and perception do vary individually and e) how the results of such experiments would compare to the results of group studies that were conducted and published in the past; the quality of audio equipment and synthesized sound has changed, and so has perhaps also the way how we are conditioned to listen. • Complete the development phase of the custom software “life-cycle”. The software has been developed to comparably high software-engineering standards, for reliability and to allow further extensions and improvements. With some wrapping-up, it could be released, under appropriate licence terms, to others. • The Qvinta (sequencing) software application should allow, either directly, or through post-processing in other systems, synthesis of sound intended for music performance or audio production, rather that the current version that is (intentionally) suited for the purpose of experiments. • Continue with the development of the Qvorum (analysis) software for transcription of recorded “mixed” music.

117 • Extend and improve the calculated consonance method and the scripting language implemented in the real-time Qvin environment, to allow their feasible use in interactive (human-machine) music. The real-time nature will require a number of adjustments compared to the “oﬄine” use of the method (as used for making the C53 Suite); namely, there seems to be a necessity to use of what could be called a “hear-ahead” and “hear-behind” (in time) behaviour of the “machine musician”. • Depart from the “straight-jacket” harmonic plan (such as the one enforced by the Prelude) in favour of more freely evolving (yet still constrained by proper tuning) sonorities. I have made a few experiments that lead in that direction, with several promising results. • Include tones with pitch contours (controlled modulation of pitches, such as the known “pitch bend” or glissando) and explore the possibility of morphing (blended transitions) between resulting sonorities, guided by the principles of calculated consonance. • Once (and this is highly speculative) the ears get accustomed to the sound of 5-limit textures on the outskirts of the 12-tone system, perhaps 7-limit timbres and the related intervals can be applied, with much caution and attention to tuning, to create microtonal music that would be still relatively acceptable to listeners used to traditional music.

118 Appendix A: C53 Suite

This page is intentionally left blank.

119 1. Prelude in C53

C Dm7/C G7/B C

5 Am/C D7/C G/B Cmaj7/B

9 C6/A D7 G Gdim7

13 Dm/F Ddim7/F C/E Fmaj7/E

17 Dm7 G7 C C7

21 Fmaj7 F♯dim7 A♭dim7 G7

25 C/G G7sus G7 Adim7/G

29 C/G G7sus G7 C7

33 Fmaj7/C F6/C G7/C C 2. Rusty Reflections

8va... 3 4 8va... 3 4 C F G7 C Am 3 4

D7 G Cmaj7 Am D7

G A7 Dm G7 C Fmaj7

17 3

Dm7 G7 C C7

Fmaj7 F7 F6 G7 C

121 3. Farmer's Fuguetta

2 4 8 2 4 8 2 4 8 9

8 13

8 17

8 20

8 25

122 4. Martian Mazurek

8va... 3 4

3 4

24 5. Genaues Glockenspiel

8va...

124 16

125 6. Turing Toccata

8va... 2 2

2 2

C Dm7/C 2 2

2 2

G7/B C

Am/C D7/C

G/B Cmaj7/B

C6/A D7

G Gdim7

126 25

Dm/F Ddim7/F

C/E Fmaj7/E

Dm7 G7

C C7

Fmaj7 F♯dim7

A♭dim7 G7

127 49

C/G G7sus

G7 Adim7/G

C/G G7sus

G7 C7

Fmaj7/C F6/C G7/C

128 Appendix B: The pitch spiral

The pitch spiral is a software tool for experiments with tones of speciﬁc pitch (frequency) and harmonic content, and their combinations that form various intervals and chords.

Prerequisites

• An HTML5-compliant web browser. The pitch spiral has been tested in Chrome, Firefox, and Safari. • A working local installation of Csound, version 5, available from http://www.csounds. com/. • A current version of Java Runtime (http://www.java.com/). • Either MacOSX or Linux. The running instance of Csound is controlled in real time through a Unix-type pipe, a standard technique that is not supported by Windows. Only the graphical part of the spiral, but not sound synthesis, can be used on Windows.

Installation The pitch spiral has two components:

1. A web-based graphical user interface, and 2. a local sound synthesis component made of a WebSocket bridge to a running Csound instance and a collection of Csound source ﬁles.

The graphical component of the pitch spiral is available at http://qvin.in/CC/PitchSpiral. On the bottom of the page is a link “local ﬁles” to download an archive of ﬁles (CSpiral.zip) that are needed for sound synthesis.

• Download and unpack the archive CSpiral.zip. Open a terminal application and navigate into the new folder (cd CSpiral). • Make the ﬁle runCWS.sh (“run CSound-WebSocket”) executable (chmod +x runCWS.sh). • Run it (./runCWS.sh). Refresh the web page with the pitch spiral in the browser. You should hear a short beep through the audio interface (headphones) which indicates that the connection between the browser and Csound has been established.

129 The communication loop

The communication is started by running the shell script runCWS.sh:

while true; do java -jar EchoWS.jar -p 53053 | \ csound --messagelevel=6 --nodisplays --noheader \ --score-in=stdin -odac HH.csd done

EchoWS.jar is a simple one–way “echo server”.1 It accepts incoming WebSocket connections on the speciﬁed port (53053, arbitrarily chosen), performs the required handshake and after that passes all incoming data directly to the standard output. The running instance of Csound accepts its “score” commands from the standard input and plays the synthesized sound through the default audio output. HH.csd contains the required orchestra and initial score deﬁnition.

Whenever the web page (the client in the WebSocket communication) is re-loaded, the WebSocket connection is closed and both the echo server and the running Csound instance terminate. The surrounding endless loop in the shell script then restarts them and the echo server is ready to accept a new incoming connection.

Usage Refer eiter to Fig. B.1 that shows a screenshot of the spiral in its initial state, or directly to a running instance of the spiral in a web browser.

The curve of the spiral represents a reasonably limited pitch continuum, from C1 to C8. Each turn of the spiral corresponds to a pitch diﬀerence of one octave. The small red circle

indicates the position of C4. There are ﬁve discs in the centre of the spiral: blue, green, orange, yellow, and white, that represent the fundamental harmonics of ﬁve tests tones. The discs can be dragged by the mouse to any location on the spiral. Each tone has an associated set of controls in the panel that is to the right of the spiral; their function is as follows, listed in the left-to-right, top-to-bottom order:

• A toggle-button of the same colour as the corresponding tone disc. The button switches 1The Java source code is available at https://github.com/dunna/jMinWS.

130 Figure B.1: The pitch spiral

the tone (its sound synthesis in Csound) on and oﬀ. The blue, green, orange, and yellow buttons are numbered 0–3 (for no particular reason at all); the white button is labelled ‘p’, as a reminder that the white tone can be used as a “probe” tone. • A push-button with a snowﬂake icon. Pushing the button aligns the “spokes” that show justly tuned intervals (if they are displayed, see ‘just’ below) with the fundamental harmonic of the respective tone. • A slider that controls the loudness of the tone. • A number box that displays or controls the fundamental frequency of the tone. • A slider that controls the harmonic content of the tone between one and six harmonics. The upper harmonics are shown as progressively smaller discs of the same colour. • A number box that displays or controls the fundamental pitch of the tone. Note that the pitch values follow the Csound convention and therefore, for example, the pitch of

C4 is shown as 8. • A slider that pans the tone from left to right in the stereo ﬁeld. • A button with a diamond sign that pans the tone to the centre.

The panel ‘division’ controls various divisions of the octave that can be displayed as “spokes”

131 (radial lines or series of dots) on the spiral. The radio buttons in the very left column select the active division; the remaining controls in the selected row modify the selected division:

• P5: A chain of fifths. The three number boxes control the number of fifths in the chain, their tempering, and rotational offset of the whole system. • M3: A chain of major thirds; the number boxes work the same as above. • m3: A chain of minor thirds; the number boxes work the same as above. • just: Just chromatic scale. The checkbox on the right switches on and off the display of spokes that show the justly tuned intervals: the chromatic and diatonic semitones, minor and major tones, the minor third, the major third, the perfect fourth, and their inversions. • mean: The quarter-comma mean tone temperament. • 12-tet: The twelve-tone equal temperament. • 53-tet: The 53-tone equal temperament.

The checkbox in the bottom right corner, labelled ‘snap’, controls whether the tone discs snap to the positions of spokes on the currently displayed radial grid. Selecting the checkbox ‘lock’ makes all tone discs maintain their current interval relations and to move as one locked-together group. The checkbox ‘comb’ tuns on and oﬀ the calculation and display of combination (diﬀerence) tone pitches. Finally, the pushbutton ‘cons’ triggers the calculation and display of the dissonance curve.

132 Appendix C: Included media and software

All audio examples and open source components of developed software are available online on the accompanying website (see p.xvi). The included CD or a digital archive ﬁle contain a copy of the audio ﬁles and a current snapshot of the open source software components, in the following folders:

1.PitchSpiral The pitch spiral JavaScript/SVG/Csound applet: 1.PitchSpiral/index.html The main file that runs the applet (open in a browser). 1.PitchSpiral/index_files Files required by index.html. 1.PitchSpiral/Csound The local Csound files and the WebSocket bridge. 2.Sensations The “sensations of tones” experiments: 2.Sensations/mp3 Audio examples. 2.Sensations/csd Csound source files. 3.Temperaments The demonstration of various temperaments: 3.Temperaments/mp3 Audio examples. 3.Temperaments/csd Csound source files. 4.C53_Suite The variations on Prelude in C, audio files.

133 Glossary

12-tet Twelve-tone equal temperament; a division of the octave into twelve equal parts (semitones).

53-tet Fifty-three-tone equal temperament; a division of the octave into 53 equal parts (ﬁtts).

amplitude The maximum displacement of a point on a wave.

amplitude envelope The variation of amplitude over time.

chromatic “Colourful”. 1) The ancient Greek chromatic genus of a minor third and two semitones. 2) The modern chromatic scale of twelve semitones. 3) The smaller of the two semitones in just intonation.

comma A small interval, signiﬁcantly less than a semitone.

comma drift Gradual change of the reference pitch in the course of a harmonic progression in just intonation.

comma shift A microtonal change (by a comma) of the pitch of common notes in a harmonic progression; necessary to avoid comma drift.

complex tone A tone with a non-sinusoidal waveform; a sum of a number of simple tones. consonance A combination of two or more tones that sound pleasant to the ear, such as that a sensible musician would not be able to improve by microtonally adjusting their pitches. diatonic “At intervals of a tone”. 1) The ancient Greek diatonic genus of two whole tones and a semitone. 2) The modern diatonic scale of ﬁve tones and two semitones. diﬀerence tone A third, phantom tone that is created by an interference (rapid beats)

134 between two simple tones. Its frequency is equal to the difference of frequencies of the two source tones. dissonance The opposite of consonance. dissonance curve The relative dissonance between two tones, plotted as a function of their frequency difference. dissonance landscape The relative dissonance between a group of fixed tones and a “probe” tone, plotted as a function of the probe tone frequency. enharmonic “In harmony”. 1) The ancient Greek enharmonic genus of a major third and two quarter tones. 2) Two different notes that are “harmonically the same”.

1 ﬁtt A unit to measure intervals, equal to 53 of an octave. frequency The number of cycles of a periodic wave in one second; related to tone pitch. fundamental 1) Fundamental frequency is the basic frequency of a complex tone. 2) Fun- damental harmonic is the basic component of a complex tone: a simple tone with the fundamental frequency. harmonic (noun) A simple tone that is a component of a complex tone. harmonic series The series of frequencies with ratios 1:2:3:4 etc. harmonious (adjective) Forming a pleasing and consistent whole, melodious, free from discord, pleasant to the ear1. harmony Combinations of simultaneously sounding tones that produce harmonious chords. A succession of chords, and the associated motion from one chord to another, is called a chord progression or harmonic progression. Such progressions, if well constructed, produce distinct aesthetically pleasing eﬀects2,3.

1“Harmonious”. Oxford Dictionaries. 2“Harmony”. Oxford Music Online. 3“Harmony”. Oxford Dictionaries.

135 heterophony Harmony of several voices that are variations of the same melodic line.

homophony Harmony of several voices that move rhythmically together.

interval Musical interval: a diﬀerence between the pitches of two tones. key 1) Musical key: a tonal system derived from a heptatonic scale and named after its central note (the tonic). 2) A lever on the piano keyboard that is pushed down in order to strike a string.

microtonal music Music made with the use of microtonal intervals, i.e. intervals that are diﬀerent (mostly narrower) than the semitone.

note 1) A symbol that represents a tone in a musical notation. 2) A name that designates the pitch of a tone.

partial Elsewhere a synonym for a ‘harmonic’. In this text used only to refer to data from the Loris library.

pitch A perceived quality of a tone related to its frequency.

polyphony Harmony of several voices with diﬀerent rhythmical structures.

simple tone A tone with a sinusoidal waveform.

temperament Deliberate, controlled mistuning of intervals in a musical scale, in order to avoid wolf intervals.

tone 1) A tone is a sound with a periodic waveform. 2) An interval of two semitones.

136 Bibliography

Barbera, Andre, ed. The Euclidean Division of the Canon (Greek and Latin Sources). The University of Nebraska Press, 1991. Barbour, J Murray. Tuning and Temperament: A Historical Survey. New York: Da Capo Press, 1972, c1951. Barker, Andrew, ed. Greek Musical Writings. 2 vols. Cambridge Univerity Press, 1984, 1989. Benson, David J. Music: a mathematical offering. Cambridge University Press, 2006. Bostock, Mike, Jeff Heer, and Vadim Ogievetsky. D3.js. (software) A JavaScript library for manipulating documents based on data using HTML, SVG and CSS. 2009–2013. url: http://d3js.org/. Burleigh, Ian George. “Computer-assisted Tone Arrangement Using Calculated Consonance”. In: Contemporary Music Review 32.5 (2013), pp. 447–458. – Qvin. (software) A suite of software components (a scripting language, libraries, and plugin modules) for general computation, modelling, and music. 2011–2013. url: http://qvin. in/Qvin/. – Qvinin. (software) A framework for making C++/Qt desktop applications. 2011–2013. url: https://github.com/qvinin/Qvinin. – Qvinta. (software) An experimental system for microtonal sound sequencing. 2009–2013. url: http://qvin.in/Qvin/. – Qvorum. (software) An experimental system for music transcription. 2011–2013. url: http://qvin.in/Qvin/. – Vinci. (software) A JavaScript library to make simple machines. 2011–2013. url: https: //github.com/dunna/vinci/. Cannam, Chris et al. Sonic Visualiser. (software) An application for viewing and analysing the contents of music audio files. 2006–2013. url: http://www.sonicvisualiser.org/. Chion, Michel. Guide to Sound Objects. Trans. by John Dack and Christine North. Trans- lation of: Guide des objets sonores: Pierre Schaeffer et la recherche musicale. Institut na- tional de l’audiovisuel & Editions´ Buchet/Chastel, Paris (1994). Self-published, 2009. url: http://www.scribd.com/doc/19239704/Chion-Michel-Guide-to-Sound-Objects. Cohen, H. Floris. Quantifying music: the science of music at the first stage of the Scientific Revolution, 1580–1650. Vol. 23. University of Western Ontario series in philosophy of science. Dordrecht, the Netherlands: Reidel, 1984.

137 Cope, David. Experiments in musical intelligence. A-R Editions, Inc., 1996. url: http : //arts.ucsc.edu/faculty/cope/experiments.htm. – Virtual Music: Computer Synthesis of Musical Style. MIT Press, 2001. d’Alembert, Jean Lerond. Elémensde´ musique, théoriqueet pratique, suivant les principes de m. Rameau. Paris: Charles-Antoine Jombert, 1759. Descartes, René. Compendium Musicae. Utrecht: Zÿll& Ackersdÿck, 1618. Ellis, Alexander J. “The History of Musical Pitch”. In: Nature 21.545 (1880), pp. 550–554. Encyclopædia Britannica Online. Encyclopædia Britannica, Inc., 2013. url: http://www. britannica.com/. Euler, Leonhard. Tentamen novae theoriae musicae. Petropoli: ex typographia Academiae Scientiarum, 1739. Fitz, Kelly and Sean Fulop. “A Unified Theory of Time-Frequency Reassignment”. In: CoRR abs/0903.3080 (2009). arXiv: 0903.3080. url: http://arxiv.org/abs/0903.3080. Fitz, Kelly and Lippold Haken. Loris. (software) Software for sound modeling, morphing, and manipulation. 1990–2013. url: http://www.cerlsoundgroup.org/Loris/. – “On the Use of Time: Frequency Reassignment in Additive Sound Modeling”. In: Journal of the Audio Engineering Society 50.11 (2002), pp. 879–893. Folland, Nicole A. et al. “Processing simultaneous auditory objects: infants’ ability to detect mistuning in harmonic complexes”. In: Journal of the Acoustical Society of America 131.1 (2012), pp. 993–997. Galilei, Vincenzo. Dialogue on Ancient and Modern Music (English translation 2004). Yale University Press, 1581. Goetschius, Percy. The structure of music. Philadelphia, PA: Theodore Presser Co., 1934. Grattan-Guinness, Ivor. The Fontana History of Mathematical Sciences. Fontana Press, 1998. Haluˇska, Ján. The Mathematical Theory of Tone Systems. Vol. 262. Pure and Applied Math- ematics. New York; Bratislava: Marcel Dekker; Ister Science, 2003. Helmholtz, Hermann. On the sensations of tone as a physiological basis for the theory of music. Trans. by Alexander J. Ellis. Translation of: Die Lehre von den Tonempfindun- gen. Second English edition, conformable to the fourth German edition (1877). London: Longmans, Green, and Co., 1885. Henz, Martin, Stefan Lauer, and Detlev Zimmermann. “COMPOzE — Intention-based Music Composition through Constraint Programming”. In: Proceedings of the 8th IEEE Interna- tional Conference on Tools with Artificial Intelligence. Toulouse, France: IEEE Computer Society Press, 1996, pp. 118–121.

138 Hofstadter, Douglas R. “Creativity, Cognition, and Knowledge: An Interaction”. In: ed. by Terry Dartnall. Greenwood Publishing Group, 2002. Chap. Staring Emmy straight in the eye — and doing my best not to flinch, pp. 67–104. – Gödel,Escher, Bach: An Eternal Golden Braid. New York: Basic Books, Inc., 1999. Janáˇcek,Leoˇs. Jabluˇnka(Apple Tree). Moravian Folk Poetry in Songs (JW 5/2). Performed by Iva Bittováand Skampaˇ Quartet. CD recording, Supraphon SU 3794-2. 2004. – Tuˇzba (Desire). Moravian Folk Poetry in Songs (JW 5/2). Performed by Iva Bittováand Skampaˇ Quartet. CD recording, Supraphon SU 3794-2. 2004. Keenan, Dave, George Secor, and Gene Ward Smith. The Sagittal notation system. A comprehensive system for notating musical pitch. 2001–2004. url: http://www.sagittal.org/. Landy, Leigh et al. The ElectroAcoustic Resource Site (EARS). url: http://ears.dmu.ac. uk/. Levy, Ernst. A theory of harmony. Ed. by Siegmund Levarie. Albany: State University of New York Press, 1985. Loy, Gareth. Musimathics: The Mathematical Foundations of Music, Volume 1 and 2. MIT Press, 2006, 2007. Mazzola, Guerino, Stefan Göller,and Stefan Müller. The topos of music: geometric logic of concepts, theory, and performance. Birkhäuser,2002. Mazzoni, Dominic and Roger Dannenberg. Audacity. (software) A software for recording and editing sounds. 2000–2013. url: http://audacity.sourceforge.net/. McCartney, James. “Rethinking the Commputer Music Language: SuperCollider”. In: Com- puter Music Journal 26.4 (2002), pp. 61–68. McCartney, James et al. SuperCollider. (software) An environment and programming language for real time audio synthesis and algorithmic composition. 1996–2013. url: http: //supercollider.sourceforge.net/. Mersenne, Marin. Harmonie universelle. Paris: Pierre Ballard, 1636. url: http://archive. org/details/imslp-universelle-mersenne-marin. NIME: International Conference on New Interfaces for Musical Expression. url: http : //www.nime.org/. Oxford Dictionaries. Oxford University Press, 2013. url: http://oxforddictionaries. com/. Oxford Music Online. Oxford University Press, 2013. url: http://www.oxfordmusiconline. com/.

139 Parncutt, Richard. Harmony: A Psychoacoustical Approach. Berlin: Springer Verlag, 1989. url: http://www.uni-graz.at/richard.parncutt/hapa.html. Partch, Harry. “Barbs and Broadsides”. In: Percussive Notes, Research Edition 18.3 (1979). Ed. by Danlee Mitchell and Jonathan Glasier. url: http://www.corporeal.com/barbs. html. – Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments. 2nd ed. New York: Da Capo Press, 1974. Plomp, Reinier and Willem J. Levelt. “Tonal Consonance and Critical Bandwidth”. In: The Journal of the Acoustical Society of America 38.4 (Oct. 1965), pp. 548–560. Rameau, Jean-Philippe. Treatise on Harmony. Trans. by Philip Gossett. Translation of: Traitéde l’harmonie réduiteàses principes naturels. Paris: Pierre Ballard (1722). Mineola, NY: Dover Publications, 1972. Reynolds, Craig W. Boids. Background and update. http://www.red3d.com/cwr/boids/. 1986. Roads, Curtis. Microsound. MIT Press, 2004. – The Computer Music Tutorial. MIT Press, 1996. Sallis, Friedemann and Ian George Burleigh. “Seizing the Ephemeral: Recording Luigi Nono’s “A Pierre dell’azzurro silenzio, inquietum. A piùcori” (1985) at the Banff Centre”. In: 5th Conference EMS08: Musique concrète- 60 years later. Electroacoustic Music Studies Network. Paris, 2008. Sallis, Friedemann, Ian George Burleigh, and Laura Zattra. “Seeking Virtual Voices in Luigi Nono’s A Pierre (1985) through a Study of a Performance and the Creative Process”. Presented at: Analyser la musique mixte, IRCAM, Paris. Apr. 2012. url: http://www. ircam.fr/fileadmin/sites/WWW_Ircam/fichiers/Programmes/conference-1012/ colloque-musique-mixte.pdf. Schillinger, Joseph. The Schillinger System of Musical Composition. Carl Fischer, Inc., 1941. Schulter, Margo. Why are there 12 notes per octave on typical keyboards? Yahoo! ‘tuning’ group, message 19881. 2001. url: http://launch.groups.yahoo.com/group/tuning/ messages/19881. Sethares, William. Tuning, Timbre, Spectrum, Scale. London: Springer Verlag, 2005. Shiloah, Amnon. The Theory of Music in Arabic Writings c. 900–1900. München: G. Henle Verlag, 1979.

140 Stevens, Stanley Smith, John Volkman, and Edwin Newman. “A Scale for the Measurement of the Psychological Magnitude Pitch”. In: Journal of the Acoustical Society of America 8.3 (1937), pp. 185–190. Tartini, Giuseppe. Trattato di musica secondo la vera scienza dell’armonia. In Padova: Nella Stamperia del Seminario, 1754. url: http://archive.org/details/trattatodimusica00tart. Trainor, Laurel J. and Andrea Unrau. “Human Auditory Development”. In: ed. by Lynne Werner, Richard R. Fay, and Arthur N. Popper. Springer Handbook of Auditory Research. New York: Springer, 2011. Chap. Development of Pitch and Music Perception, pp. 223– 254. Turner, R. Steven. “Dictionary of Scientific Biography”. In: ed. by C.C. Gillispie. Vol. VI. New York: Charles Scribner’s Sons, 1972. Chap. Helmholtz, Hermann von, pp. 241–253. Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Com- mon Practice. Oxford University Press, 2001. Varèse,Edgard and Alcopley. “Edgard Varèseon Music and Art: A Conversation between Varèseand Alcopley”. In: Leonardo 1.2 (Apr. 1968), pp. 187–195. url: http://www. jstor.org/stable/1571960. Vercoe, Barry and Dan Ellis. “Real-Time CSOUND: Software Synthesis with Sensing and Control”. In: Proceedings of the International Computer Music Conference. Glasgow: In- ternational Computer Music Association, 1990, pp. 209–211. Vercoe, Barry et al. Csound. (software) A sound design, audio synthesis, and signal processing system. 1990–2013. url: http://www.csounds.com/. Vogel, Martin. On the relations of tone. Ed. by Carl A. Poldy. Trans. by Vincent J. Kissel- bach. Translation of: Lehre von den Tonbeziehungen (1975). Bonn-Bad Godesberg: Verlag fürSystematische Musikwissenschaft GmbH, 1993. Walshaw, Chris et al. The ABC Music project. (software) The text-based music notation system. 1991–2013. url: http://abc.sourceforge.net/. Wightman, Frederic L. and David M. Green. “The Perception of Pitch”. In: American Sci- entist 62.2 (1974), pp. 208–215. Wilson, Ervin. Moments of Symmetry. Handwritten letter to John Chalmers. 1975. url: http://www.anaphoria.com/wilson.html. Wilson, Jeremy et al. JazzStandards.com — the first and only centralized information source for the songs and instrumentals jazz musicians play most frequently. 2013. url: http: //www.jazzstandards.com/. Wolfram MathWorld™. url: http://mathworld.wolfram.com/.

141 WolframTones™. url: http://tones.wolfram.com/. Xenakis, Iannis. “Determinacy and indeterminacy”. In: Organised Sound 1.3 (Dec. 1996), pp. 143–155. Zattra, Laura, Ian George Burleigh, and Friedemann Sallis. “Studying Luigi Nono’s A Pierre. Dell’azzurro silenzio, inquietum (1985) as a Performance Event”. In: Contemporary Music Review 30.6 (2011), pp. 449–470.

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