Tuning Timbre Spectrum Scale
Table of Contents

by William A. Sethares

Here is the complete table of contents for Tuning Timbre Spectrum Scale. Click for a quick rundown of the central ideas, here for a brief overview or here to read the first chapter. The book also includes a set of sound examples on CD.

The Octave is Dead... Long Live the Octave

Introducing a dissonant octave--almost any interval can be made consonant or dissonant by proper choice of timbre.

  1. A Challenge
  2. A Dissonance Meter
  3. New Perspectives
  4. Nonwestern Musics
  5. New Scales
  6. New Sounds
  7. New `Music Theories'
The Science of Sound

"Sound" as a physical phenomenon and "sound" as a perceptual phenomena are not the same thing. Definitions and results from acoustics are compared and contrasted to the appropriate definitions and results from perception research and psychology. Auditory perceptions such as loudness, pitch, and timbre can often be correlated with physically measurable properties of the sound wave. Some basic ideas from psychoacoustics are required to understand the interaction between timbre (or spectrum) and sensory consonance and dissonance.

  1. What is Sound?
  2. Spectrum and Timbre
  3. Prisms, Fourier Transforms, and Ears
  4. Analytic vs. Holistic Listening: Tonal Fusion
  5. What is Timbre?
  6. Multidimensional Scaling
  7. Analogies with Vowels
  8. Spectrum and the Synthesizer
  9. What is Timbre?
  10. Frequency and Pitch
Sound on Sound
All is clear when dealing with a single sine wave of reasonable amplitude and duration. The measured amplitude is correlated with the perceived loudness, the measured frequency is correlated with the perceived pitch, and the phase is essentially undetectable by the ear. Little is clear when dealing with large clusters of sine waves such as those that give rise to ambiguous virtual pitches. This chapter explores the in-between case where two sinusoids interact to produce interference, beating, and roughness. This is the simplest setting in which sensory dissonance occurs.
  1. Pairs of Sine Waves
  2. Interference
  3. Beats
  4. Critical Band and JND
  5. Sensory Dissonance
  6. Counting Beats
  7. Ear vs. Brain
Musical Scales

People have been organizing, codifying, and systematizing musical scales with numerological zeal since antiquity. Scales have proliferated like tribbles in quadra-triticale: just intonations, equal temperaments, scales based on overtones, scales generated from a single interval or pair of intervals, scales without octaves, scales arising from arcane mathematical formulas, scales that reflect cosmological or religious structures, scales that "come from the heart." Each musical culture has its own preferred scales, and many have used different scales at different times in their history. This chapter reviews a few of the more common organizing principles, and then discusses the question "what makes a good scale?"

  1. Why Use Scales?
  2. Pythagoras and the Spiral of Fifths
  3. Equal Temperaments
  4. Just Intonations
  5. Partch
  6. Meantone and Well Temperaments
  7. Overtone Scales
  8. Real Tunings
  9. Gamelan Tunings
  10. My Tuning Is Better Than Yours
  11. A Better Scale?
Consonance and Dissonance of Harmonic Sounds
Just as a tree may crash silently (or noisily) to the ground, depending on the definition of sound, the terms "consonance" and "dissonance" have both a perceptual and a physical aspect. There is also a dichotomy between attitude and practice, between the way theorists talk about consonance and dissonance and the ways that performers and composers use consonances and dissonances in musical situations. This chapter explores five different historical notions of consonance and dissonance in an attempt to avoid confusion and to place sensory consonance in its historical perspective. Several different explanations for consonance are reviewed, and curves drawn by Helmholtz, Partch, and Plomp for harmonic timbres are explored.
  1. A Brief History
  2. Melodic Consonance
  3. Polyphonic Consonance
  4. Contrapuntal Consonance
  5. Functional Consonance
  6. Sensory Consonance
  7. Explanations of Consonance and Dissonance
  8. Small Is Beautiful
  9. Fusion
  10. Difference Tones
  11. Cultural Conditioning
  12. Which Consonance Explanation?
  13. Harmonic Dissonance Curves
  14. Helmholtz and Sensory Dissonance
  15. Partch's One-Footed Bride
  16. Erlich's Harmonic Entropy
  17. Sensory Consonance and Critical Bandwidth
  18. A Simple Experiment
Related Spectra and Scale

Sensory dissonance is a function of the interval and the spectrum of a sound. A scale and a spectrum are related if the dissonance curve for the spectrum has minima (points of maximum sensory consonance) at the scale steps. This chapter shows how to calculate dissonance curves, and gives examples that verify the perceptual validity of the calculations, and others that demonstrate their limits. The idea of related spectra and scales unifies and gives insight into a number of previous musical and psychoacoustic investigations, and some general properties of dissonance curves are derived. Finally, the idea of the dissonance curve is extended to multiple sounds, each with its own spectrum. Many of these ideas were first presented in a paper in the Journal of the Acoustical Society of America, and later in the less technical Relating Tuning and Timbre, the full text of which is available online. Computer programs for drawing dissonance curves are also available.

  1. Dissonance Curves and Spectrum
  2. From Spectrum to Tuning
  3. From Tuning to Spectrum
  4. Realization and Performance
  5. Drawing Dissonance Curves
  6. A Consonant Tritone
  7. Past Explorations
  8. Pierce's Octotonic Spectrum
  9. Stretching Out
  10. Is Stretched Music Viable?
  11. Plastic City: A Stretched Journey
  12. The Bohlen-Pierce Scale
  13. Found Sounds
  14. Carlos' Graphical Method
  15. A Tuning for Ideal Bars
  16. Tunings for Bells
  17. Tuning for FM Spectra
  18. Properties of Dissonance Curves
  19. Dissonance Curves for Multiple Spectra
  20. Dissonance "Surfaces"
A Bell, A Rock, A Crystal
To bring the relationship between tuning and spectrum into sharper focus, this chapter looks at three examples in detail: an ornamental hand bell, a resonant rock from Chaco Canyon, and an "abstract" sound created from a morphine crystal. All three are discussed at length, and each step is detailed so as to highlight the practical issues, techniques, and trade-offs that arise when applying the ideas to real sounds making real music. The bell, rock and crystal were used as the basis for three compositions: Tingshaw, The Chaco Canyon Rock, and Duet for Morphine and Cymbal, which appear on the accompanying CD.
  1. Tingshaw: A Simple Bell
  2. Chaco Canyon Rock
  3. Sounds of Crystals
  4. The Sound of Data
Adaptive Tunings

Throughout the centuries, composers and theorists have wished for musical scales that are faithful to the consonant simple integer ratios (like the octave and fifth) but that can also be modulated to any key. Inevitably, with a fixed (finite) scale, some intervals in some keys must be compromised. But what if the notes of the "scale" are allowed to vary? This chapter presents a method of adjusting the pitches of notes dynamically, an adaptive tuning, that maintains fidelity to a desired set of intervals and can be modulated to any key. The adaptive tuning algorithm changes the pitches of notes in a musical performance so as to maximize sensory consonance. The algorithm can operate in real time, is responsive to the notes played, and can be readily tailored to the spectrum of the sound. This can be viewed as a generalized "dynamic" just intonation, but it can operate without specifically musical knowledge such as key and tonal center, and is applicable to timbres with nonharmonic spectra as well as the more common harmonic timbres. Much of this chapter first appeared in a technical paper in the Journal of the Acoustical Society of America.

  1. Fixed vs. Variable Scales
  2. The Hermode Tuning
  3. Spring Tuning
  4. Consonance-Based Adaptation
  5. Behavior of the Algorithm
  6. The Sound of Adaptive Tunings

A Wing, An Anomaly, A Recollection

The adaptive tuning of the last chapter adjusts the pitches of notes in a musical performance to minimize the sensory dissonance of the currently sounding notes. This chapter presents a real-time implementation called Adaptun (written in the Max programming language and available on the CD in the software folder) that can be readily tailored to the timbre (or spectrum) of the sound. Several tricks for sculpting the sound of the adaptive process are discussed. Wandering pitches can be tamed with an appropriate context, an (inaudible) collection of partials that are used in the calculation of dissonance within the algorithm, but that are not themselves adapted or sounded. The overall feel of the tuning is effected by whether the adaptation converges fully before sounding (or whether intermediate pitch bends are allowed). Whether adaptation occurs when currently sounding notes cease (or only when new notes enter) can also have an impact on the overall solidity of the piece. Several compositional techniques are explored in detail, and a collection of sound examples and musical compositions highlight both the advantages and weaknesses of the method. Much of this chapter first appeared in a paper in the Journal of New Music Research.

  1. Practical Adaptive Tunings
  2. A Real-Time Implementation in Max
  3. The Simplified Algorithm
  4. Context, Persistence, and Memory
  5. Examples
  6. Compositional Techniques and Adaptation
  7. Toward an Aesthetic of Adaptation
  8. Implementations and Variations

The Gamelan

The gamelan "orchestras" of Central Java in Indonesia are one of the great musical traditions. The gamelan consists of a large family of nonharmonic metallophones that are tuned to either the five note slendro or the seven tone pelog scales. Neither scale lies close to the familiar 12-tet. The nonharmonic spectra of certain instruments of the gamelan are related to the unusual intervals of the pelog and slendro scales in much the same way that the harmonic spectra of instruments in the Western tradition is related to the Western diatonic scale.

  1. A Living Tradition
  2. An Unwitting Ethnomusicologist
  3. The Instruments
  4. Tuning the Gamelan
  5. A Tale of Two Gamelans
  6. Spectrum and Tuning

Consonance-Based Musical Analysis

The measurement of (sensory) consonance and dissonance is applied to the analysis of music using dissonance scores. Comparisons with a traditional score-based analysis of a Scarlatti sonata show how the contour and variance of the dissonance score can be used to concretely describe the evolution of dissonance over time. Dissonance scores can also be applied in situations where no musical score exists, and two examples are given: a xenharmonic piece by Carlos, and a Balinese gamelan performance. Another application, to historical musicology, attempts to reconstruct probable tunings used by Scarlatti from an analysis of his extant work. The Scarlatti work was done in conjunction with John Sankey, and first appeared in a paper in the Journal of the Acoustical Society of America.

  1. A Dissonance "Score"
  2. Reconstruction of Historical Tunings
  3. Total Dissonance
  4. What's Wrong With This Picture?
From Tuning to Spectrum

The related scale for a given spectrum is found by drawing the dissonance curve and locating the minima. The complementary problem of finding a spectrum for a given scale is not as simple, since there is no single "best" spectrum for a given scale. But it is often possible to find "locally best" spectra which can be specified as the solution to a certain constrained optimization problem. For some kinds of scales, such as n-tet, properties of the dissonance curves can be exploited to directly solve the problem. A general "symbolic method" for constructing related spectra works well for scales built from a small number of successive intervals. Some of this material also appeared in a technical paper in the Journal of the Acoustical Society of America.

  1. Looking for Spectra
  2. Spectrum Selection as an Optimization Problem
  3. Spectra for Equal Temperaments
  4. Symbolic Computation of Spectra
  5. Spectra for Tetrachords
Spectral Mappings
A spectral mapping is a transformation from a `source' spectrum to a `destination' spectrum. One application is to transform nonharmonic sounds into harmonic equivalents. More interestingly, it can be used to create nonharmonic instruments that retain much of the tonal quality of familiar (harmonic) instruments. Musical uses of such timbres are discussed, and forms of (nonharmonic) modulation are presented. A more detailed presentation will soon appear in a paper in the Computer Music Journal.
  1. The Goal: Life-like Nonharmonic Sounds
  2. Mappings Between Spectra
  3. Examples
  4. Discussion
  5. Robustness of Sounds Under Spectral Mappings
  6. Timbral Change
  7. Related Perceptual Tests
  8. Increasing Consonance
  9. Consonance Based Modulations
A "Music Theory" for 10-tet

Dissonance curves provide a starting point for the exploration of nonharmonic sounds when played in unusual tunings by suggesting suitable intervals, chords, and scales. This chapter makes a first step towards a description of 10-tet, using dissonance curves to help define an appropriate "music theory." Most previous studies (such as those of Blackwood, Carlos, Douthett, Hall, Krantz, and Yunik) explore equal temperaments by comparing them to the just intervals or to the harmonic series. In contrast, this new music theory is based on properties of the 10-tet scale and related 10-tet spectra. Possibilities for modulations between 10-tet "keys" are evident, and simple progressions of chords are available. Together, these show that this xentonal 10-tet system is rich and varied. The theoretical ideas are demonstrated in two compositions, showing that the claimed consonances exist, and that the xentonal motions are perceptible to the ear.

  1. What is 10-tet?
  2. 10-tet Keyboard
  3. Spectra for 10-tet
  4. 10-tet Chords
  5. Neutral Chords
  6. Circle of Thirds
  7. "I-IV-V"
  8. The Tritone Chord
  9. 10-tet Scales
  10. A Progression
Classical Music of Thailand and 7-tet
Thai classical music is played on a variety of indigenous instruments (such as the xylophone-like renat and pong lang) in a scale containing seven equally spaced tones per octave. This chapter shows how the timbres of these instruments (in combination with a harmonic sound) are related to the 7-tet scale, and then explores a variety of interesting sounds and techniques useful in 7-tet.
  1. Introduction to Thai Classical Music
  2. Tuning of Thai Instruments
  3. Timbre of Thai Instruments
  4. Exploring 7-tet
Speculation, Correlation, Interpretation, Conclusion

Tuning Timbre Spectrum Scale began with a review of basic psychoacoustic principles and the related notion of sensory dissonance, introduced the dissonance curve, and then applied it across a range of disciplines. Most of the book stays fairly close to "the facts," without undue speculation. This final chapter ventures further.

  1. The Zen of Xentonality
  2. Coevolution of Tunings and Instruments
  3. To Boldly Listen
  4. New Musical Instruments?
  5. Silence Hath No Beats
  6. Coda
  1. Mathematics of Beats: Where beats come from
  2. Ratios Make Cents: Conversions from ratios to cents and back again
  3. Speaking of Spectra: How to use and interpret the FFT
  4. Additive Synthesis: Generating sound directly from the sine wave representation: a simple computer program
  5. How to Draw Dissonance Curves: Detailed derivation of the dissonance model, and computer programs to carry out the calculations
  6. Properties of Dissonance Curves: General properties help give an intuitive feel for dissonance curves
  7. Analysis of the Time Domain Model
  8. Behavior of Adaptive Tunings
  9. Harmonic Entropy
  10. Tables of Scales: Miscellaneous tunings and tables.
Sound Examples (including downloadable mp3's!)

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