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