TIMBRE AS A PSYCHOACOUSTIC PARAMETER FOR HARMONIC
ANALYSIS AND COMPOSITION
John MacCallum, Jeremy Hunt, and Aaron Einbond
Center for New Music and Audio Technology (CNMAT)
University of California Berkeley
{johnmac, jrmy, einbond} @berkeley.edu
ABSTRACT
Timbre can affect our subjective experience of musical dissonance and harmonic progression. To this end,
we have developed a set of algorithms to measure roughness (sensory dissonance), and pitch correlation between
sonorities, taking into account the effects of timbre and
microtonal inflection. We proceed from the work of
Richard Parncutt and Ernst Terhardt, extending their algorithms for the psychoacoustic analysis of harmony to include spectral data from actual instrumental sounds. This
allows for the study of a much wider variety of timbrallyrich acoustic or electronic sounds which was not possible
with the previous algorithms. Further, we generalize these
algorithms by working directly with frequency rather than
a tempered division of the octave, making them available
to the full range of microtonal harmonies. The new algorithms, by yielding different roughness estimates depending on the orchestration of a sonority, confirm our intuitive understanding that orchestration affects sensory dissonance. This package of tools presents rich possibilities
for composition and analysis of music that is timbrallydynamic and microtonally-complex.
1. INTRODUCTION
Beginning in the 1970's Ernst Terhardt proposed a psychoacoustic model of harmony [8, 9, 10]. Proceeding from
Rameau and Helmholtz, he described "musical
consonance" as the product of the co-operative perception
of "sensory consonance" (the absence of sensory dissonance, or roughness) and "harmony" (or harmonicity, the
similarity of a sound to a harmonic series) [9].
Musicologist Richard Parncutt has further extended and
developed Terhardt's theory [3, 4]. In particular, Parncutt
describes a measure of roughness of individual sonorities
and of pitch commonality between two sonorities. Although other harmonic theories have taken perceptual data
into account, a major advantage of Parncutt's algorithms is
that they avoid biases towards pre-existing musical styles
or techniques, with the exception that they are designed
for equally-termpered music. Therefore they are promising tools for the composition and analysis of new, perceptually coherent, post-tonal music.
However, the work of Terhardt and Parncutt accounts
for instrumental timbre in only a limited way. Composers
and analysts have become increasingly interested in timbre as a conveyor of musical meaning. It has long been
acknowledged that timbre and orchestration have effects
on our perception of dissonance and even harmonic relationships [7]. However, no harmonic theory has attempted
to quantify these differences. Currently a variety of tools,
such as Diphone and AudioSculpt, are available to create acoustical analyses in the versatile Sound Description
Interchange Format (SDIF) [11]. Using SDIF data, we
extend Terhardt's and Parncutt's measures to take timbre
into account.
2. ACCOUNTING FOR TIMBRE
2.1. Virtual Fundamental
According to Terhardt, our ability to match a sonority to
the harmonic series is one of the components of our perception of musical consonance [8]. In the terminology of
Parncutt, by matching the pure tones of a sonority to an
harmonic model, we may sense a complex tone, or virtual
fundamental. The higher the frequency of the virtual fundamental, and the better its harmonics match the sonority, the more harmonic the sonority. Terhardt's algorithm
[10] does not require adjustment to account for instrumental timbre or microtonal frequencies. But we propose
inputting to the algorithm not merely an idealized list of
pitches, but a list of timbrally-complex sounds each with
many pure-tone components. For example, we can take
several instrumental notes, each with its own spectrum,
and ask what is the virtual fundamental of the spectra together.
2.2. Sensory Dissonance
Parncutt built into his model a rudimentary framework
for including the effects of timbre, distinguishing between
only three types of tones: pure-tones, harmonic-complex
tones, and octave-spaced (Shepard) tones. The harmoniccomplex tones are meant to model a general instrument
timbre-they contain the first 10 partials of the harmonic
series (rounded to semitones) with a roll-off that varies
as the inverse of the partial number. Although these generalized timbres already give roughness data that correspond to our psychoacoustic experience better than pure
