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
Top of page Top of page