ABSTRACT
Medieval music theory was dominated by ancient Pythagorean cosmic numerology; the more down-to-earth practical approach of Aristoxenus took a back seat or was forgotten. Even today, many musicians, theorists and composers (e.g. microtonalists) are still theorizing with complex number ratios. But interval ratios only make psychological sense if the numbers correspond to audible harmonics in complex tones, so they should be limited to 10 or perhaps 16 (an interval of “64:81” is meaningless). Musical intervals are part of an aural cultural tradition; their sizes and meanings are determined by a complex psychohistorical process. The perceived intervals between the harmonics of a complex tone, and corresponding intervals in music performance, are variable in size (plus or minus some 5 to 50 cents!) and slightly stretched (for physical, psychological or musical reasons). There is no “perfect” tuning for any interval; even octaves tend to exceed 1:2. Differences in interval size that are smaller than a few cents are perceptually irrelevant. Every tuning is a compromise among competing psychoacoustic or musical criteria such as maximizing harmonicity and familiarity, minimizing roughness (by minimizing beating), allowing for interval stretch, anticipating voice leading, or manipulating tuning for expressive purposes. There is no such thing as “perfect tuning”; in practice, tuning is optimal when pitches approach the centres of perceptual pitch categories and expressive mistunings are deliberate and musically sensible. Most musical intervals have two theoretical ratios (Pythagorean and just), both of which lie within a continuous range of acceptable tunings. Different tuning systems in common use are not inherently better or worse than each other. The piano is not inherently out of tune; 12-tone equal temperament is successful not only because all keys are available and sound the same, but also because it compromises just and Pythagorean tunings, which incidentally makes it the most likely hypothetical tuning for pre-12ET vocal polyphony. In psychoacoustics, pitch is a one-dimensional experiential variable; the underlying neurophysiology involves an inextricable mix of temporal and spectral (tonotopic) processing. There is no clear evidence for a neural mechanism that is sensitive to musical frequency ratios. In summary, the apparent relationship between number ratios and musical intervals is indirect and mediated by music history, performance constraints and complex tone perception. I will consider the implications of these conclusions for the future of music theory and its interaction with music psychology.
(I thank composer Graham Hair, Professor Emeritus, Glasgow University and Visiting Professor, Manchester Metropolitan University for his feedback during the development of these ideas.)
ABOUT RICHARD PARNCUTT
Richard Parncutt is Professor of Systematic Musicology at the University of Graz, Austria. His publications address musical structure (pitch, consonance, harmony, tonality, tension, rhythm, meter, accent), music performance (psychology, piano, applications), the origins of tonality and of music, and musicological interdisciplinarity. He holds qualifications in music and physics from the University of Melbourne and a PhD from the University of New England, Australia. He is or was a board member of all leading music psychology journals, founding academic editor of the Journal of Interdisciplinary Music Studies (JIMS), and (co-) founder of three conference series: Conference on Interdisciplinary Musicology (CIM), Conference on Applied Interculturality Research (cAIR), and International Conference of Students of Systematic Musicology (SysMus).