Keynote Speakers

KEYNOTE 01 | June 18th

35 Years of Well-formed Scale Theory: Origins and Development

We present an anecdotal history of the studies that led to our first publication, the 1989 article “Aspects of Well-formed Scales” in Music Theory Spectrum, and give a brief introduction to the concepts. We explore the mathematics behind the concepts: continued fractions, distributions modulo 1, and symmetries of cyclic groups. Our precursor studies included the region notion, suppressed in the initial article, but developed in the context of the history of music theory in our 1996 Journal of Music Theory article, “Regions: A Theory of Tonal Spaces in Early Medieval Treatises.” The region concept was subsequently reinterpreted relative to central words in algebraic combinatorics on words, and was the point of departure for Carey’s 2013 “Lambda Words” in Journal of Integer Sequences. We present the connection between our work and that of our principal precursor, John Clough, who with his co-author Gerald Myerson introduced Myhill’s Property (MP). In our 1996 Perspectives of New Music article, “Self-similar Pitch Structures, Their Duals, and Rhythmic Analogues,” we generalized MP and proved that it is equivalent to non-degenerate well-formedness. This article set the stage for the word-theoretical turn in well-formed scale theory, with an abstract scale (determined only by its cardinality N and a scale-step interval multiplicity g) represented by a circular word over a two-letter alphabet, and with examples of infinite words, later determined to be Sturmian words. It was another decade before the connections with algebraic combinatorics on words came to light, and were explored principally by Thomas Noll and Clampitt. We cover briefly the equivalence between Christoffel words and their conjugates on the one hand, and well-formed scales and their modes on the other. Carey and Clampitt used the word theory perspective to re-prove Clough and Myerson’s 1985 Cardinality Equals Variety theorem in a different mathematical context, in a special issue of Journal of Mathematics and Music, and in Theoretical Computer Science, both in 2018.

(ongoing work with Wayne Vitale)

Gong kebyar is a widely performed style of bronze gamelan in Bali, Indonesia, that features a rich variety of tunings. Andrew Toth (1948-2007), a passionate researcher and enthusiast of Balinese music, collected tuning measurements of about 150 keys for each of 49 gamelan during his decades of fieldwork. His data, now preserved in the Wesleyan University Library, offers a valuable resource for studying the tuning practices and musical aesthetics of Balinese gamelan. In this talk, we present a novel way of accessing and analyzing Toth’s data using an interactive computer program, a Max/MSP Patch called the "Gamelan Tuning Explorer." This program allows users to listen to and manipulate the tuning data using sound, visualization, and adjustable parameters. We demonstrate how the Gamelan Tuning Explorer can reveal some of the key features of gamelan tuning, such as the ombak (the distinctive beating effect of two paired notes played in "unison""), the octave treatment (the degree of alignment or divergence between octaves), and the octave stretching or compression (the deviation from the equal-tempered octave). We hope that this program will enhance the understanding and appreciation of the diversity and complexity of Balinese gamelan tuning among scholars, musicians, and listeners.