William Sethares

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William A. Sethares
Born (1955-04-19) April 19, 1955 (age 66)
Alma materCornell University
Known forConsonance
Scientific career
FieldsSignal processing and music theory
InstitutionsUniversity of Wisconsin–Madison

William A. Sethares (born April 19, 1955) is an American music theorist and professor of electrical engineering at the University of Wisconsin. In music, he has contributed to the theory of Dynamic Tonality and provided a formalization of consonance.

Consonance and dissonance[edit]

Among the earliest musical traditions, musical consonance was thought to arise in a quasi-mystical manner from ratios of small whole numbers. (For instance, Pythagoras made observations relating to this, and the ancient Chinese Guqin contains a dotted scale representing the harmonic series.) The source of these ratios, in the pattern of vibrations known as the harmonic series, was exposed by Joseph Sauveur the early 18th century and even more clearly by Helmholtz in the 1860s.

In 1965, Plomp and Levelt[1] showed that this relationship could be generalized beyond the harmonic series, although they did not elaborate in detail.

In the 1990s, Sethares began exploring Plomp and Levelt's generalization, both mathematically and musically. His 1993 paper On the relationship between timbre and scale[2] formalized the relationships between a tuning's notes and a timbre's partials that control sensory consonance. A more accessible version also appeared in Experimental Musical Instruments as "Relating Tuning and Timbre"[3] These papers were followed by two CDs, Xenotonality and Exomusicology (some songs from which can be freely downloaded here), which explored the application of these ideas to musical composition.

In his 1998 book Tuning, Timbre, Spectrum, Scale,[4] Sethares developed these ideas further, using them to expose the intimate relationship between the tunings and timbres of Indonesian and Thai indigenous music, and to explore other novel combinations of related tunings and timbres. Where microtonal music was previously either dissonant (due to being played with harmonic timbres to which it was not "related"), or restricted to the narrow range of harmonically related tunings (to retain sensory consonance), Sethares's mathematical and musical work showed how musicians might explore microtonality without sacrificing sensory consonance.

As one reviewer of the second edition[5] of this book wrote, "Physics had built a prison round music, and Sethares set it free."[6] Another reviewer wrote that it "is not only the most important book about tuning written to date, but it is the most important book about music theory written in human history."[7]

Dynamic tonality[edit]

In 2003, Sethares began an informal collaboration with Andrew Milne and Jim Plamondon, whimsically called the Isomorphic Conspiracy. Its aim was to explore the musical relationships exposed by isomorphic keyboards, which are an unusual design of two-dimensional keyboards that have the intriguing characteristic of having transpositional invariance[8]—that is, "the same fingering in every key."

By early 2006, the Isomorphic Conspiracy had discovered that such keyboards also had the same fingering in every tuning, too—or, at least, every tuning of what the Conspiracy came to call the syntonic temperament.[9] This consistency of fingering across tunings, which they called tuning invariance, enables a performer, using an isomorphic keyboard and compatible synthesizer (or software synthesizer), to play a given tonal piece in any of a wide range of tunings. The Conspiracy turned its attention to identifying the means by which different isomorphic keyboards could be compared,[10] and identified the Wicki/Hayden keyboard as being optimal for the syntonic temperament's wide tuning range.

In addition to being able to play in any fixed tuning with consistent fingering, the tuning invariance of isomorphic keyboards enables performers to change a piece's tuning on the fly, along the syntonic temperament's smooth tuning continuum, while retaining consonance (or, optionally, while introducing dissonance to any tonal structure). It soon became clear that this dynamic tonality offered an entirely novel means of controlling tension and release.[11]

To put these ideas into practice, Sethares (with collaborators and students) developed a freely available software-based synthesizer, the TransFormSynth, which enables a performer to bend tunings polyphonically during performance. In April 2008, Sethares used the TransFormSynth to compose and record the first musical piece that used dynamic tonality, which he called "C to Shining C" (although the piece, as recorded, is not actually in any musical key). A single chord is played throughout the piece, yet it gains a feeling of tension and release through its tuning progression from 19-tone equal temperament tuning to 5-tone equal temperament tuning and back, complemented by a slower timbre progression from a fully harmonic timbre to a fully tuning-aligned timbre.

In 2009, Sethares led the Isomorphic Conspiracy's extension of Dynamic tonality to include a wider variety of tunings, including[12]

All of these different tunings can be controlled with identical fingering and full consonance using a dynamic-tonality-capable synthesizer such as the TransFormSynth and an isomorphic keyboard. (The TransFormSynth maps the Wicki/Hayden note-layout to the standard QWERTY keyboard, so that no special keyboard is necessary.)

Musica Facta[edit]

Sethares' conception of consonance is one of the foundation-stones of a new research program called Musica Facta.[13]

See also[edit]


  1. ^ R. Plomp and W. J. M. Levelt (October 1965). "Tonal Consonance and Critical Bandwidth". Journal of the Acoustical Society of America. 38 (4): 548–560. doi:10.1121/1.1909741. hdl:2066/15403. PMID 5831012.
  2. ^ Sethares, William (September 1993). "Local consonance and the relationship between timbre and scale". Journal of the Acoustical Society of America. 94 (3): 1218–1228. doi:10.1121/1.408175.
  3. ^ Sethares, William (September 1992). "Relating Tuning and Timbre". Experimental Musical Instruments. IX (2).
  4. ^ Sethares, William (January 1998). Tuning, Timbre, Spectrum, Scale (1st ed.). New York: Springer. ISBN 978-3-540-76173-0.
  5. ^ Sethares, William (November 2004). Tuning, Timbre, Spectrum, Scale (2nd ed.). New York: Springer. ISBN 978-1-85233-797-1.
  6. ^ Luca Turin (September 2004). "The sound of impossible objects". NZZ Folio.
  7. ^ Scott, X. J. "nonoctave.com / tuning / book reviews". Retrieved 2009-09-20.
  8. ^ Keislar, D. (April 1988). History and Principles of Microtonal Keyboard Design (PDF). Center for Computer Research in Music and Acoustics, Stanford University. Report No. STAN-M-45.
  9. ^ Milne, Andrew; Sethares, W.A.; Plamondon, J. (December 2007). "Invariant Fingerings Across a Tuning Continuum". Computer Music Journal. 31 (4): 15–32. doi:10.1162/comj.2007.31.4.15. S2CID 27906745.
  10. ^ Milne, Andrew; Sethares, W.A.; Plamondon, J. (March 2008). "Tuning Continua and Keyboard Layouts". Journal of Mathematics and Music. 2 (1): 1–19. CiteSeerX doi:10.1080/17459730701828677. S2CID 1549755.
  11. ^ Plamondon, Jim; Milne, A.; Sethares, W.A. (2009). "Dynamic Tonality: Extending the Framework of Tonality into the 21st Century" (PDF). Proceedings of the Annual Conference of the South Central Chapter of the College Music Society.
  12. ^ Sethares, William; Milne, A.; Tiedje, S.; Prechtl, A.; Plamondon, J. (2009). "Spectral Tools for Dynamic Tonality and Audio Morphing". Computer Music Journal. 33 (2): 71–84. CiteSeerX doi:10.1162/comj.2009.33.2.71. S2CID 216636537. Retrieved 2009-09-20.
  13. ^ Musica Facta: http://musicafacta.org

Further reading[edit]

External resources[edit]