Gene Ward Smith

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Gene Ward Smith (1947–2021) was an American mathematician, music theorist, and composer.

In mathematics, he worked in the areas of Galois theory and Moonshine theory.

In music theory, he introduced wedge products as a way of classifying regular temperaments. In this system, a temperament is specified by means of a wedgie, which may technically be identified as a point on a Grassmannian. He had long drawn attention to the relationship between equal divisions of the octave and the Riemann zeta function.[1][2][3] He early on identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the regular mapping paradigm.

In the 1970s, Gene experimented with musical compositions using a device with four square-wave voices, whose tuning was very stable and accurate, being controlled by a crystal oscillator. The device in turn was controlled by HP 9800 series desktop computers, initially the HP 9830A, programmed in HP Basic, later the 9845A. Using this, he explored both just intonation with a particular emphasis on groups of transformations, and pajara.

Gene had a basic understanding of the regular mapping paradigm during this period, but it was limited in practice since he was focused on the idea that the next step from meantone should keep some familiar features, and so was interested in tempering out 64/63 in place of 81/80. He knew 7-limit 12 and 22 had tempering out 64/63 and 50/49 in common, and 12 and 27 had tempering out 64/63 and 126/125 in common, and thought these would be logical places to progress to, blending novelty with familiarity. While he never got around to working with augene, he did consider it. For pajara, he found tempering certain JI scales, the 10 and 12 note highschool scales, led to interesting (omnitetrachordal) results, and that there were also closely related symmetric (MOS) scales of size 10 and 12 for pajara; he did some work with these, particularly favoring the pentachordal decatonic scale.

Gene was among the first to consider extending the Tonnetz of Hugo Riemann beyond the 5-limit and hence into higher dimensional lattices. In three dimensions, the hexagonal lattice of 5-limit harmony extends to a lattice of type A3 ~ D3. He is also the first to write music in a number of exotic intonation systems.

Historical interest

See also


  1. Rusin, Dave. "Why 12 tones per octave?"
  2. OEIS. Increasingly large peaks of the Riemann zeta function on the critical line: OEIS: A117536.
  3. OEIS. Increasingly large integrals of the Z function between zeros: OEIS: A117538.