Gene Ward Smith

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Gene Ward Smith (born 1947) is an American mathematician and music theorist. In mathematics he has worked in the areas of Galois theory and Moonshine theory. In music theory, he is noted for a number of innovations in the theory of musical tuning, such as the introduction of multilinear algebra and for being the first to write music in a number of exotic intonation systems.[1] A boyhood friend of Steven Spielberg, a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.[2]


In mathematics, Smith's most notable achievement is the construction of what has been called the Smith generic cyclic polynomial.[3] For any integer n not divisible by eight, this constructs a polynomial which, upon specializing the values, gives all of the cyclic extensions of any given base field with characteristic prime to n. This can then be extended to metacyclic extensions, such as dihedral groups.

Smith was also a member of the Amdahl Six group, which held the record for the largest known prime number from 1989-1992.[4]

Music theory[edit]

Smith introduced wedge products as a way of classifying regular temperaments, and of dealing with the problem of torsion.[5][6][7][8] In this system, a temperament is specified by means of a wedgie, which technically may be identified as a point on a Grassmannian.[9]

Smith has long been drawing attention to the relationship between equal divisions of the octave and the Riemann zeta function.[10][11][12]

Smith was among the first to consider extending the Tonnetz of Leonhard Euler 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.[13]

See also[edit]


  1. ^ Erlich, Paul (2006). "A Middle Path. Between Just Intonation and the Equal Temperaments, part 1" (PDF). Xenharmonikôn. Frog Peak Music. 18: 159–199. 
  2. ^ McBride, Joseph (1999). Steven Spielberg: A Biography. Da Capo Press. ISBN 0-306-80900-1. 
  3. ^ Jensen, Christian U.; Ledet, Arne; Yui, Noriko (2002). Generic Polynomials: Constructive Aspects of the Inverse Galois Problem (PDF). Cambridge: Cambridge University Press. ISBN 0-521-81998-9. 
  4. ^ The Amdahl Six
  5. ^ Smith, Gene (2001-11-26). "Wedge products and the torsion mess". tuning-math (Mailing list). 
  6. ^ Smith, Gene (2001-11-27). "Unlocking the mysteries of the wedge invariant". tuning-math (Mailing list). 
  7. ^ 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. 
  8. ^ Milne, Andrew; Sethares, W.A.; Plamondon, J. (March 2008). "Tuning Continua and Keyboard Layouts". Journal of Mathematics and Music. 2 (1): 1–19. doi:10.1080/17459730701828677. 
  9. ^ Breed, Graham (2008-03-14). "Prime Based Error and Complexity Measures" (PDF). pp. 19–20. 
  10. ^ Rusin, Dave. "Why 12 tones per octave?". Archived from the original on October 21, 2007. 
  11. ^ Sequence OEISA117536 Increasingly large peaks of the Riemann zeta function on the critical line, On-Line Encyclopedia of Integer Sequences.
  12. ^ Sequence OEISA117538 Increasingly large integrals of the Z function between zeros, On-Line Encyclopedia of Integer Sequences
  13. ^ "The Seven Limit Symmetrical Lattices",

External links[edit]