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Journal of Mathematics and Music
Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance
Volume 5, 2011 - Issue 3
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Original Articles

Algorithmic and computational approaches to pure-tone approximations of equal-tempered musical scales

Richard Krantz Metropolitan State College of Denver, Denver, CO, USACorrespondence[email protected]
&
Jack Douthett University of New Mexico, Albuquerque, NM, USA
Pages 171-194 | Received 29 Mar 2010, Accepted 15 Nov 2011, Published online: 14 Dec 2011

References

  • Douthett , J. and Krantz , R. 2007 . Continued fractions, least measurements, and musical scales and intervals . J. Math. Music , 1 : 47 – 70 .
  • Polansky , L. , Rockmore , D. , Johnson , K. , Repetto , D. and Pan , W. 2009 . A mathematical model for optimal tuning systems . Perspect. New Music , 47 : 69 – 110 .
  • Ya Khinchin , A. “ Continued Fractions ” . Chicago : The University of Chicago Press . 1964 and 1992.
  • Douthett , J. , Entringer , R. and Mullhaupt , A. 1992 . Musical scale construction: A continued fraction compromise . Utilitas Mathematica , 42 : 97 – 113 .
  • Bohlen , H. 1978 . 13 Tonstufen in der Duodezime . Acustica , 39 : 76 – 86 . (English translation: H. Bohlen, 13 Tone Steps in the Twelfth, acta acustica, 87 (2001), pp. 617–624).
  • Mathews , M. , Pierce , J. , Reeves , A. and Roberts , L. 1988 . Theoretical and experimental explorations of the Bohlen–Pierce scale . J. Acoust. Soc. Am , 84 : 1214 – 1222 .
  • Krantz , R. and Douthett , J. 1994 . A measurement of the reasonableness of equal-tempered musical scales . J. Acoust. Soc. Am , 95 : 3642 – 3650 .
  • Carey , N. and Clampitt , D. 1989 . Aspects of well-formed scales . Music Theor. Spectrum , 11 : 187 – 206 .
  • Clough , J. and Douthett , J. 1991 . Maximally even sets . J. Music Theory , 36 : 93 – 173 .
  • Carey , N. and Clampitt , D. 1996 . Self similar pitch structure, there duals and rhythmic analogues . Perspect. New Music , 34 : 62 – 87 .
  • Sos , V. T. 1958 . On the distribution mod 1 of the sequence nα . Ann. Univ. Sci. Butapest Eotvos Sect. Math , 1 : 127 – 134 .
  • Swierczkowski , S. 1958 . On successive settings of an Arc on the circumference of a circle . Fund. Math , 36 : 187 – 189 .
  • Halton , J. H. 1965 . The distribution of the sequence (n=0, 1, 2, …) . Prec. Camb. Phil. Soc , 61 : 665 – 670 .
  • Slater , N. B. 1967 . Gaps and steps for the sequence nθ mod 1 . Proc. Camb. Phil. Soc , 63 : 1115 – 1123 .
  • Barbour , J. M. 1951 . “ Tuning and Temperament ” . East Lansing Mich : Michigan State Press .
  • Randell , D. M. 1986 . “ The New Harvard Dictionary ” . Edited by: Randell , D. M. Cambridge, MA : Belkamp .

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