Distributional Analysis of n-Dimensional Feature Space for 7-Note Scales in 22-TET

  • Gareth M. HearneEmail author
  • Andrew J. Milne
  • Roger T. Dean
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11502)


Many scale features have been defined in an effort to account for the ubiquity of the diatonic scale in tonal music. In 12-TET, their relative influences have been difficult to disentangle. In 22-TET however, the features are spread differently across different scales. We sought here to to establish a set of 7-note scales in 22-TET that represent the major clusters within the whole population of scales. We first calculate numerous features of every 7-note scale in 22-TET that may relate to their perception in harmonic tonality. This feature space is then reduced by the step-by-step removal of features which may be most completely expressed as linear combinations of the others. A k-medoids cluster analysis leads finally to the selection of 11 exemplar scales, including approximations of four different tunings of the diatonic scale in just intonation.


Diatonic scale features K-medoids cluster analysis 22-TET 


  1. 1.
    Amiot, E.: David Lewin and maximally even sets. J. Math. Music 1(3), 157–172 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amiot, E.: Discrete Fourier transform and Bach’s good temperament. Music Theory Online 15(2), (2009). Accessed 10 Jan 2019
  3. 3.
    Balzano, G.J.: The pitch set as a level of description for studying musical pitch perception. In: Clynes, M. (ed.) Music, Mind, and Brain, pp. 321–351. Springer, Boston (1982). Scholar
  4. 4.
    Carey, N.: On coherence and sameness, and the evaluation of scale candidacy claims. J. Music Theory 46(1/2), 1–56 (2002)CrossRefGoogle Scholar
  5. 5.
    Carey, N.: Coherence and sameness in well-formed and pairwise well-formed scales. J. Math. Music 1(2), 79–98 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Carey, N., Clampitt, D.: Aspects of well-formed scales. Music Theory Spectr. 11(2), 187–206 (1989)CrossRefGoogle Scholar
  7. 7.
    Clampitt, D.: Mathematical and musical properties of pairwise well-formed scales. In: Klouche, T., Noll, T. (eds.) MCM 2007. CCIS, vol. 37, pp. 464–468. Springer, Heidelberg (2009). Scholar
  8. 8.
    Clampitt, D.L.: Pairwise Well-formed Scales: Structural and Transformational Properties (1998)Google Scholar
  9. 9.
    Clough, J., Douthett, J.: Maximally even sets. J. Music Theory 35(1/2), 93–173 (1991)CrossRefGoogle Scholar
  10. 10.
    Clough, J., Douthett, J., Ramanathan, N., Rowell, L.: Early indian heptatonic scales and recent diatonic theory. Music Theory Spectr. 15(1), 36–58 (1993)CrossRefGoogle Scholar
  11. 11.
    Clough, J., Engebretsen, N., Kochavi, J.: Scales, sets, and interval cycles: a taxonomy. Music Theory Spectr. 21(1), 74–104 (1999)CrossRefGoogle Scholar
  12. 12.
    Clough, J., Myerson, G.: Variety and multiplicity in diatonic systems. J. Music Theory 29(2), 249–270 (1985)CrossRefGoogle Scholar
  13. 13.
  14. 14.
    Daniélou, A.: Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. Inner Traditions/Bear, Rochester (1995)Google Scholar
  15. 15.
    Erlich, P.: Tuning, tonality, and twenty-two-tone temperament. Xenharmonikon 17, 12–40 (1998)Google Scholar
  16. 16.
    Erlich, P.: Private communication (2017)Google Scholar
  17. 17.
    Milne, A.J., Dean, R.T.: Computational creation and morphing of multilevel rhythms by control of evenness. Comput. Music J. 40(1), 35–53 (2016)CrossRefGoogle Scholar
  18. 18.
    Rothenberg, D.: A mathematical model for perception applied to the perception of pitch. In: Storer, T., Winter, D. (eds.) Formal Aspects of Cognitive Processes. LNCS, vol. 22, pp. 126–141. Springer, Heidelberg (1975). Scholar
  19. 19.
    Rothenberg, D.: A model for pattern perception with musical applications part II: the information content of pitch structures. Math. Syst. Theory 11(1), 353–372 (1977)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rousseeuw, P.J.: Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J. comput. Appl. Math. 20, 53–65 (1987)CrossRefGoogle Scholar
  21. 21.
    Wilson, E.: Letter to John Chalmers petertaining to Moments of Symmetry/Tanabe Cycle (1975).
  22. 22.
    Wilson, E.: On the development of intonational systems by extended linear mapping. Xenharmonikon 3 (1975). Accessed 10 Jan 2019

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Gareth M. Hearne
    • 1
    Email author
  • Andrew J. Milne
    • 1
  • Roger T. Dean
    • 1
  1. 1.The MARCS Institute for Brain, Behaviour and DevelopmentWestern Sydney UniversityPenrithAustralia

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