Tonal Scales and Minimal Simple Pitch Class Cycles

  • David Meredith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6726)


Numerous studies have explored the special mathematical properties of the diatonic set. However, much less attention has been paid to the sets associated with the other scales that play an important rôle in Western tonal music, such as the harmonic minor scale and ascending melodic minor scale. This paper focuses on the special properties of the class, \(\mathcal{T}\), of sets associated with the major and minor scales (including the harmonic major scale). It is observed that \(\mathcal{T}\) is the set of pitch class sets associated with the shortest simple pitch class cycles in which every interval between consecutive pitch classes is either a major or a minor third, and at least one of each type of third appears in the cycle. Employing Rothenberg’s definition of stability and propriety, \(\mathcal{T}\) is also the union of the three most stable inversional equivalence classes of proper 7-note sets. Following Clough and Douthett’s concept of maximal evenness, a method of measuring the evenness of a set is proposed and it is shown that \(\mathcal{T}\) is also the union of the three most even 7-note inversional equivalence classes.


Diatonic set Pitch class sets Pitch class cycles Scales Minor scales 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Piston, W.: Harmony. Gollancz, London (1991)Google Scholar
  2. 2.
    Clough, J., Engebretsen, N., Kochavi, J.: Scales, Sets and Interval cycles: A Taxonomy. Music Theory Spectrum 21(1), 74–104 (1999)CrossRefGoogle Scholar
  3. 3.
    Cross, I., West, R., Howell, P.: Cognitive Correlates of Tonality. In: Howell, P., West, R., Cross, I. (eds.) Representing Musical Structure, pp. 201–243. Academic Press, London (1991)Google Scholar
  4. 4.
    Rahn, J.: Coordination of Interval Sizes in Seven-Tone Collections. Journal of Music Theory 35(1/2), 33–60 (1991)CrossRefGoogle Scholar
  5. 5.
    Hauptmann, M.: Die Natur der Harmonik und der Metrik zur Theorie der Musik. Breitkopf und Härtel, Leipzig (1853)Google Scholar
  6. 6.
    McCune, M.: Moritz Hauptmann: Ein haupt Mann in Nineteenth Century Music Theory. Indiana Theory Review 7(2), 1–28 (1986)Google Scholar
  7. 7.
    Schenker, H.: Harmony. University of Chicago Press, Chicago (1954)Google Scholar
  8. 8.
    Clough, J., Douthett, J.: Maximally Even Sets. Journal of Music Theory 35(1/2), 93–173 (1991)CrossRefGoogle Scholar
  9. 9.
    Clough, J., Myerson, G.: Variety and Multiplicity in Diatonic Systems. Journal of Music Theory 29(2), 249–270 (1985)CrossRefGoogle Scholar
  10. 10.
    Rothenberg, D.: A Model for Pattern Perception with Musical Applications. Part I: Pitch Structures as Order-Preserving Maps. Mathematical Systems Theory 11, 199–234 (1978a)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rothenberg, D.: A Model for Pattern Perception with Musical Applications. Part II: The Information Content of Pitch Structures. Mathematical Systems Theory 11, 353–372 (1978b)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Balzano, G.J.: The Group-Theoretic Description of 12-Fold and Microtonal Pitch Systems. Computer Music Journal 4(4), 66–84 (1980)CrossRefGoogle Scholar
  13. 13.
    Carey, N.: On Coherence and Sameness, and the Evaluation of Scale Candidacy Claims. Journal of Music Theory 46(1/2), 1–56 (2002)CrossRefGoogle Scholar
  14. 14.
    Block, S., Douthett, J.: Vector Products and Intervallic Weighting. Journal of Music Theory 38(1), 21–41 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • David Meredith
    • 1
  1. 1.Aalborg UniversityDenmark

Personalised recommendations