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A “tertial element” is a primary interval (y, z) such that z is a generic “klang generator” e of a non-empty harmonic tone system (Sect. 8.1). A “proto-tonal system” is a harmonic tone system such that the quintic element (c, d) satisfies c is as close as possible to alog2(3/2); moreover, assuming a non-empty system, there exists a tertial element (y, z) such that y is as close as possible to alog2(5/4). It is proven that a proto-tonal system satisfies a = 12, b = 7 = c, d = 4, and (if the system is not empty) e = 2. Section 8.2 revisits the theory of categorical equal temperament from a historical and historical-theoretical perspective, now that a = 12 has been established. Finally, Sect. 8.3 considers an alternative to the theory of proto-tonality, based on Clough and Douthett’s (1991) theory of “maximally even sets.” It is shown that the alternative theory is weaker than the proposed theory in terms of explanatory power.

Original languageEnglish
Title of host publicationComputational Music Science
PublisherSpringer Nature
Number of pages17
StatePublished - 2013

Publication series

NameComputational Music Science
ISSN (Print)1868-0305
ISSN (Electronic)1868-0313

Bibliographical note

Publisher Copyright:
© 2013, Springer-Verlag Berlin Heidelberg.


  • Categorical Principle
  • Complex Tone
  • Harmonic System
  • Pitch Interval
  • Tone System


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