Noneuclidean Harmony

  • Scott Kim


One of the outstanding achievements of the ancient Greeks was the formation of a deductive system of tonal harmony constructed on the principal sounds of everyday experience. This system was set forth in the 13 books of Euclid’s Elements of Harmony, otherwise known as the “Harmonic Series”. Starting from just 5 axioms:
  1. 1

    Given two notes, there is an interval that joins them. (Axiom of Arpeggiation)

  2. 2

    An interval can be prolonged indefinitely. (Axiom of the Fermata)

  3. 3

    A progression can be constructed when its key center and one of its chords are given. (Axiom of the Sequence)

  4. 4

    All tritones are equal. (The Devil’s Axiom)

  5. 5

    If a melodic line accompanying two melodic lines a fifth apart makes the interior intervals on the same side less than two tritones, the two melodic lines, if extended indefinitely, resolve on that side where the intervals are less than the two tritones. (The Fifth postulate)



Deductive System Harmonic Series Music Theoretician Outstanding Achievement Tuning System 
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Copyright information

© Wadsworth International 1981

Authors and Affiliations

  • Scott Kim
    • 1
  1. 1.Stanford UniversityUSA

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