The Harmonic Aspect

Part of the Computational Music Science book series (CMS)


Harmony is a magic word in music theory and practice. It is not only a technical term but also refers to more transcendental layers of existence. In ancient Greece, harmony was the ultimate ground of the universe. In Pythagorean philosophy, the ultimate formula of the universe was their tetractys, a triangular arrangement of ten points starting at the base with 4, then 3, then 2, then 1 on top. You have already seen in the previous chapter on intervals that the successive ratios of these triangular numbers—2/1, 3/2, 4/3—are the intervals of an octave, perfect fifth, and perfect fourth. So the tetractys was the universal law behind both music and the universe. This formula was an expression of harmony since these intervals were considered to be particularly agreeable, or “consonant,” as music theorist call them. In opposition to consonant intervals, they speak of “dissonant” intervals; for example, the minor second or the tritone are dissonant and sound less agreeable to many persons—but not all. We come back to these interval qualities in tutorial 11 about counterpoint.


Harmonic Function Music Theory Musical Object Traditional Music Magic Word 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Guerino Mazzola
    • 1
    • 2
  • Joomi Park
    • 3
  • Florian Thalmann
    • 1
  1. 1.School of MusicUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of InformaticsUniversity of ZurichZurichSwitzerland
  3. 3.McNally Smith College of MusicSaint PaulUSA

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