2:3:4 Harmony within the tritave. Issue 3 (2nd September 2019)

• Record Type: Journal Article
• Authors: Schmidmeier, Markus
• Is Part Of: Journal of mathematics and music. Volume 13:Issue 3(2019)
• Page Start: 222
• Page End: 244
• Publication Date: 2019-09-02
• Subjects: Harmony -- scales -- composition -- continued fractions -- Bohlen-Pierce -- tritave -- neo-Riemannian Tonnetz; 00A65 (primary) -- 11A55 -- 05E99; Music theory -- Periodicals; Music -- Mathematical models -- Periodicals; Music -- Mathematics -- Periodicals; 781.2
• DOI: 10.1080/17459737.2019.1605626
• Languages: English
• ISSNs: 1745-9737

Abstract

Abstract : In the Pythagorean tuning system, the fifth is used to generate a scale of 12 notes per octave. In this paper, we use the octave to generate a scale of 19 notes per tritave; one can play this scale on a traditional piano. In this system, the octave becomes a proper interval and the 2:3:4 chord a proper chord. We study harmonic properties obtained from the 2:3:4 chord, in particular composition elements using dominants, inversions, major and minor chords, and diminished chords. The Tonnetz (array notation) turns out to be an effective tool to visualize the harmonic development in a composition based on these elements. 2:3:4 harmony may sound pure, yet sparse, as we illustrate in a short piece.