2:3:4-Harmony within the Tritave
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, subdominants, higher dominants, associated minor chords, inversions, 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.
Cite
@article{arxiv.1709.00375,
title = {2:3:4-Harmony within the Tritave},
author = {Markus Schmidmeier},
journal= {arXiv preprint arXiv:1709.00375},
year = {2019}
}
Comments
In the new version, some remarks about music perception have been added. 29 pages, 22 figures, 7 tables. To appear in the Journal of Mathematics and Music