TL;DR
This paper introduces a novel 19-note scale within the Pythagorean tuning system based on the tritave, exploring its harmonic properties and visualizing its structure using the Tonnetz, with implications for composition.
Contribution
It proposes a new 19-note scale in the Pythagorean system based on the tritave, expanding harmonic analysis and visualization tools for this tuning.
Findings
The 2:3:4 chord forms a proper harmonic unit.
The Tonnetz effectively visualizes harmonic development in this system.
A short composition demonstrates the sound and sparsity of 2:3:4-harmony.
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.
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