2:3:4 Harmony within the tritave
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.
A Mathematical Model for Optimal Tuning Systems
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An Experimental Study of Internal Interval Standards in Javanese and Western Musicians
Six Javanese and six Western musicians performed a magnitude-estimation task using 36 melodic intervals ranging from 60 to 760 cents at 20-cent increments. Several musicians displayed well-defined regions of confusion in which a range of intervals was assigned approximately equal magnitude estimates. The results suggest that these listeners assimilate the intervals to a set of internal interval standards. No evidence for assimilation was found for other musicians in both groups, some of whom made highly accurate estimates. For the Javanese musicians who showed assimilation to internal interval standards, the regions corresponded to the two Javanese tuning systems, slendro and pelog. For the Western musicians, the regions corresponded to the equal-tempered scale. The relatively wider regions of confusion for the Javanese musicians may reflect the greater variability of intonation in Java. In addition, the Javanese musicians seemed able to choose between internal interval standards based on the two tuning systems.