Theorem of (upwards) Harmonic Subdivision

Published: Dec 5, 2018
String Aliquot fingering / fretting Flute / finger-hole Interval ratio Musical scale Arithmetic / harmonic sequence Harmonic mean
Demetrios E. Lekkas

The purpose of this paper is to study the theoretical outcome of an aliquot equidistant fingering / fretting pattern on a string, within standard formality, a. by stating and proving a theorem expressing the exact resulting intervallic structure mathematically –a set of interval ratios between adjacent terms of a mirrored arithmetic sequence, b. by pursuing the complete mathematical properties of this structure, c. by discussing its various applications, with emphasis put on drillings on flutes, d. by calling upon practice in order to spot actual historical manifestations of the theorem and its discussion, e. by hypothesizing the rôle of the particular conditions implicit in the theorem towards the historical emergence and evolution of two primeval prototypes of the most prominent and universally disseminated intonational systems / “ur-scales”: an anhemitonic pentatonic one ([C, D, E¼↑, G, A+]) and the lower fifth of a “smooth” heptatonic one ([D, E¼↓, F+, G, A+]).

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Author Biography
Demetrios E. Lekkas, The Hellenic Open University, Patras Greece

Ex Academic, The Hellenic Open University, Patras Greece

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