Transposition of Notations in Just Intonation
David Ryan

TL;DR
This paper introduces a notation system for rational frequencies in Just Intonation, enabling transposition through multiplication and inversion, with practical examples and considerations for computational implementation.
Contribution
It defines a four-component notation system for rational frequencies in Just Intonation, facilitating transposition and arithmetic operations, and discusses extensions and simplifications.
Findings
Defined multiplication and inversion for notations
Validated notation operations with examples
Discussed computational shortcuts and notation extensions
Abstract
A notation system was previously presented which can notate any rational frequency in free Just Intonation. Transposition of music is carried out by multiplying each member of a set of frequencies by a single frequency. Transposition of JI notations up by a fixed amount requires multiplication to be defined for any two notations. Transposition down requires inversion to be defined for any notation, which allows division to also be defined for any two notations. Each notation splits into four components which in decreasing size order are octave, diatonic scale note, sharps or flats, rational comma adjustment. Multiplication can be defined for each of the four notation components. Since rational number multiplication is commutative, this leads to a definition of multiplication for frequencies and thus notations. Examples of notation inversion and multiplication are given. Examples of…
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Taxonomy
TopicsPhonetics and Phonology Research
