Transposition of Notations in Just Intonation

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 transposing melodies are given. These are checked for accuracy using the rational numbers which each notation represents. Calculation shortcuts are considered which make notation operations quicker to carry out by hand. A question regarding whether rational commas should be extended from 5-rough rational numbers to all rational numbers is considered which would greatly simplify notation multiplication. This approach is rejected since it leads to confusion about octave number. The four component notation system is recommended instead. Extensions to computer notation systems and stave representations are briefly mentioned.