An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation

TL;DR

This paper introduces a new algorithm for assigning prime commas in Just Intonation, enabling a compact and universal notation system called Rational Comma Notation that improves upon existing methods.

Contribution

A novel algorithm for prime comma assignment in Just Intonation is developed, enhancing the Rational Comma Notation system's compactness and universality.

Findings

The DR algorithm effectively assigns prime commas for primes below 1400.

Results show the DR algorithm's advantages over SAG and KG2 algorithms.

Rational Comma Notation allows for consistent translation between different free-JI notations.

Abstract

Musical frequencies in Just Intonation are comprised of rational numbers. The structure of rational numbers is determined by prime factorisations. Just Intonation frequencies can be split into two components. The larger component uses only integer powers of the first two primes, 2 and 3. The smaller component decomposes into a series of microtonal adjustments, one for each prime number 5 and above present in the original frequency. The larger 3-limit component can be notated using scientific pitch notation modified to use Pythagorean tuning. The microtonal adjustments can be notated using rational commas which are built up from prime commas. This gives a notation system for the whole of free-JI, called Rational Comma Notation. RCN is compact since all microtonal adjustments can be represented by a single notational unit based on a rational number. RCN has different versions depending on the choice of algorithm to assign a prime comma to each prime number. Two existing algorithms SAG and KG2 are found in the literature. A novel algorithm DR is developed based on discussion of mathematical and musical criteria for algorithm design. Results for DR are presented for primes below 1400. Some observations are made about these results and their applications, including shorthand notation and pitch class lattices. Results for DR are compared with those for SAG and KG2. Translation is possible between any two free-JI notations and any two versions of RCN since they all represent the same underlying set of rational numbers.