Mathematical representations of musical tunings

John S. Allen

On rabbits, mathematics and musical scales
2, 5, 7, 12, 19 -- these numbers are the basis of
scale structure in the music of many cultures. And they reflect
a simple mathematical relationship.

Equal temperaments defined as mathematical series
Different instruments generate their tunings differently, making
them more or less adaptable to other, unconventional tunings. The first chapter of
an examination of the relationship between tunings, instrument architectures and keyboard designs.

Defining octaves separately from steps within the octave
It is useful to define musical musical tunings as matrices
in two variables, in which one variable generally corresponds to
the octave. The architecture of many electronic instruments reflects
this organization.

Some applications of the matrix
Many electronic instruments' tuning systems are
organized accorcing to the matrix defining octaves separately
from other intervals. The matrix may be used to implement just intonation
and is also the basis for tuning by ear.

Matrix Tunings
There are many musical tunings which require a two-dimensional
matrix to describe them mathematically. These include
practical and historical tunings such as the Pythagorean and
mean-tone tunings, as well as others which hold promise
for use with electronic instruments.