A mathematical approach to musical scales
I learned the following from Rudolf Wille, who first wrote about this in his state examination thesis, Frankfurt 1962, and later wrote an article for the 1976 book "Musik und Zahl".
Let be a set of labels, and be a map which assigns a frequency to each element of . The pair is a scale!
For example, setting and we get the modern equal temperamental tuning, or and yields a tuning used by K.H. Stockhausen in his "Studie II".
A chord could then be a subset of , but Wille uses ordered tuples like here, to make it easier to refer to ratios like . Now we can call a scale tonal, if all its ratios are rational.
We can capture what an octave is by defining a relation which relates two chords and if they're shifted by an octave, that is if and we can always find and such that and for some integers .
Because is an equivalence relation, we can now obtain the quotient of the set of all possible chords of a scale by , and call the result harmonies of the scale. So, forget about octaves, let's only talk about things that can happen within a single octave.
Say, we want to talk about minor and major chords. To do that, we still need to define what a transposition is, so we can recognize a major chord wherever it may be shifted.
[ to be continued ]
Harry Partch liked to work in a 43 tone scale, for example for this 1952 piece Castor and Pollux:
I learned of his work from Tom Jenkins aka Squarepusher when he shared his favorites stepping in for Stuart in BBC6' Freak Zone in 2016.