Generating Scales

Here is a new piece in 31edo. 31edo is very close the quarter-comma meantone, which was used in Europe int the 16th and 17th Centuries. So it is easy to make very conventional music in 31edo. The scale I have chosen for this piece, however, pushes the sound in a rather unconventional direction!

A simple way to construct a musical scale is to select an interval as a generator. Start at some first pitch in the scale, then move up by the generating interval and add to the scale the pitch one arrives at. Repeat this process some number of times, and one has a scale with some basic structure. For example, the conventional diatonic scale is generated with the perfect fifth. Starting at F, one visits C, then G, then D, A, E, and B.

How many notes should be in the scale? Key circulation provides an answer. When the next pitch in the sequence is so close to the starting pitch that none of the pitches in the scale so far are between the next pitch and the starting pitch, that is a good place to stop. Leave out the next pitch! Then for a key change, one can remove the starting pitch from the scale and add the next pitch. So, for example, after B the next pitch in the sequence would be F#. This is very close to the starting pitch of F. To shift the scale to a new key, remove the F and add the F#.

Another good stopping point would be at A. The next pitch is E, which is very close to F. This earlier stopping point yields a pentatonic scale. To shift the pentatonic scale, one removes the F and adds the E.

Another fertile property of a scale is its support for traversing one or more commas. When pairs of pitches in the scale are separated by intervals that are not simply stacks of the generating interval, so there are two different harmonic relationships between the two pitches, this means that the tunings system has tempered out a comma. For example, one can move from F to A by a stack of four perfect fifths. But one can move from F to A directly by a major third. The pentatonic and diatonic scales temper out the syntonic comma.

This scale for this new piece is not a diatonic scale! Instead of generating the scale with the perfect fifth, a 3:2 frequency ratio, this scale is generated with the interval 35:24. I see that this interval has been called a septimal sub-fifth. In 31edo this interval is represented by 17 microsteps. Moving by this interval 20 times brings one back very close to the starting point, so a circulating scale can be constructed with 20 pitch classes. This is the scale used in this piece.

In exploring 42edo, which gives good approximations for the intervals 5:3 and 7:4, I noticed that the combination of these intervals, 35:24, supports traversing the comma 10616832:10504375. Moving by 35:24 four times, one arrives at a pitch that can also be reached directly from the starting pitch via the interval 7:4. 31edo supports the same scale, but approximates many intervals better than 42edo, so I decided to try this scale in 31edo instead.

Anyway, this is an unconventional scale in a conventional tuning system! The key properties of the scale are 1) it circulates, and 2) it supports traversal of commas.