Circulation

In yesterday's post I discussed a type of musical instrument that presents a fixed number of keys the player can strike or press etc. to produce pitches. These pitches would be from a section of the chain of fifths. This is a very traditional meantone approach, but it can be generalized to a chain of whatever other interval is of interest. The instruments I described would have some kind of auxiliary control to allow the section to be shifted along the chain. The idea is that by shifting one step along the chain, the pitch of a single key is changed by some relatively small amount, so the pitch order of the keys is not altered. An example of such an instrument is a harp, which has seven strings per octave. Pedals allow the player to sharpen or flatten strings to follow key signature changes.

This only works for sections of suitable length. For example, a section of five pitches along the chain of fifths would be F C G D A. The next pitch on the chain is E. To shift the section along the chain one step, the F must be replace by E. This does not change the pitch order of the keys, so this works.

If one tries to use a section of six pitches, F C G D A E, to move the section one step along the chain of fifths, one would have to replace the F by B. This is a very large change in pitch that severely disrupts the pitch order of the keys of the instrument and so would be be quite unwieldy.

A seven pitch section is again practical. To shift F C G D A E B along the chain of fifths, the F needs to be replaced with F#, which again does not disrupt the pitch order of the keys of the instrument.

Observe that sometimes moving the section up a fifth along the chain involves sharpening the changing pitch, as with the seven pitch section, and sometimes involves flattening the changing pitch, as with the five pitch section.

Which size sections of the chain will preserve the pitch order when shifting along the chain depends on the exact size of the fifth. For the just tuned fifth of Pythagorean tuning, sections of size 7, 12, and 53 shift by sharpening a pitch; sections of size 5, 17, 29, 41, and 94 shift by flattening a pitch. To take 1/5 comma meantone as a contrasting example, sections of size 7, 19, 31, 74, and 117 shift by sharpening a pitch; 5, 12, and 43 shift by flattening a pitch.

These various lengths all correspond to useful ways to divide octaves to form practical tuning systems. The chain of fifths of length n needs to wrap around the octave circle to come back very close to where it started, if shifting the chain is to involve a small pitch change. So n fifths must be very close to m octaves. So m/n needs to be good approximation to the size of the fifth relative to the size of the octave. This means that dividing octaves into n equal parts will provide a good approximation to the fifth, which will be at m steps of this division.

In celebration of the beauty of these mathematical aspects of music, here is an algorithmic composition using a 19 pitch section of the fifths of 43edo: 43edo scale 19.