Tuesday, March 22, 2022

Sliding Schismas

For some years now I have been exploring music and tuning, through algorithmic composition. I have a computer program that I tweak, to change tuning systems, scales, etc. How much of the tweaking that I do is actually reflected in the output, in any perceptible way? That's a question worth examining!

Here is a set of musical pieces. The only thing I changed in the software from one piece to the next is that I changed the seed for the random number generator. The random numbers it generates control very many choices in the execution of the program, so these pieces will vary quite a bit. But the primary choice in question is the harmonic movement involved, the key changes. Some of the pieces don't have any movement at all. Others have a progression that is six steps long. Some of the pieces move along the progression in the forward order, other move in the reverse order.

So the question here is: can you divide these pieces into three groups, one group with no key changes, another group that moves in one direction, and a final group that moves in the opposite direction. Can I tell the difference? (The names of the pieces are the seeds I used to initialize the random number generator for each piece.)

These pieces all use the 53edo tuning system, where octaves are divided into 53 equal steps rather than the conventional 12. These pieces all work with a schisma[17] scale, where 17 notes are selected in each octave out of the full set of 53. In the pieces with no key changes, the scale is constant throughout the piece. In the other pieces, the key changes in a regular pattern, shifting every measure. With six key changes, the scale returns to the starting scale.

Each row in this picture shows which notes are in the scale in one of the keys. In the pieces with key changes, from one measure to the next the scale will shift to the next row up or down in the diagram; in some pieces the key changes move up in the diagram, in other pieces the key changes move down. I repeated the sequence three times in the diagram, and also extended the scale a bit beyond an octave, just to make clear that the pattern continues smoothly through time and up and down the pitch space.

One could play the pieces with no key changes on a piano reasonably accurately. There are five pairs of notes that are very close togther, just one step apart of the 53 per octave. These would correpond to spit keys on a deluxe piano, a slightly sharp version of a note and a slightly flat version. Thinking of the split note as just two versions of a single note, then there are twelve coarse notes per octave, very close to a conventional piano.

The sequence of key changes in the other pieces involve two different shifts in the scale. Moving along the sequence in one direction, the scale shifts five times by a minor third, and once by a minor sixth. Moving in the other direction, the shifts are the inversions, i.e. five major sixths and one major third. In the 53edo tuning system, this combination of key changes brings the scale back to its starting point.

When the scale is shifted by a minor third, the new position of the scale include eight of the notes of the scale before the shift. The shift by a minor sixth has a similar amount of overlap. This overlap allows for good continuity of musical phrases across the shift.

Listen to the pieces above: some have key shifts, and some don't. Can you tell the difference?