## Tuesday, July 16, 2024

### Chord Progression

Here is a new piece in 171edo. 171edo, the tuning system that divides octaves into 171 equal steps, provides very precise approximation to the just intervals 3:2 (a perfect fifth), 5:4 (a major third), and 7:4 (an unconventional interval). 3:2 is approximated by 100 steps of 171edo, 5:4 by 55 steps, and 7:4 by 138 steps. If one starts at any pitch, and moves up a perfect fifth, then up five major thirds, and then up again by a 7:4 interval, the total movement will be 100 + 5*55 + 138 = 513 steps, which is exactly three octaves, equivalent to the starting point. This piece moves around this loop 36 times, once per 63 seconds. All 36 cycles are superimposed in this score:

This piece has five voices, which form relatively complex chords. In constructing this piece, the chord shapes have been constrained:

This is a fragment of the Tonnetz diagram for 171edo. It shows the three dimensional network of relationships among the pitch classes. Horizontal neighbors are connected by perfect fifths, vertical neighbors by major thirds, and the third dimension, in and out of the page, shows pitch classes related by 7:4. The green and purple boxes here have that same shape: the purple box is simply shifted to the right. Each box encloses 8 pitch classes. These boxes represent the constraint on chord shape. At any instant in time, the pitch classes assigned to the five voices must be contained in a box of this size and shape. Picking 5 points out of a total set of 8 allows for 56 different chord shapes.

What fascinates me at the moment is the relationship between the chord constraint and the harmonic movement driven by the 63 second cycle. With the five voices often starting and stopping at different times, much of the time the pitch class of just one voice will change at a time. The cube shaped chord constraint used here will allow unbounded harmonic movement even with this kind of overlap. The green box and the purple box in the diagram include four pitch classes in their intersection: 7, 40, 123, and 156. A five note chord might add pitch class 78, which would be allowed because all five pitch classes are in the green box. But then the voice sounding the 78 could switch to pitch class 52, which would be valid because all the pitch classes are in the purple box. The other voices could all move within the purple box to set up another move to the right. The same tactic works for movment in the other directions.