1029:1024

Dividing octaves into 87 equal steps, 87edo, has some quite precise approximations for consonant intervals. But the details of how these approximations combine, the algebra of the intervals, is worth exploring too.

This is a tonnetz diagram for 87edo. Each cell represents a pitch class, i.e. a pitch with octaves treated as equivalent. To the right of a 0 cell is a 51 cell: a perfect fifth is 51 steps of 87edo. Above a 0 cell is a 28 cell: a major third is 28 steps of 87edo. 87edo also has a good approximation for the less conventional 8:7 interval: 17 steps. A nice feature of all this is that 3*17 = 51, i.e. moving by 8:7 three times, one arrives at 3:2, at least as those intervals are approximated by 87edo. This is the same as saying that 87edo tempers out the comma 1029:1024.

This feature of 87edo can be used to construct a nice scale, as highlighted in the diagram above. This scale is generated by the 17 step approximation of 8:7. The scale has 26 notes per octave, with the gaps between the scale notes being 2 and 9 steps of 87edo. 87edo also tempers out 245:243, which this scale also traverses. So there are nine different major thirds in the scale: three clusters of three major thirds. Each cluster is in the shape of a diatonic scale. So this scale looks like three diatonic scales with a bit of connective tissue.

87edo does not temper out the syntonic comma, so conventional music will not work very well in this tuning system. The diatonic clusters have some unresolved tension. We can associate the white keys on a piano with some pitch classes on this tonnetz diagram: F=0, C=51, G=15, A=28, E=79, B=43. The challenge is with the D. It could be 64, to make it a perfect fourth from A. Or it could be 66, to make it a perfect fifth from G. This gap of two steps of 87edo is how 87edo represents the syntonic comma.

Some sound: 87edo scale 26.