Extended Consonance

Here's a new piece: 494edo scale 17.

This piece is in a 17 note scale, in the 494edo tuning system. The scale is diagrammed above. Green arrows, e.g. from pitch class 459 to pitch class 254, represent perfect fifths, a frequency ratio of 3:2, or at least the best approximation available in the 494edo tuning system. Blue arrows, e.g. from 300 to 459, represent major thirds, 5:4. Red arrows, e.g. from 459 to 364, represent the 7:4 interval which is not so conventional. Orange arrows, e.g. from 0 to 227, represent the even less conventional 11:8 interval. Dark purple arrows, e.g. from 148 to 0, represent the yet less conventional interval 13:8. I didn't use a strict division between consonant vs. dissonant intervals in constructing this piece, but a more flexible scoring system. For this piece, 13:8 is treated as more consonant than e.g. 81:50, which is a more complex interval but built up from simpler primes.

This scale contains loops such as 459, 300, 205, 432, 227, 22, 170, 459. This loop traverses the comma 2080:2079. The loop travels along three green arrows, one red arrow, and one orange arrow in the forward direction, and along one blue and one purple arrow in the reverse direction. If these intervals were all tuned to precise rational intervals, the loop would not return to the start, but would have shifted by that comma 2080:2079. The tempered scale 494edo approximates these intervals, adjusting them slightly so the loop returns to the starting pitch class.

494edo might seem like a rather arbitrary choice for a tuning system, but the above diagram shows how it is not. The diagram shows the tuning errors for a variety of intervals. For example, in the column labeled 5 and the row labeled 3 appears the number 0.061. This is the error in the approximation of 494edo for the interval 5:3. This error number is given in terms of a single step of 494edo. 494edo divides octaves into 494 equal steps, so these steps are very small. But the error for 5:3 is only 0.061 of one of these small steps. This table has many such small errors. The way I chose 494edo was quite simple: I just computed these errors for a wide range of edo possibilities, and then searched through the results for the tuning system that had small errors for all the intervals I wanted to use. The table shows that 494edo does not approximate e.g. 17:16 very well, but that is not an interval I wanted to use here.