Tempering 4096:4095

Here is a new piece in 140edo.

This piece traverses the comma 4096:4095 eighteen times. The repeating staircase pattern in the above score is indicative of this looping structure.

I wanted to construct a piece that traversed the comma 4096:4095, and went searching for a good tuning system, a good equal division of octaves, to do this.

This table guided me to 140edo. I looked through thousands of possible equal divisions of the octave. First I filtered out just those that temper out 4096:4095. 4096 is a power of 2. Most of my software treats octaves as equivalent notes, so powers of 2 tend to disappear. The graphical score above works this way: the vertical axis is pitch class, i.e. the pitch with the octaves erased. 4095 = 3*3*5*7*13, so the search for a good tuning system involves those primes. In 140edo, the prime number 3 is approximated by 222 steps, 5 by 325 steps, 7 by 393 steps, and 13 by 518 steps. So the composite number 4095 is represented by 222 + 222 + 325 + 393 + 518 = 1680 steps. But 1680 = 12 * 140, i.e. 1680 is exactly 12 octaves. 4095 and 4096 are both exactly 1680 steps in 140edo, so we say that 140edo tempers out 4096:4095.

This table also shows the error involved in the various tunings. E.g., the prime number 3 is actually 221.8948... steps of 140edo, which the tuning will approximate by 222 steps. The error for the prime 3 is thus 0.1052 of a step. The table shows the errors for all the primes involved, and then a combined score. I sorted the tunings by this combined score. 441edo is a bit better than 140edo, but then it is also nice to have a smaller division of the octave, so I decided to use 140edo instead.