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.

Harmonic Layers

Here's a new piece in 270edo. 270edo is the tuning system that divides octaves into 270 equal parts.

The main thing I am exploring here is how the meaning of unusual intervals can be clarified by a rich harmonic context. This piece pushes my usual pattern in two ways. This piece has six voices; usually I limit myself to four. Also, this piece uses intervals that approximate frequency ratios built with the prime numbers 2, 3, 5, 7, 11, and 13; I rarely go beyond 7. Intervals such as 14:13 and 15:14 have been permitted in the construction process of this piece. 14:13 is approximated by 29 steps of 270edo; 15:14 by 27 steps. So the tuning system has the precision to distinguish between these intervals. I suspect that if these intervals were heard in isolation, they would be practically impossible to distinguish. But if a richer context is provided, perhaps in the form of tetrads 9:11:13:14 and 9:11:14:15, the ear would have more information and would be able to hear the difference. Anyway, that's what I am trying to do here. With six voices, the more esoteric intervals will occur in combination with less esoteric intervals, and the ear will be able to make some sense of what is happening.

This is a graph of the piece, with time in seconds on the horizontal axis, and pitch class on the vertical axis. The pitch class is the pitch folded into a single octave. The vertical axis labels give the fraction of that single octave.

This graph shows that there is further structure to this piece, that should help clarify the meaning of the intervals. This piece traverses the comma 2080:2079 twelve times.

270edo is such a precise tuning system, one might think it to be effectively equivalent to just intonation, where the intervals are exact rational frequency ratios, rather than approximations. But 270edo does temper out many commas, as indeed any edo, any equal division of the octave, must. Another way to think about tempering out 2080:2079 is that 270edo maps 77:65 and 32:27 to the same interval.