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.
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