PAUL ERLICH 72-TET TABLE

This table shows which 72-tET intervals best approximate the just ratios of the 17-limit within half an octave. "17- limit" here means the largest odd factor of either of the numbers forming the ratio does not exceed 17, as in Harry Partch's Tonality Diamonds. Intervals larger than half an octave are easily calculated: for example, 7:4 is an octave minus 8:7, so it's 72 - 14 = 58 steps.

The table shows that all 11-limit ratios are represented uniquely; that is, no 72-tET interval has to represent more than one of them. What the table does not show is that the maximum error of the 72-tET approximations of the 11-limit ratios is 4 cents. The lowest ET that improves on that is 118-tET where the largest error is 3 cents.

72-tET is consistent through the 17- limit. To explain what this means, let me show why 72-tET is not consistent in the 19-limit. In 72-tET, the best approximation of 13:8 is 50 steps, the best approximation of 19:8 is 90 steps, and the best approximation of 19:13 is 39 steps. That means that if you try to play the chord 8:13:19 in 72- tET, you cannot use the best approximation of all the intervals involved. That difficulty will never arise in a chord all of whose intervals are within the 17-limit, thus the term "consistency". 58-tET is also consistent through the 17-limit, but with worse approximations; the lowest ET that is consistent through the 19-limit is 80-tET, and the smallest that improves on 72-tET's 17-limit approximations is 94-tET, which is also consistent through the 23-limit.