
PAUL ERLICH 72TET TABLE 

This table shows which 72tET intervals best approximate the just ratios of the 17limit 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 11limit ratios are represented uniquely; that is, no 72tET interval has to represent more than one of them. What the table does not show is that the maximum error of the 72tET approximations of the 11limit ratios is 4 cents. The lowest ET that improves on that is 118tET where the largest error is 3 cents. 72tET is consistent through the 17 limit. To explain what this means, let me show why 72tET is not consistent in the 19limit. In 72tET, 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 17limit, thus the term "consistency". 58tET is also consistent through the 17limit, but with worse approximations; the lowest ET that is consistent through the 19limit is 80tET, and the smallest that improves on 72tET's 17limit approximations is 94tET, which is also consistent through the 23limit.
Wed, 30 Jun 1999 

December
4, 2003
