Consistent

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If N-edo is an equal division of the octave, and if for any interval r, N(r) is the best N-edo approximation to r, then N is consistent with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of q odd limit intervals, consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be q limit consistent. If each interval in the q-limit is mapped to a unique value by N, then it said to be uniquely q limit consistent.

See also this list of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And this list of edos, with the largest odd limit that this edo is consistent or uniquely consistent in.

Examples

An example for a system that is not consistent in a particular odd limit is 25edo:

The best approximation for the interval of 7/6 (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the perfect fifth 3/2 is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = 7/4, the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

An example for a system that is consistent in the 7-odd-limit is 12edo: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the 9-odd-limit, but not in the 11-odd-limit.

One notable example: 46edo is not consistent in the 15 odd limit. The 15:13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the functional 15/13 (the difference between 46edo's versions of 15/8 and 13/8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-integer-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.

Examples on consistency vs. unique consistency: In 12edo the 7-odd-limit intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the 9-odd-limit, it is uniquely consistent only up to the 5-odd-limit. Another example or non-unique consistency is given by the intervals 14/13 and 13/12 in 72edo where they are both mapped to 8 steps. Although 72edo is consistent up to the 17-odd-limit, it is uniquely consistent only up to the 11-odd-limit.

Generalization to non-octave scales

It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v.

This also means that the concept of octave inversion no longer applies: in this example, 13/9 is in S, but 18/13 is not.

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