Consistent
If N is an Equal-step Tuning, and if for any interval r, N(r) is the best N 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). In case of edo systems 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 odd-limit consistent. If each interval in the q-limit is mapped to a unique value by N, then it said to be uniquely q odd-limit consistent. In case of edt systems this is considered when S is the set of q 3reduced-limit intervals, consisting of everything of the form 3^n u/v, where u and v are 3reduced integers less than or equal to q. N is then said to be q 3reduced-limit consistent. If each interval in the q-limit is mapped to a unique value by N, then it said to be uniquely q 3reduced-limit consistent.
See also Minimal consistent EDOs of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And Consistency levels of small EDOs of edos, with the largest odd limit that this edo is consistent or uniquely consistent in. For 3reduced-limit edts, Minimal consistent EDTs, Consistency levels of small EDTs. For notwos-3reduced-limit edts, Minimal notwos consistent EDTs, Consistency levels of notwos small EDTs.
Examples[edit]
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 very closer 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 9 of the harmonic series. Alternatively, 73edt is also 18-integer-limit consistent, but also 17-3reduced-limit consistent due to tritave equivalence.
Generalization to non-octave scales[edit]
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. In equal divisions per tritave we use a 3reduced-limit, while in notwos systems we use notwos-limit, and in nontwo edt we use notwos-3reduced-limit. 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.