Consistency
An edo represents the q-odd limit consistently if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. 12edo is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19.
A different formulation is that an edo approximates a chord C consistently if there exists an approximation of the chord in the edo such that:
- every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 shouldn't be approximated using two different sizes of fifths), and
- no interval within the chord is off by more than 50% of an edo step.
(If such an approximation exists, it must be the only such approximation, since changing one interval would make that interval go over the 50% threshold.)
In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.
The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators.
Stated more mathematically, 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.
The page Minimal consistent EDOs shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page Consistency levels of small EDOs shows the largest odd limit that a given 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 just 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.
Consistency to distance d
Non-technically, a chord is consistent to distance d in an edo, if the chord is consistent and you can "walk away" up to distance m from the chord consistently. So an approximation consistent to distance d would play more nicely in a regular temperament-style subgroup context.
Formally, if d ≥ 0, a chord C is consistent to distance d in N-edo if there exists an approximation C' of C in N-edo such that:
- every instance of an interval in C is mapped to the same size in C', and
- no interval within C' has relative error 1/(2(d+1)) or more.
"Consistent to distance 0" is equivalent to "consistent". (The 1/(2(m+1)) threshold is meant to allow stacking d dyads that occur in the chord without having the sum of the dyads have over 50% relative error.)
Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of that one chord to check distance-d consistency.
For example, 4:5:6:7 is consistent to distance 2 in 31edo. However, 4:5:6:7:11 is only consistent to distance 0 because 11/5 is mapped too inaccurately (rel error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.
Examples of more advanced concepts that build on this are telicity and maximal consistent neighborhoods.
Maximal consistent set
(Under construction)
Non-technically, a maximal consistent set (MCS) is a chord in a JI subgroup such that when you add another interval which is adjacent to the chord, then the chord becomes inconsistent in the edo.
Formally, given N-edo, a chord C and a JI subgroup G generated by the octave and the dyads in C, a maximal consistent set is a connected set S (connected via dyads that occur in C) such that adding another interval adjacent to S via a dyad in C results in a chord that is inconsistent in N-edo. The maximal connected neighborhood (MCN) of C is a maximal consistent set containing C.
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.