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 the following hold for the best approximation C' of the chord in the edo:

1. 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
2. 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. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in 80edo

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 (assuming the generator is an irrational fraction of the octave).

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 span d

Non-technically, a chord is consistent to distance d in an edo, if the chord is consistent and error accrues slowly enough that you can move up to distance d from the chord consistently. So an approximation consistent to some reasonable distance would play more nicely in a regular temperament-style subgroup context. "Consistent to span 1" is equivalent to "consistent".

For example, 4:5:6:7 is consistent to span 3 in 31edo. However, 4:5:6:7:11 is only consistent to span 1 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.

Formally, if d ≥ 0, a chord C is consistent to span d in N-edo if there exists an approximation C' of C in N-edo such that:

1. every instance of an interval in C is mapped to the same size in C', and
2. no interval within C' has relative error 1/(2(d)) or more.

(The 1/(2(d)) threshold is meant to allow stacking d+1 chords, including the original chord, via dyads that occur in the chord without having the sum of the dyads have over 50% relative error. That is, you can make a copy of a chord up to distance d away from the original chord without inconsistency.)

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 span-d consistency.

Examples of more advanced concepts that build on this are telicity and maximal consistent sets.

Maximal consistent set

(Under construction)

Non-technically, a maximal consistent set (MCS) is a piece of a JI subgroup such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece 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 might 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.

Alternatively, we can use "modulo-n limit" if the equave is n/1. Thus the tritave analogue of odd limit would only allow integers not divisible by 3 under a given limit, assuming that tritave equivalence and tritave inversion applies.

Links

• consistent (TonalSoft encyclopedia)
• Consistency and relative error of EDO