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, meaning for the set of 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:
- every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 should not 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 1:3:5:7:9:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in 80edo. This is a feature, not a bug, as the distinction can be useful in some circumstances.
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, edo (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, edo (N, ab) = edo (N, a) + edo (N, b). Normally this is considered when S is the set of q-odd-limit intervals, consisting of everything of the form 2n 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.
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
A chord is consistent to span d in an edo (or other equal division) iff all of the following are true:
- The chord is "consistent", meaning every instance of an interval in the chord is represented using the same number of steps.
- Error accrues slowly enough that any 0 to d intervals can be combined (multiplied or divided) in any order without accruing 50% (i.e. half a step) or more of relative error, as long as all the intervals chosen are ones present in the chord. (Note that you may use the same interval d times even if only one instance of that interval is present in the chord.)
For the mathematically/geometrically inclined, you can think of the set of all n distinct intervals in the chord as forming n (mutually perpendicular) axes of length 1 that form a (hyper)cubic grid of points (existing in n-dimensional space) representing intervals. Then moving in the direction of one of these axes by 1 unit of distance represents multiplying by the corresponding interval once, and going in the opposite direction represents division by that interval. Then, to be consistent to span d means that all points that are a taxicab distance of at most d from the origin (which represents unison) have the direct mapping of their associated intervals agree with the sum of the steps accumulated through how they were reached in terms of moving along axes, with each axis representing the whole number of steps that closest fits the associated interval present in the chord.
Therefore, consistency to large spans represent very accurate (relative to the step size) subgroup interpretations because a large "space" of the arithmetic is captured "correctly" (without causing contradictions; consistently). Approximations consistent to some reasonable distance (ideally at least 2) would play more nicely in a regular temperament-style subgroup context where you might prefer a larger variety of low complexity intervals to be consistent to a lesser degree rather than focusing on high-span consistency of a small number of intervals.
Note that if the chord comprised of all the odd harmonics up to the q-th is "consistent to span 1", this is equivalent to the EDO (or EDk) being consistent in the q-odd-limit, and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it is intuitively equivalent to the idea of "consistency" with respect to a set of "basis intervals" (intervals you can combine how you want up to d times); namely, the ones present in the chord.
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 (relative 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, for some real d > 0, a chord C is consistent to span d in n EDk if there exists an approximation C' of C in n EDk such that:
- every instance of an interval in C is mapped to the same size in C', and
- all intervals in C' are off from their corresponding intervals in C by less than 1/(2d) EDk.
This more formal definition also provides an interesting generalisation of d from the naturals to the positive reals, as consistency to distance 1/2 can be interpreted as meaning that all intervals in C are at worst represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case C' is said to be a "semiconsistent" representation/approximation of C.
Theorem: A 1/(2d) EDk threshold can be interpreted as allowing stacking d copies of a chord C, including the original chord, via dyads that occur in the chord, so that the resulting chord will always satisfy condition #2 of chord consistency.
Proof: Consider a dyad D = {x, y} on two notes x and y that occur in the resulting chord C' = C1 ∪ C2 ∪ … ∪ Cd in the equal temperament, where the Ci are copies of the (approximations of) chord C. We may assume that the notes x and y belong in two different copies of C, Ci and Ci + m, where 1 ≤ i ≤ i + m ≤ d. Suppose Ci, Ci + 1, …, Ci + m are separated by dyads D1, D2, …, Dm that occur in C. Let x' be the interval x + D1 + D2 + … + Dm, the counterpart of x in Ci + m. Since m ≤ d - 1, by consistency to span d, each dyad Dj have relative error 1/(2d) since Di occurs in C. the relative error on D1 + D2 + … + Dm relative to their just counterparts is < (d - 1)/2d, and again by assumption of consistency to span d, the dyad y - x' has error 1/(2d). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.
Question: Will the resulting chord always satisfy condition #1 as well?
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 2n 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.