Consistency
An edo (or other equal-step tuning) represents the q-odd-limit consistently if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An equal-step tuning is distinctly consistent (aka uniquely consistent) in the q-odd-limit if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent in the 7-odd-limit if it maps 7/5 and 10/7 to the same step. (This would correspond to tempering 50/49, and in the case of edos, would mean the edo must be a multiple of (aka superset of) 2edo).
Note that we are looking at the direct approximation (i.e. the closest approximation) for each interval, and trying to find a val to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.
While the term consistency is most frequently used to refer to some odd-limit, sometimes one may only care about 'some' of the intervals in some odd-limit; this situation often arises when working in JI subgroups. We can also "skip" certain intervals when evaluating consistency. For instance, 12edo is consistent in the no-11's, no-13's 19-odd-limit, meaning for the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, and 19, where we deliberately skip 11 and 13.
In general, we can say that some edo is consistent relative to a chord C, or that chord C is consistent in some edo, if its best approximation to all the notes in the chord, relative to the root, also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the q-odd limit if and only if it is consistent relative to the chord 1:3:…:(q - 2):q.
The concept only makes sense for equal-step tunings and not for unequal multirank tunings, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitrary precision by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).
The page Minimal consistent edos shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page Consistency limits of small edos shows the largest odd limit that a given edo is consistent or distinctly consistent in.
Mathematical definition
Formally, if N-edo is an equal division of the octave, and if for any interval r, N-edo (r) is the closest N-edo approximation to r, then N is consistent with respect to a set of intervals S if for any two intervals ri and rj in S where rirj is also in S, N-edo (rirj) = N-edo (ri) + N-edo (rj).
- Alternative formulation using val
If for any interval r, N-edo (r) is the closest N-edo approximation to r, and if V (r) is r mapped by a val V, then N is consistent with respect to a set of intervals S if there exists a val V such that N-edo (r) = V (r) for any r in S.
The ratio r mapped by the val V is the tempered step number V (r) = V·m. There is the following identity:
[math]V\cdot(\vec {m_i} + \vec {m_j}) = V\cdot\vec {m_i} + V\cdot\vec {m_j}[/math]
Hence,
[math]V (r_i r_j) = V (r_i) + V (r_j)[/math]
If N-edo satisfies
[math]\operatorname {\mathit {N}-edo} (r_i r_j) = \operatorname {\mathit {N}-edo} (r_i) + \operatorname {\mathit {N}-edo} (r_j)[/math]
then N-edo is a member of the function space formed by all vals. Therefore, there exists a val V such that N-edo (r) = V (r) for any r in S. [math]\square[/math]Normally, S is considered to be some 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-odd-limit is mapped to a unique value by N, then it is said to be uniquely q-odd-limit consistent.
Examples
An example for a system that is not consistent in a particular odd limit is 25edo:
The closest approximation for the interval of 7/6 (the septimal subminor third) in 25edo is 6 steps, and the closest 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 closest 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.
An example of the difference between consistency vs distinct 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 distinctly consistent only up to the 5-odd-limit. Another example of non-distinct 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 distinctly consistent only up to the 11-odd-limit.
Consistency to distance d
A chord is consistent to distance d ≥ 1 or consistent to d copies in an edo (or other equal division) iff the following holds: 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 d ≥ 1, this implies consistency in the ordinary sense.
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 distance 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 distances 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 long-range 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 distance 1", this is equivalent to the edo (or edk) being consistent in the q-odd-limit, and more generally, as "consistent to distance 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, 4:5:7 is consistent to distance 10 in 31edo. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.
Formally, for some real d > 0, a chord C is consistent to distance d in n-edk if the consistent approximation C' of C in n-edk satisfies the property that all intervals in C' are off from their corresponding intervals in C by less than 1/(2d) steps of n-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: Consistency to distance d 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 (the union of the d copies) will always be consistent in the temperament (no matter which intervals are used to stack the d copies).
Proof: Consider the union C' = C1 ∪ C2 ∪ … ∪ Cd in the equal temperament, where the Ci are copies of the (approximations of) chord C. We need to show that this chord is consistent.
Consider any dyad D = {x, y} consisting of two notes x and y that occur in 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. Thus x and y are separated by a path of at most d steps (at most d − 1 for the different copies of C, and 1 for the additional step within C). By consistency to distance d, each dyad Dj in the path has relative error 1/(2d). Hence by the triangle inequality, the total relative error ε on D is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of D must have relative error 1 - ε > 1/2 and 1 + ɛ respectively as approximations to the JI dyad D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. QED.
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 consistent 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-step tunings. 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 and v ≤ q (q is the largest integer harmonic in S).
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 below a given number, assuming tritave equivalence and tritave invertibility.