User:Currywurst44/Consistency Rewrite

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At the end is the unrewritten original article with line numbers

1c An edo 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, if the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5.

2c An edo is distinctly consistent (or uniquely consistent) in the q-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep.

3c For example, an edo cannot be distinctly consistent in the 7-odd-limit if it maps 7/5 and 10/7 to the same step (in this case, the semi-octave of 2edo, tempering out 50/49).

9cThe concept is only defined for equal-step tunings and not for unequal, multirank tunings, since for most 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).

10c 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.

Motivation

Consistency is a widely used metric to quickly appraise edos and other equal step tunings.

Consistency can be interpreted as giving a recommendation for limits in which it makes sense to utilize a tuning. Although tunings often still make sense to use in limits they are inconsistent in.

During composition consistency ensures that there is no reason for intervals to be a different size (because of a better approximation) no matter how they were reached or how other intervals were stacked.

Combined with the step size of the tuning, consistency guarantees that the error of all intervals is below a certain threshold. A high consistency limit usually means that a tuning has low average error across all the intervals of that limit even if individual intervals may be close to 50% error. Thus a record consistency often marks tunings that are especially efficient in approximating low complexity intervals.

By demanding that every interval uses a different step, distinct consistency ensures that all intervals that are different on paper actually sound different as well.

Different intervals mapped to the same step imply a certain error for those intervals. By forbidding this, distinct consistency usually improves upon the average error of a tuning.

Mathematical definition

12a S shall be a set of intervals and M a tuning's pitch mapping of these intervals. r₁ and r₂ shall be in S with r₁ * r₂ = r₃ also in S. A tuning is consistent to distance d when the error of all M(r₁ * r₂) < 1/(2d) and the interval mapping is linear with M(r₁ * r₂) = M(r₁) + M(r₂).

A tuning is distinctly consistent if all M(r₃) are different.

Examples

20c An example for a system that is not consistent in the 7-odd-limit is 25edo:

21c 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.

22c 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.

23c Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit.

24c The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

27c 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.

28c12edo is also consistent in the 9-odd-limit, but not in the 11-odd-limit.

29cAn example of the difference between consistency vs distinct consistency:

30c 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.

26edo integer vs odd distinct consistency

Application

The default when talking about consistency is in regards to edos in an odd-limit to distance 1. This simplifies consistency to require that the odd-limit intervals within an octave have less than 50% relative error by patent val.

Interval Sets

Depending on the application, a variety of interval sets may be checked for consistency. Below are common ones but other sets are possible.

65a! An odd-limit, integer-limit, >1-integer-limit or chord of the same limit are all equivalent when used with an edo in regards to consistency but differ for distinct consistency.

Odd Limit

58a Odd limits are infinite sets so a tuning without pure octaves will always accrue infinite error as well and be inconsistent. Because of this it is only applicable to EDOs.

Other Equave Limits

64c It is possible to extend the concept of odd limits to other equave limits, such as the "q-throdd-limit" with 3/1 (tritave) equivalence, but because an edt that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets.

Integer Limit

25c 46edo is not consistent in the 15-integer-limit.

26c 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.

63aIf we compress the octave of 46edo slightly (by about a cent), we end up with an 18-integer-limit consistent system.

>1-Integer Limit

It might be desirable to investigate a tunings consistency above some root instead of considering arbitrarily removed modulations. This is accomplished by only including intervals larger than 1/1 in the interval set. It can be proven that >1-integer-limit consistency can be checked by looking at the relative error of intervals with a denominator two or more less less than the numerator. In the 6-limit for example {3/1, 4/2=2/1, 4/1, 5/3, 5/2, 5/1, 6/1}

Chords

6,7,8a! An odd harmonics chord (1:3:5:7:9:…) provides an alternative construction for odd-limit consistency. Using all possible differences between notes of the chord produces an interval set on which consistency is equivalent to the full odd-limit consistency for edos. For example in the 9-limit {9/7, 9/5, 9/1, 7/5, 7/3, 7/1, 5/3, 5/1, 3/1, 1/1}. This construction provides an interval set that has a few intervals as possible while still being equivalent to consistency of other common interval sets. Distinct consistency for an odd-harmonic-chord-set can be interpreted as the tuning being distinct in a non-over-1 context.

Using a full harmonic series chord (1:2:3:4:5:6:..) simply produces all intervals of the corresponding integer limit.

JI-Subgroups

4,5a! Interval sets are often modified to be compatible with a JI-subgroup by removing all intervals that contain a certain prime in its factorization. Removing all powers of 2 results in the same interval set as an odd-harmonic-chord-set.

Integer Harmonics

Consistency to distance d over the integer harmonics {1/1, 2/1, 3/1, 4/1, 5/1, 6/1,…} implies consistency to distance d/2 in the corresponding integer-limit. A distance of 2 or 25% relative error is sometimes called pure consistency^{[idiosyncratic term]} and implies consistency to distance 1 in the same odd/integer-limit.

Unrewritten Original Article

C=copy, A=alter, D=delete

1. C An edo 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, if the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5.
2. C An edo is distinctly consistent (or uniquely consistent) in the q-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep.
3. C For example, an edo cannot be distinctly consistent in the 7-odd-limit if it maps 7/5 and 10/7 to the same step (in this case, the semi-octave of 2edo, tempering out 50/49).
4. A 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.
5. A We can also skip certain intervals when evaluating consistency. For instance, 12edo is consistent in the no-11's, no-13's 21-odd-limit, meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, and 21, where we deliberately skip 11 and 13.
6. A In general, we can say that some equal-step tuning is consistent relative to a chord C, or that a chord C is consistent in some equal-step tuning, if its best approximation to all the notes in the chord, relative to an arbitrarily chosen root, also gives the best approximation to all of the intervals between the pairs of notes in the chord.
7. A 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.
8. A By convention, when assessing a tuning's consistency limit, this type of odd-integer harmonic series chord (limited to an odd limit) is used in edos, while in other equal-step tunings the unmodified harmonic series (limited to an integer limit) is used instead.
9. C The concept is only defined for equal-step tunings and not for unequal, multirank tunings, since for most 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).
10. C 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.
11. C ==Mathematical definition==
12. A Formally, if T is an equal tuning, and if for an interval r, T(r) is the closest approximation to r in T, then T is consistent with respect to a set of intervals S if for any two intervals r_i and r_j in S where r_ir_j is also in S, T(r_ir_j) = T(r_i) + T(r_j).
13. This is equivalent to looking at the direct approximation (i.e. the closest approximation) for each interval, and trying to find a val that does the same approximation, so that the intervals are lined up by the val. If there is such a val, then the edo is consistent within that odd limit, otherwise it is inconsistent.
14. Alternative formulation using val
15. If for any interval r, T(r) is the closest approximation to r in T, and if V(r) is r mapped by a val V, then T is consistent with respect to a set of intervals S if there exists a val V such that T(r) = V(r) for any r in S.
16. Proof for equivalence

    Let us denote the monzo of any ratio r by m. Due to the linearity of the interval space, for any intervals r_i, r_j, and r_ir_j in S, their monzos are m_i, m_j, and m_i + m_j, respectively. The ratio r mapped by the val V is the tempered step number V(r) = V·m, with the following identity:[math]\displaystyle{ V\cdot(\vec {m_i} + \vec {m_j}) = V\cdot\vec {m_i} + V\cdot\vec {m_j} }[/math]Hence, [math]\displaystyle{ V (r_i r_j) = V (r_i) + V (r_j) }[/math]If T satisfies [math]\displaystyle{ T (r_i r_j) = T (r_i) + T (r_j) }[/math]then T is an element of the function space formed by all vals {V}. Therefore, there exists a val V such that T(r) = V(r) for any r in S. [math]\displaystyle{ \square }[/math]
17. 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. T is then said to be q-odd-limit consistent.
18. If each interval in the q-odd-limit is mapped to a unique value by T, then it is said to be uniquely q-odd-limit consistent.
19. C ==Examples==
20. CAn example for a system that is not consistent in a particular odd limit is 25edo:
21. C 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.
22. C 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.
23. C Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit.
24. C The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
25. C As another notable example, 46edo is not consistent in the 15-odd-limit.
26. C 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.
27. C 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.
28. C12edo is also consistent in the 9-odd-limit, but not in the 11-odd-limit.
29. CAn example of the difference between consistency vs distinct consistency:
30. C 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.
31. D 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.
32. D Although 72edo is consistent up to the 17-odd-limit, it is distinctly consistent only up to the 11-odd-limit.
33. ==Generalizations==
34. =Pure consistency=
35. Going even further than consistency, an equal-step tuning is purely consistent^{[idiosyncratic term]} if it approximates all integer harmonics from 1 up to and including q within one-quarter of a step (in other words, maintaining relative interval errors of no greater than than 25%).
36. Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.
37. =Consistency to distance d=
38. A chord is consistent to distance d ≥ 1 or consistent to d copies in an equal-step tuning iff the following holds:
39. 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.
40. (Note that you may use the same interval d times even if only one instance of that interval is present in the chord.)
41. For d ≥ 1, this implies consistency in the ordinary sense.
42. For the 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.
43. 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.
44. 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 approximation 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.
45. 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).
46. 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.
47. Note that if the chord comprised of the harmonic series up to q is "consistent to distance 1", this is equivalent to the tuning being consistent in the q-integer-limit (as well as the (2⌈q/2⌉ − 1)-odd-limit if it is an edo);
48. more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept 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) – in this case, intervals between the "basis" harmonics of a truncated harmonic series (an integer limit).
49. For example, 4:5:7 is consistent to distance 10 in 31edo.
50. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%).
51. This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.
52. Formally, for some real d > 0, a JI chord c is consistent to distance d in an equal tuning T if the consistent approximation C of c in T satisfies the property that all intervals in C are off from their corresponding intervals in c by less than 1/(2d) steps of T.
53. 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.
54. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case C is said to be a "semiconsistent" representation/approximation of c.
55. Consistency to distance d can be interpreted as allowing stacking d copies of a chord C, including the original chord, via intervals 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).

    Consider the union C = C₁ ∪ C₂ ∪ … ∪ C_d in the equal tuning, where the C_i are copies of the (approximations of) chord C. We need to show that this chord is consistent.Consider any interval 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, C_i and C_{i + 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 interval D_j 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 + ε respectively as approximations to the JI interval D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. [math]\displaystyle{ \square }[/math]
56. Examples of more advanced concepts that build on this are telicity and maximal consistent sets.
57. ==For non-octave tunings==
58. A In non-octave equal-step tunings, octaves are not perfectly tuned, and thus an infinite odd limit cannot fully be consistently represented.
59. Instead, we measure consistency in the q-integer-limit, which is simply the set S consisting of all intervals u/v where u ≤ q and v ≤ q (and q is the largest integer harmonic in S).
60. Accordingly, the consistency limit of an edo describes the highest odd limit it represents consistently, while the consistency limit of any other equal-step tuning (or equal temperament without an exact octave) instead describes the highest integer limit it represents consistently.
61. The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, 13/10 and 11/7 are in the 16-integer-limit, but 20/13 and 22/7 are not.
62. As a result, octave stretch and compression can be employed to improve an equal tuning's consistency limits:
63. A if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-integer-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series.
64. C It is possible to extend the concept of odd limits to other equaves, such as the "q-throdd-limit" with 3/1 (tritave) equivalence, but because an edt that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets.
65. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the q-odd-limit will be consistent to the (q + 1)-integer-limit and vice versa) unless intervals or primes are skipped or if a JI subgroup is used.