Consistency: Difference between revisions

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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 [[~~EDO|~~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 [[~~Odd_limit~~|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''.

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 [[Odd limit|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''.

Page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.

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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 [[~~3/2|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.

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]].

@@ Line 9: / Line 9: @@
 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 [[EDO|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 [[Odd_limit|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''.
+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 [[Odd limit|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''.
 Page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.
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 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 [[3/2|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.
+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]].