Consistency: Difference between revisions

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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 irrational).

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 [[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''.

@@ Line 14: / Line 14: @@
 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 irrational).
+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 [[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''.