Consistency: Difference between revisions

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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. [[~~17edo~~]] is consistent in the no-5's 13-odd limit, i.e. the odd harmonics 3, 7, 9, 11, and 13. 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.

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, i.e. the odd harmonics 3, 5, 7, 9, 17, and 19. 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''.

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-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. [[17edo]] is consistent in the no-5's 13-odd limit, i.e. the odd harmonics 3, 7, 9, 11, and 13. 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.
+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, i.e. the odd harmonics 3, 5, 7, 9, 17, and 19. 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''.