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.

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 3, 5, 7, 9, 15, 17, and 19, where we deliberately skip 11 and 13.

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 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''' 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.

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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.
-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 3, 5, 7, 9, 15, 17, and 19, where we deliberately skip 11 and 13.
+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 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''' 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.