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.

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. (Note: This does not necessarily refer to being consistent with a [[patent val]] of the edo.)

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 [[Just_intonation_subgroup|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.

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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.
+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. (Note: This does not necessarily refer to being consistent with a [[patent val]] of the edo.)
 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 [[Just_intonation_subgroup|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.