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

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 that we aren't using the 'patent' val for the EDO when making these approximations, but rather looking at the best approximation for each interval directly, rather than just the primes. If everything lines up, then the EDO is consistent within that odd-limit, otherwise it is inconsistent.

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. (Note: This does not necessarily refer to being consistent with a [[patent val]] of the edo.)
+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 that we aren't using the 'patent' val for the EDO when making these approximations, but rather looking at the best approximation for each interval directly, rather than just the primes. If everything lines up, then the EDO is consistent within that odd-limit, otherwise it is inconsistent.
 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.