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. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19. A different formulation: an edo represents a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:

# the same interval in C is always mapped to the same size in C', and

# no interval within the chord ~~has to be~~ off by more than 50% of an edo step.

# no interval within the chord is off by more than 50% of an edo step.

In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.

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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. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19. A different formulation: an edo represents a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:
 # the same interval in C is always mapped to the same size in C', and
-# no interval within the chord has to be off by more than 50% of an edo step.
+# no interval within the chord is off by more than 50% of an edo step.
 In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.