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 approximates a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:

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 approximates a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:

# every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 shouldn't be approximated using two different sizes of fifths), and

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

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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 approximates a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:
+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 approximates a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:
 # every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 shouldn't be approximated using two different sizes of fifths), and
 # no interval within the chord is off by more than 50% of an edo step.