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]], meaning for the set of the odd harmonics 3, 5, 7, 9, 15, 17, and 19.

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.

~~Stated more mathematically, if~~ ''N''-~~edo is an~~ [[~~equal division of the octave~~]], ~~and if for any interval ''r'', edo (''N'', ''r'')~~ is the ~~best ''N''~~-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals ''a'' and ''b'' in S where ''ab'' is also in S, edo (''N'', ''ab'') = edo (''N'', ''a'') + edo (''N'', '~~'b''). Normally this is considered when S is the set of~~ [[~~odd limit|''q''~~-odd-limit ~~intervals~~]], ~~consisting of everything~~ of the ~~form 2<sup>''n''</sup> ''u''/''v''~~, where ~~''u''~~ and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''. If each interval in the ''q''-limit is mapped to a unique value by ''N'', then it said to be ''uniquely q-odd-limit consistent''.

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.

~~A different formulation~~ is ~~that an edo approximates~~ a chord C ~~'''consistently~~''' if the ~~following hold for~~ the best approximation C' of the ~~chord in~~ the ~~edo:~~

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.

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

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

(~~If such an~~ approximation ~~exists~~, ~~it must be~~ the ~~only such approximation~~, ~~since changing one~~ interval ~~would make that interval go over~~ the ~~50% error threshold~~.)

Stated more mathematically, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', edo (''N'', ''r'') is the best ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals ''a'' and ''b'' in S where ''ab'' is also in S, edo (''N'', ''ab'') = edo (''N'', ''a'') + edo (''N'', ''b''). Normally this is considered when S is the set of [[odd limit|''q''-odd-limit intervals]], consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''. If each interval in the ''q''-limit is mapped to a unique value by ''N'', then it said to be ''uniquely q-odd-limit consistent''.

In this formulation, 12edo represents the chord 1:3:5:7:9:17:19 consistently. Question: Is using this formulation with a chord C equivalent to using the first formulation with S = the [[diamond function]] applied to C?

The concept only makes sense for equal temperaments and not for unequal rank-2 (or higher) temperaments, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

The concept only makes sense for ~~edos~~ and not for ~~non-edo~~ rank-2 (or higher) temperaments, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.

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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]], meaning for the set of the odd harmonics 3, 5, 7, 9, 15, 17, and 19.
+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.
-Stated more mathematically, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', edo (''N'', ''r'') is the best ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals ''a'' and ''b'' in S where ''ab'' is also in S, edo (''N'', ''ab'') = edo (''N'', ''a'') + edo (''N'', ''b''). Normally this is considered when S is the set of [[odd limit|''q''-odd-limit intervals]], consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''. If each interval in the ''q''-limit is mapped to a unique value by ''N'', then it said to be ''uniquely q-odd-limit consistent''.
+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.
-A different formulation is that an edo approximates a chord C '''consistently''' if the following hold for the best approximation C' of the chord in the edo:
+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.
-# every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 should not be approximated using two different sizes of fifths), and
-# no interval within the chord C' is off by more than 50% of an edo step.
-(If such an approximation exists, it must be the only such approximation, since changing one interval would make that interval go over the 50% error threshold.)
+Stated more mathematically, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', edo (''N'', ''r'') is the best ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals ''a'' and ''b'' in S where ''ab'' is also in S, edo (''N'', ''ab'') = edo (''N'', ''a'') + edo (''N'', ''b''). Normally this is considered when S is the set of [[odd limit|''q''-odd-limit intervals]], consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''. If each interval in the ''q''-limit is mapped to a unique value by ''N'', then it said to be ''uniquely q-odd-limit consistent''.
-In this formulation, 12edo represents the chord 1:3:5:7:9:17:19 consistently. Question: Is using this formulation with a chord C equivalent to using the first formulation with S = the [[diamond function]] applied to C?
+The concept only makes sense for equal temperaments and not for unequal rank-2 (or higher) temperaments, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).
-The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).
 The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.