Consistency: Difference between revisions

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An [[edo]] represents the q-[[~~odd limit|~~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. 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.

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:

# 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

# 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 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% threshold.)

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The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings 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).

Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, 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, N(ab) = N(a) + 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^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.

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''.

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.

==Examples==

== Examples ==

An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:

The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives 3/2 × 7/6 = [[7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. [[12edo]] is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

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-An [[edo]] represents the q-[[odd limit|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. 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.
 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:
-# 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
+# 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 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% threshold.)
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 The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings 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).
-Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, 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, N(ab) = N(a) + 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^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.
+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''.
 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.
-==Examples==
+== Examples ==
 An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:
-The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
+The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives 3/2 × 7/6 = [[7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
 An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. [[12edo]] is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].