Consistency: Difference between revisions

Line 10:

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.

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 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 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.

== Mathematical Definition ==

Formally, 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 S is considered to be some 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''.

== Examples ==

@@ Line 10: / Line 10: @@
 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.
-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 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 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.
+== Mathematical Definition ==
+Formally, 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 S is considered to be some 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''.
 == Examples ==