Consistency: Difference between revisions
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An [[ | 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. | ||
Note that we aren't using the 'patent' val for the EDO when making these approximations, but rather looking at the best approximation for each interval directly, rather than just the primes. If everything lines up, then the EDO is consistent within that odd-limit, otherwise it is inconsistent. | Note that we aren't using the 'patent' val for the EDO when making these approximations, but rather looking at the best approximation for each interval directly, rather than just the primes. If everything lines up, then the EDO is consistent within that odd-limit, otherwise it is inconsistent. | ||
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 [[Just_intonation_subgroup|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. | 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 [[Just_intonation_subgroup|subgroups]]. We can also "skip" certain intervals when evaluating consistency. For instance, [[12edo|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. | ||
In general, we can say that some EDO is '''consistent relative to a chord C''', or that '''chord C is consistent in some EDO''', 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. In particular, an | In general, we can say that some EDO is '''consistent relative to a chord C''', or that '''chord C is consistent in some EDO''', 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. In particular, an EDO is consistent in the ''q''-odd limit if and only if it is consistent relative to the chord 1:3:...:q-2:q. | ||
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 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 | 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 == | == Mathematical Definition == | ||
Formally, if ''N''- | 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''. | 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''. | ||
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== Examples == | == Examples == | ||
An example for a system that is ''not'' consistent in a particular odd limit is [[25edo|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. | |||
An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo|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]]. | |||
One notable example: [[46edo|46EDO]] is not consistent in the 15-odd-limit. The 15:13 interval is slightly closer to 9 degrees of 46EDO than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46EDO's versions of [[15/8]] and [[13/8]]) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series. | |||
An example of the difference between consistency vs. unique consistency: In 12EDO the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12EDO is consistent up to the [[9-odd-limit]], it is uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo|72EDO]] where they are both mapped to 8 steps. Although 72EDO is consistent up to the [[17-odd-limit]], it is uniquely consistent only up to the [[11-odd-limit]]. | |||
An example of the difference between consistency vs. unique consistency: In | |||
== Consistency to distance ''d'' == | == Consistency to distance ''d'' == | ||
A chord is '''consistent to distance''' ''d'' ≥ 1 or '''consistent to''' ''d'' '''copies''' in an | A chord is '''consistent to distance''' ''d'' ≥ 1 or '''consistent to''' ''d'' '''copies''' in an EDO (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to d intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.) | ||
For ''d'' ≥ 1, this implies consistency in the ordinary sense. | For ''d'' ≥ 1, this implies consistency in the ordinary sense. | ||
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Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to distance 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to distance 1" means that the direct mappings agree with how the intervals are reached arithmetically, it is intuitively equivalent to the idea of "consistency" with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times); namely, the ones present in the chord. | Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to distance 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to distance 1" means that the direct mappings agree with how the intervals are reached arithmetically, it is intuitively equivalent to the idea of "consistency" with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times); namely, the ones present in the chord. | ||
For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that | For example, 4:5:7 is consistent to distance 10 in [[31edo|31EDO]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31EDO is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11. | ||
Formally, for some real ''d'' > 0, a chord C is consistent to distance ''d'' in ''n'' ED''k'' if the consistent approximation C' of C in ''n'' ED''k'' satisfies the property that all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n'' ED''k''. | Formally, for some real ''d'' > 0, a chord C is consistent to distance ''d'' in ''n'' ED''k'' if the consistent approximation C' of C in ''n'' ED''k'' satisfies the property that all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n'' ED''k''. | ||
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(''Under construction'') | (''Under construction'') | ||
Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the | Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the EDO. | ||
Formally, given ''N''- | Formally, given ''N''-EDO, a chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the dyads in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via dyads that occur in C) such that adding another interval adjacent to ''S'' via a dyad in ''C'' results in a chord that is inconsistent in ''N''-EDO. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''. | ||
== Generalization to non-octave scales == | == Generalization to non-octave scales == | ||
It is possible to generalize the concept of consistency to non-EDO equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we might use an integer limit, and the term 2<sup>''n''</sup> in the above equation is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ ''q'' and ''v'' ≤ ''q'' (''q'' is the largest integer harmonic in S). | |||
It is possible to generalize the concept of consistency to non- | |||
This also means that the concept of octave inversion no longer applies: in this example, [[13/9]] is in S, but [[18/13]] is not. | This also means that the concept of octave inversion no longer applies: in this example, [[13/9]] is in S, but [[18/13]] is not. | ||