Consistency: Difference between revisions

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An [[edo]] (or other [[equal-step tuning]]) 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. An [[equal-step tuning]] is '''distinctly~~/uniquely~~ consistent''' in the ''q''-[[odd-limit]] if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent ~~(aka uniquely consistent)~~ in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step. (This would correspond to tempering [[50/49]], and in the case of edos, would mean the edo must be a multiple of (aka superset of) 2edo).

An [[edo]] (or other [[equal-step tuning]]) represents the ''q''-[[odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An [[equal-step tuning]] is '''distinctly consistent''' (aka uniquely consistent) in the ''q''-[[odd-limit]] if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step. (This would correspond to tempering [[50/49]], and in the case of edos, would mean the edo must be a multiple of (aka superset of) 2edo).

Note that we are looking at the closest ~~direct~~ approximation for each interval, and trying to find a val to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.

Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, 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 [[subgroup]]s. 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 1, 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 [[subgroup]]s. 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 1, 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 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-step tuning]]s and not for unequal multirank tunings, since for some choices of generator sizes in these temperaments, you can get any ratio you want to ~~arbitary~~ precision by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

The concept only makes sense for [[equal-step tuning]]s and not for unequal multirank tunings, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitrary precision 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 limits of small edos]]'' shows the largest odd limit that a given edo is consistent or ~~uniquely~~ consistent in.

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

== Mathematical definition ==

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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 closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest 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 closest 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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One notable example: [[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]] 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 distinct 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 distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]].

== Consistency to distance ''d'' ==

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-An [[edo]] (or other [[equal-step tuning]]) 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. An [[equal-step tuning]] is '''distinctly/uniquely consistent''' in the ''q''-[[odd-limit]] if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent (aka uniquely consistent) in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step. (This would correspond to tempering [[50/49]], and in the case of edos, would mean the edo must be a multiple of (aka superset of) 2edo).
+An [[edo]] (or other [[equal-step tuning]]) represents the ''q''-[[odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An [[equal-step tuning]] is '''distinctly consistent''' (aka uniquely consistent) in the ''q''-[[odd-limit]] if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step. (This would correspond to tempering [[50/49]], and in the case of edos, would mean the edo must be a multiple of (aka superset of) 2edo).
-Note that we are looking at the closest direct approximation for each interval, and trying to find a val to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.
+Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, 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 [[subgroup]]s. 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 1, 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 [[subgroup]]s. 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 1, 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 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-step tuning]]s and not for unequal multirank tunings, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitary precision by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).
+The concept only makes sense for [[equal-step tuning]]s and not for unequal multirank tunings, since for some choices of generator sizes in these temperaments, you can get any ratio you want to arbitrary precision 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 limits of small edos]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.
+The page ''[[Minimal consistent edos]]'' shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page ''[[Consistency limits of small edos]]'' shows the largest odd limit that a given edo is consistent or distinctly consistent in.
 == Mathematical definition ==
@@ Line 48: / Line 48: @@
 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 closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest 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 closest 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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 One notable example: [[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]] 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 distinct 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 distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]].
 == Consistency to distance ''d'' ==