Consistency: Difference between revisions

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For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.

Formally, for some real ''d'' > 0, a chord ''C'' is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation ''C~~' '~~' of ''C'' in ''n'' ED''k'' such that:

Formally, for some real ''d'' > 0, a chord C is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation C' of C in ''n'' ED''k'' such that:

* every instance of an interval in C is mapped to the same size in C', and

* all intervals in ''C~~' '~~' are off from their corresponding intervals in ''C'' by less than (1/2''d'') ED''k''.

* all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') ED''k''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of ''C''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.

'''Theorem:''' A 1/(2''d'') ED''k'' threshold can be interpreted as allowing stacking ''d'' copies of a chord, including the original chord, via dyads that occur in the chord, so that the resulting chord will always satisfy condition #2 of chord consistency.

Proof: Consider a dyad ''D'' = (''x'', ''y'') that occurs in the resulting chord ''C' = C1 ∪ C2 ∪ ~~...~~ ∪ Cd'' in the equal temperament. We may assume that ''x'' and ''y'' belong in two different copies of ''C'', ''~~C_i~~'' ~~and~~ ''~~C_i+~~m'', where 1 <= i <= i+m <= d. Suppose ~~C_i~~, ~~C_i~~+1, ~~...~~, ~~C_i~~+m are separated by dyads ''D''1, ''D''2, ~~...~~, ''Dm'' that occur in ''C''. Let ''x' '' be the interval ''x'' + ''D''1 + ''D''2 + ~~...~~ + ''Dm'', the counterpart of ''x'' in ~~C_i~~+m. Since m <= d-1, by consistency to span ''d'', each dyad ''D''j have relative error 1/(2d) since Di occurs in C. the relative error on ''D''1 + ''D''2 + ~~...~~ + ''Dm'' relative to their just counterparts is < (d-1)/2d, and again by assumption of consistency to span ''d'', the dyad y-x' has error 1/(2d). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.

Proof: Consider a dyad D = (''x'', ''y'') that occurs in the resulting chord C' = C1 ∪ C2 ∪ … ∪ C''d'' in the equal temperament. We may assume that ''x'' and ''y'' belong in two different copies of C, C''i'' and C''i'' + ''m'', where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''. Suppose C''i'', C''i'' + 1, …, C''i'' + ''m'' are separated by dyads D1, D2, …, D''m'' that occur in C. Let ''x' '' be the interval ''x'' + D1 + D2 + … + D''m'', the counterpart of ''x'' in C''i'' + ''m''. Since ''m'' ≤ ''d'' - 1, by consistency to span ''d'', each dyad D''j'' have relative error 1/(2''d'') since D''i'' occurs in C. the relative error on D1 + D2 + … + D''m'' relative to their just counterparts is < (''d'' - 1)/2''d'', and again by assumption of consistency to span ''d'', the dyad ''y'' - ''x' '' has error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.

Question: Will the resulting chord always satisfy condition #1 as well?

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== 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 2n in the above equation is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ q ≥ v.

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''n'' in the above equation is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ ''q'' ≥ ''v''.

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.

@@ Line 46: / Line 46: @@
 For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.
-Formally, for some real ''d'' > 0, a chord ''C'' is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation ''C' '' of ''C'' in ''n'' ED''k'' such that:
+Formally, for some real ''d'' > 0, a chord C is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation C' of C in ''n'' ED''k'' such that:
 * every instance of an interval in C is mapped to the same size in C', and
-* all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than (1/2''d'') ED''k''.
+* all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') ED''k''.
-This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of ''C''.
+This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.
 '''Theorem:''' A 1/(2''d'') ED''k'' threshold can be interpreted as allowing stacking ''d'' copies of a chord, including the original chord, via dyads that occur in the chord, so that the resulting chord will always satisfy condition #2 of chord consistency.
-Proof: Consider a dyad ''D'' = (''x'', ''y'') that occurs in the resulting chord ''C' = C1 ∪ C2 ∪ ... ∪ Cd'' in the equal temperament. We may assume that ''x'' and ''y'' belong in two different copies of ''C'', ''C_i'' and ''C_i+m'', where 1 <= i <= i+m <= d. Suppose C_i, C_i+1, ..., C_i+m are separated by dyads ''D''1, ''D''2, ..., ''Dm'' that occur in ''C''. Let ''x' '' be the interval ''x'' + ''D''1 + ''D''2 + ... + ''Dm'', the counterpart of ''x'' in C_i+m. Since m <= d-1, by consistency to span ''d'', each dyad ''D''j have relative error 1/(2d) since Di occurs in C. the relative error on ''D''1 + ''D''2 + ... + ''Dm'' relative to their just counterparts is < (d-1)/2d, and again by assumption of consistency to span ''d'', the dyad y-x' has error 1/(2d). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.
+Proof: Consider a dyad D = (''x'', ''y'') that occurs in the resulting chord C' = C<sub>1</sub> ∪ C<sub>2</sub> ∪ … ∪ C<sub>''d''</sub> in the equal temperament. We may assume that ''x'' and ''y'' belong in two different copies of C, C<sub>''i''</sub> and C<sub>''i'' + ''m''</sub>, where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''. Suppose C<sub>''i''</sub>, C<sub>''i'' + 1</sub>, …, C<sub>''i'' + ''m''</sub> are separated by dyads D<sub>1</sub>, D<sub>2</sub>, …, D<sub>''m''</sub> that occur in C. Let ''x' '' be the interval ''x'' + D<sub>1</sub> + D<sub>2</sub> + … + D<sub>''m''</sub>, the counterpart of ''x'' in C<sub>''i'' + ''m''</sub>. Since ''m'' ≤ ''d'' - 1, by consistency to span ''d'', each dyad D<sub>''j''</sub> have relative error 1/(2''d'') since D<sub>''i''</sub> occurs in C. the relative error on D<sub>1</sub> + D<sub>2</sub> + … + D<sub>''m''</sub> relative to their just counterparts is < (''d'' - 1)/2''d'', and again by assumption of consistency to span ''d'', the dyad ''y'' - ''x' '' has error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2. Since D is arbitrary, we have proved condition #2 of chord consistency. QED.
 Question: Will the resulting chord always satisfy condition #1 as well?
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 == 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 ≥ v.
+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'' ≥ ''v''.
 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.