Consistency: Difference between revisions

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== Consistency to ''d'' copies ==

A chord is '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] all of the following are true:

* The chord is "consistent~~", meaning every instance of an interval~~ in the ~~chord is represented using the same number of steps.~~

* The chord is consistent in the ordinary sense, and

* 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.)

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For example, 4:5:6:7 is consistent to 3 copies in [[31edo]]. However, 4:5:6:7:11 is only consistent to 1 copy 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 ''d'' copies 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 ''d'' copies 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''.

* 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'') steps of ''n'' ED''k''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to 1/2 copies'' 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 1/2 copies" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.

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'''Theorem:''' Consistency to ''d'' copies can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads that occur in the chord, so that the resulting chord (the union of the ''d'' copies) will always be consistent in the temperament (no matter which intervals are used to stack the ''d'' copies).

'''~~Theorem:~~''' ~~A (1/(2~~''d''~~) steps of~~ ''n'' ED''k'~~') threshold can be interpreted as allowing stacking~~ ''d'' copies of a chord C~~, 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''} consisting of two notes ''x'' and ''y'' that occur in the union C' = C1 ∪ C2 ∪ … ∪ C''d'' in the equal temperament, where the Ci are copies of the (approximations of) chord C. We need to show that this chord is consistent.

~~Proof: Consider a dyad D = {~~''x''~~, ''y''} consisting of two notes ''x'~~' and ''y'' that occur in the resulting chord C' = C1 ∪ C2 ∪ … ∪ C''d'' in the equal temperament, where the Ci are copies of the (approximations of) chord C. We may assume that the notes ''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 ''d'' copies, each dyad D''j'' have relative error 1/(2''d'') since D''i'' occurs in C. The relative error on D1 + D2 + … + D''m'' compared to its just counterpart is < (''d'' - 1)/(2''d''), and again by assumption of consistency to ''d'' copies, 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.

We may choose an arbitrary root ''r'' ∈ C' and assume that the notes ''x'' and ''y'' belong in two different copies of C, C''i'' and C''i'' + ''m'', where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''.

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

There is a path from ''r'' to ''x'' that uses at most ''d'' steps, and a path from ''r'' to ''y'' that uses at most ''d'' steps.

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 ''d'' copies, each dyad D''j'' have relative error 1/(2''d'') since D''i'' occurs in C. The relative error on D1 + D2 + … + D''m'' compared to its just counterpart is < (''d'' - 1)/(4''d''), and again by assumption of consistency to ''2d'' copies, the dyad ''y'' - ''x' '' has error 1/(4''d''). Hence the total relative error on D is strictly less than 1/4. Since the adjacent intervals to the approximation of D must have relative error >75% and > 25% respectively as apporixmations to the JI dyad D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. QED.

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Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s.

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 == Consistency to ''d'' copies ==
 A chord is '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] all of the following are true:
-* The chord is "consistent", meaning every instance of an interval in the chord is represented using the same number of steps.
+* The chord is consistent in the ordinary sense, and
 * 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.)
@@ Line 47: / Line 47: @@
 For example, 4:5:6:7 is consistent to 3 copies in [[31edo]]. However, 4:5:6:7:11 is only consistent to 1 copy 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 ''d'' copies 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 ''d'' copies 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''.
-* 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'') steps of ''n'' ED''k''.
 This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to 1/2 copies'' 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 1/2 copies" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.
+<!--
+'''Theorem:''' Consistency to ''d'' copies can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads that occur in the chord, so that the resulting chord (the union of the ''d'' copies) will always be consistent in the temperament (no matter which intervals are used to stack the ''d'' copies).
-'''Theorem:''' A (1/(2''d'') steps of ''n'' ED''k'') threshold can be interpreted as allowing stacking ''d'' copies of a chord C, 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''} consisting of two notes ''x'' and ''y'' that occur in the union C' = C<sub>1</sub> ∪ C<sub>2</sub> ∪ … ∪ C<sub>''d''</sub> in the equal temperament, where the C<sub>i</sub> are copies of the (approximations of) chord C. We need to show that this chord is consistent.
-Proof: Consider a dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in the resulting chord C' = C<sub>1</sub> ∪ C<sub>2</sub> ∪ … ∪ C<sub>''d''</sub> in the equal temperament, where the C<sub>i</sub> are copies of the (approximations of) chord C. We may assume that the notes ''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 ''d'' copies, 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> compared to its just counterpart is < (''d'' - 1)/(2''d''), and again by assumption of consistency to ''d'' copies, 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.
+We may choose an arbitrary root ''r'' ∈ C' and assume that the notes ''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''.
-Question: Will the resulting chord always satisfy condition #1 as well?
+There is a path from ''r'' to ''x'' that uses at most ''d'' steps, and a path from ''r'' to ''y'' that uses at most ''d'' steps.
+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 ''d'' copies, 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> compared to its just counterpart is < (''d'' - 1)/(4''d''), and again by assumption of consistency to ''2d'' copies, the dyad ''y'' - ''x' '' has error 1/(4''d''). Hence the total relative error on D is strictly less than 1/4. Since the adjacent intervals to the approximation of D must have relative error >75% and > 25% respectively as apporixmations to the JI dyad D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. QED.
+-->
 Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s.