Consistency: Difference between revisions

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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 ~~without causing~~ the resulting chord ~~to be inconsistent~~.

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, and the resulting chord will satisfy condition #2 of chord consistency.

Proof: Consider a dyad ''D'' = (''x'', ''y'') that occurs in the resulting chord ''C' = C1 ∪ C2 ∪ ... ∪ Cd''. 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.

Proof: Consider a dyad ''D'' = (''x'', ''y'') that occurs in the resulting chord ''C' = C1 ∪ C2 ∪ ... ∪ Cd''. 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, which is condition #2 of chord consistency.

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

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 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 without causing the resulting chord to be inconsistent.
+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, and the resulting chord will satisfy condition #2 of chord consistency.
-Proof: Consider a dyad ''D'' = (''x'', ''y'') that occurs in the resulting chord ''C' = C1 ∪ C2 ∪ ... ∪ Cd''. 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.
+Proof: Consider a dyad ''D'' = (''x'', ''y'') that occurs in the resulting chord ''C' = C1 ∪ C2 ∪ ... ∪ Cd''. 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, which is condition #2 of chord consistency.
 Examples of more advanced concepts that build on this are [[telicity]] and [[Consistent#Maximal consistent set|maximal consistent set]]s.