Consistency: Difference between revisions

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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'' in the ~~edo~~. 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.

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.

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

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 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'' in the edo. 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.
+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.
 Question: Will the resulting chord always satisfy condition #1 as well?