Consistency: Difference between revisions

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Proof: Consider 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.

Consider any dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in 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''. Thus ''x'' and ''y'' are separated by at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each dyad D''j'' in the path has relative error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2 (50%). ~~Assuming the relative error ε has 0 < ɛ < +50%,~~ the adjacent intervals to the approximation of D must have relative error 1 - ε > 1/2 and 1 + ɛ respectively as approximations to the JI dyad D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. QED.

Consider any dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in 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''. Thus ''x'' and ''y'' are separated by at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each dyad D''j'' in the path has relative error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of D must have relative error 1 - ε > 1/2 and 1 + ɛ respectively as approximations 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.

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 Proof: Consider 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.
-Consider any dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in 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''. Thus ''x'' and ''y'' are separated by at most ''d'' steps (at most ''d'' &minus; 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each dyad D<sub>''j''</sub> in the path has relative error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2 (50%). Assuming the relative error ε has 0 < ɛ < +50%, the adjacent intervals to the approximation of D must have relative error 1 - ε > 1/2 and 1 + ɛ respectively as approximations to the JI dyad D, the approximation we got must be the best one. Since D is arbitrary, we have proved chord consistency. QED.
+Consider any dyad D = {''x'', ''y''} consisting of two notes ''x'' and ''y'' that occur in 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''. Thus ''x'' and ''y'' are separated by at most ''d'' steps (at most ''d'' &minus; 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each dyad D<sub>''j''</sub> in the path has relative error 1/(2''d''). Hence the total relative error on D is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of D must have relative error 1 - ε > 1/2 and 1 + ɛ respectively as approximations 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.