Consistency: Difference between revisions

Line 51:

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:''' Consistency to distance ''2d+2'' 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:''' Consistency to distance ''2d'' 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).

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.

Line 57:

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''.

Thus ''x'' and ''y'' are separated by at most ''d+1'' steps. By consistency to distance ''2d+2'', each dyad D''j'' in the path has relative error 1/(4''d''+4). 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.

Thus ''x'' and ''y'' are separated by at most ''d'' steps. By consistency to distance ''2d'', each dyad D''j'' in the path has relative 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.

@@ Line 51: / Line 51: @@
 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:''' Consistency to distance ''2d+2'' 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:''' Consistency to distance ''2d'' 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).
 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.
@@ Line 57: / Line 57: @@
 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''.
-Thus ''x'' and ''y'' are separated by at most ''d+1'' steps. By consistency to distance ''2d+2'', each dyad D<sub>''j''</sub> in the path has relative error 1/(4''d''+4). 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.
+Thus ''x'' and ''y'' are separated by at most ''d'' steps. By consistency to distance ''2d'', each dyad D<sub>''j''</sub> in the path has relative 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.