Consistency: Difference between revisions

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Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.

== Consistency to distance ''m'' ==

If ''m'' ≥ 0, a chord ''C'' is '''consistent to distance''' ''m'' in ''N''-edo if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:

# every instance of an interval in C is mapped to the same size in C', and

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"Consistent to distance 0" is equivalent to "consistent".

(The 1/(2(''m''+1)) threshold is meant to allow stacking ''m'' dyads that occur in the chord without having the sum of the dyads have over 50% relative error. Since "consistent to distance ''m''" conveys the idea that a local neighborhood of the consonant chord in the JI lattice is mapped nicely, an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.)

Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of one chord.

For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:11 is only consistent and not to distance 1 because 11/5 is mapped too inaccurately (rel 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.

@@ Line 17: / Line 17: @@
 Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.
+== Consistency to distance ''m'' ==
 If ''m'' ≥ 0, a chord ''C'' is '''consistent to distance''' ''m'' in ''N''-edo if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:
 # every instance of an interval in C is mapped to the same size in C', and
@@ Line 23: / Line 23: @@
 "Consistent to distance 0" is equivalent to "consistent".
 (The 1/(2(''m''+1)) threshold is meant to allow stacking ''m'' dyads that occur in the chord without having the sum of the dyads have over 50% relative error. Since "consistent to distance ''m''" conveys the idea that a local neighborhood of the consonant chord in the JI lattice is mapped nicely, an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.)
+Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of one chord.
 For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:11 is only consistent and not to distance 1 because 11/5 is mapped too inaccurately (rel 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.