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''.

''N''~~-edo~~ is '''~~strongly~~ consistent''' ~~with respect to a chord~~ ''C'' if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:

If ''m'' ≥ 1, 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

# no interval within ''C' '' has [[relative error]] ~~25%~~ or more.

# no interval within ''C' '' has [[relative error]] 1\2m or more.

(The ~~25%~~ threshold is meant to allow stacking ~~two dyads~~ ''a''~~, ''b'' once~~ without having the sum ~~''a + b''~~ of the dyads have over 50% relative error~~; thus~~ a ~~strongly~~ consistent ~~approximation~~ would play more nicely in a regular temperament-style [[subgroup]] context.)

(The 1\2m 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 small part of the JI lattice is mapped nicely, an approximation consistent to distance m would play more nicely in a regular temperament-style [[subgroup]] context.)

The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.

@@ Line 14: / Line 14: @@
 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''.
-''N''-edo is '''strongly consistent''' with respect to a chord ''C'' if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:
+If ''m'' ≥ 1, 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
-# no interval within ''C' '' has [[relative error]] 25% or more.
+# no interval within ''C' '' has [[relative error]] 1\2m or more.
-(The 25% threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have over 50% relative error; thus a strongly consistent approximation would play more nicely in a regular temperament-style [[subgroup]] context.)
+(The 1\2m 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 small part of the JI lattice is mapped nicely, an approximation consistent to distance m would play more nicely in a regular temperament-style [[subgroup]] context.)
 The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.