Consistency: Difference between revisions

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{{proof

| title=Consistency to distance ''d'' 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).

| title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via intervals 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).

| contents=Consider the union {{nowrap|''C' '' {{=}} ''C''1 ∪ ''C''2 ∪ … ∪ ''C''''d''}} in the equal temperament, where the ''C''''i'' are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.

Consider any ~~dyad~~ {{nowrap|''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 {{nowrap|C''i'' + ''m''}}, where {{nowrap|1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''}}. Thus ''x'' and ''y'' are separated by a path of 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 by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 − ε > 1/2}} and {{nowrap|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.

Consider any interval {{nowrap|''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 {{nowrap|C''i'' + ''m''}}, where {{nowrap|1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''}}. Thus ''x'' and ''y'' are separated by a path of 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 interval ''D''''j'' in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 − ε > 1/2}} and {{nowrap|1 + ɛ}} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.

}}

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Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.

Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the ~~dyads~~ in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via ~~dyads~~ that occur in C) such that adding another interval adjacent to ''S'' via ~~a dyad~~ in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''.

Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the intervals in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via intervals that occur in C) such that adding another interval adjacent to ''S'' via an interval in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''.

== For non-octave tunings ==

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 {{proof
-| title=Consistency to distance ''d'' 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).
+| title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via intervals 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).
 | contents=Consider the union {{nowrap|''C' '' {{=}} ''C''<sub>1</sub> &cup; ''C''<sub>2</sub> &cup; … &cup; ''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 {{nowrap|''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 {{nowrap|C<sub>''i'' + ''m''</sub>}}, where {{nowrap|1 &le; ''i'' &le; ''i'' + ''m'' &le; ''d''}}. Thus ''x'' and ''y'' are separated by a path of 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 by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 &minus; ε &gt; 1/2}} and {{nowrap|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.
+Consider any interval {{nowrap|''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 {{nowrap|C<sub>''i'' + ''m''</sub>}}, where {{nowrap|1 &le; ''i'' &le; ''i'' + ''m'' &le; ''d''}}. Thus ''x'' and ''y'' are separated by a path of 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 interval ''D''<sub>''j''</sub> in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 &minus; ε &gt; 1/2}} and {{nowrap|1 + ɛ}} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.
 }}
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 Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.
-Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the dyads in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via dyads that occur in C) such that adding another interval adjacent to ''S'' via a dyad in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''.
+Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the intervals in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via intervals that occur in C) such that adding another interval adjacent to ''S'' via an interval in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''.
 == For non-octave tunings ==