Consistency: Difference between revisions

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== Mathematical definition ==

Formally, if ''N''~~-edo~~ is an [[equal ~~division of the octave]]~~, and if for ~~any~~ interval ''r'', ''N''~~-edo~~ (''r'') is the closest ''N''~~-edo approximation to~~ ''r'', then ''N'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''''i'' and ''r''''j'' in ''S'' where ''r''''i''''r''''j'' is also in ''S'', ''N''~~-edo~~ (''r''''i''''r''''j'') = ''N''~~-edo~~ (''r''''i'') + ''N''~~-edo~~ (''r''''j'').

Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''''i'' and ''r''''j'' in ''S'' where ''r''''i''''r''''j'' is also in ''S'', ''T'' (''r''''i''''r''''j'') = ''T'' (''r''''i'') + ''T'' (''r''''j'').

; Alternative formulation using val

If for any interval ''r'', ''N''~~-edo~~ (''r'') is the closest ''N''~~-edo approximation to~~ ''r'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''N'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that ''N''~~-edo~~ (''r'') = ''V'' (''r'') for any ''r'' in ''S''.

If for any interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that ''T'' (''r'') = ''V'' (''r'') for any ''r'' in ''S''.

{{Template:Proof|title=Proof for equivalence|

contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''''i'', ''r''''j'', and ''r''''i''''r''''j'' in ''S'', their monzos are m''i'', m''j'', and m''i'' + m''j'', respectively.

contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''''i'', ''r''''j'', and ''r''''i''''r''''j'' in ''S'', their monzos are '''m'''''i'', '''m'''''j'', and '''m'''''i'' + '''m'''''j'', respectively.

The ratio ''r'' mapped by the val ''V'' is the tempered step number V (''r'') = ''V''·'''m'''~~. There is~~ the following identity:

The ratio ''r'' mapped by the val ''V'' is the tempered step number ''V'' (''r'') = ''V''·'''m''', with the following identity:

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If ''N''~~-edo~~ satisfies

If ''T'' satisfies

<math>~~\operatorname {\mathit {N}-edo}~~ (r_i r_j) = ~~\operatorname {\mathit {N}-edo}~~ (r_i) + ~~\operatorname {\mathit {N}-edo}~~ (r_j)</math>

then ''N''~~-edo~~ is ~~a member~~ of the function space formed by all vals. Therefore, there exists a val ''V'' such that ''N''~~-edo~~ (''r'') = V (''r'') for any ''r'' in ''S''.

then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that ''T'' (''r'') = ''V'' (''r'') for any ''r'' in ''S''.

}}

Normally, ''S'' is considered to be some set of ''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-odd-limit consistent''.

Normally, ''S'' is considered to be some set of ''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''. ''T'' is then said to be ''q-odd-limit consistent''.

If each interval in the ''q''-odd-limit is mapped to a unique value by ''N'', then it is said to be ''uniquely q-odd-limit consistent''.

If each interval in the ''q''-odd-limit is mapped to a unique value by ''T'', then it is said to be ''uniquely q-odd-limit consistent''.

== Examples ==

@@ Line 18: / Line 18: @@
 == Mathematical definition ==
-Formally, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', ''N''-edo (''r''<sub>''i''</sub>''r''<sub>''j''</sub>) = ''N''-edo (''r''<sub>''i''</sub>) + ''N''-edo (''r''<sub>''j''</sub>).
+Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', ''T'' (''r''<sub>''i''</sub>''r''<sub>''j''</sub>) = ''T'' (''r''<sub>''i''</sub>) + ''T'' (''r''<sub>''j''</sub>).
 ; Alternative formulation using val
-If for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''N'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that ''N''-edo (''r'') = ''V'' (''r'') for any ''r'' in ''S''.
+If for any interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that ''T'' (''r'') = ''V'' (''r'') for any ''r'' in ''S''.
 {{Template:Proof|title=Proof for equivalence|
-contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in ''S'', their monzos are m<sub>''i''</sub>, m<sub>''j''</sub>, and m<sub>''i''</sub> + m<sub>''j''</sub>, respectively.
+contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in ''S'', their monzos are '''m'''<sub>''i''</sub>, '''m'''<sub>''j''</sub>, and '''m'''<sub>''i''</sub> + '''m'''<sub>''j''</sub>, respectively.
-The ratio ''r'' mapped by the val ''V'' is the tempered step number V (''r'') &#61; ''V''·'''m'''. There is the following identity:
+The ratio ''r'' mapped by the val ''V'' is the tempered step number ''V'' (''r'') &#61; ''V''·'''m''', with the following identity:
 <math>V\cdot(\vec {m_i} + \vec {m_j}) &#61; V\cdot\vec {m_i} + V\cdot\vec {m_j}</math>
@@ Line 34: / Line 34: @@
 <math>V (r_i r_j) &#61; V (r_i) + V (r_j)</math>
-If ''N''-edo satisfies
+If ''T'' satisfies
-<math>\operatorname {\mathit {N}-edo} (r_i r_j) &#61; \operatorname {\mathit {N}-edo} (r_i) + \operatorname {\mathit {N}-edo} (r_j)</math>
+<math>T (r_i r_j) &#61; T (r_i) + T (r_j)</math>
-then ''N''-edo is a member of the function space formed by all vals. Therefore, there exists a val ''V'' such that ''N''-edo (''r'') &#61; V (''r'') for any ''r'' in ''S''.
+then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that ''T'' (''r'') &#61; ''V'' (''r'') for any ''r'' in ''S''.
 }}
-Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''.
+Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''T'' is then said to be ''q-odd-limit consistent''.
-If each interval in the ''q''-odd-limit is mapped to a unique value by ''N'', then it is said to be ''uniquely q-odd-limit consistent''.
+If each interval in the ''q''-odd-limit is mapped to a unique value by ''T'', then it is said to be ''uniquely q-odd-limit consistent''.
 == Examples ==