Epimorphic scale: Difference between revisions

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A JI scale ''S'' is '''epimorphic''' if on the JI subgroup ''A'' generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

A JI scale ''S'' is '''epimorphic''' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI group ''A'' is a temperament supported by its epimorphism on ''A''. Some [[exotemperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales:

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== Facts ==

=== Definition: constant structure (CS) ===

Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the set of ''k''-steps of ''S''. Then ''S'' is [[constant structure]] (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math>

Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the [[interval class|set of ''k''-steps]] of ''S''. Then ''S'' is [[constant structure]] (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math>

=== Epimorphic scales are CS ===

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Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that the <math>C_1</math> is a basis of the JI subgroup ''A'' generated by it. Then there exists an epimorphism <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).

The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is CS but not epimorphic. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].

(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is CS but not epimorphic. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].)

{{proof|contents=

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-A JI scale ''S'' is '''epimorphic''' if on the JI subgroup ''A'' generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.
+A JI scale ''S'' is '''epimorphic''' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.
 An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI group ''A'' is a temperament supported by its epimorphism on ''A''. Some [[exotemperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales:
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 == Facts ==
 === Definition: constant structure (CS) ===
-Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the set of ''k''-steps of ''S''. Then ''S'' is [[constant structure]] (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math>
+Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the [[interval class|set of ''k''-steps]] of ''S''. Then ''S'' is [[constant structure]] (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math>
 === Epimorphic scales are CS ===
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 Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that the <math>C_1</math> is a basis of the JI subgroup ''A'' generated by it. Then there exists an epimorphism <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).
-The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is CS but not epimorphic. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].
+(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is CS but not epimorphic. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].)
 {{proof|contents=