Epimorphic scale: Difference between revisions

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== Facts ==

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

Given a periodic scale ''S'', let <math>C_k</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 ''S'', let <math>C_k</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>

=== Epimorphic scales are CS ===

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=== If the steps of a CS scale are linearly independent, then the scale is epimorphic ===

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 <math>C_1</math> is a ''basis~~'' for~~ the JI ~~group~~ ''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).

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]].

@@ Line 27: / Line 27: @@
 == Facts ==
 === Definition: constant structure (CS) ===
-Given a periodic scale ''S'', let <math>C_k</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 ''S'', let <math>C_k</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>
 === Epimorphic scales are CS ===
@@ Line 37: / Line 37: @@
 === If the steps of a CS scale are linearly independent, then the scale is epimorphic ===
-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 <math>C_1</math> is a ''basis'' for the JI group ''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).
+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]].