Epimorphic scale: Difference between revisions

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</math>

where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale.

where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.

== Facts ==

=== Definition: constant structure (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 ~~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).

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 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 24: / Line 24: @@
 </math>
-where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale.
+where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.
 == Facts ==
 === Definition: constant structure (CS) ===
@@ Line 37: / Line 38: @@
 === 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 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).
+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 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]].)