Epimorphic scale: Difference between revisions
| Line 18: | Line 18: | ||
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 epimorphic val <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 group ''A'' generated by it. Then there exists an epimorphic val <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 | 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= | {{proof|contents= | ||