User:Inthar/Epimorphic temperament: Difference between revisions

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An '''epimorphic temperament''' of an [[epimorphic]] scale ''S'' on a JI group ''A'' is a temperament supported by its [[epimorphic val]] (linear map ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'') on ''G''. Some [[exotemperament]]s (including vals for small edos) can be used as epimorphic temperaments for small CS scales:
* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.
* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].
== Facts ==
=== 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 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 linear independence cannot be omitted, since otherwise the scale {5/4, 32/25, 2/1} is a counterexample.
{{proof|contents=
Define <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have
<math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i.</math>
That <math>v(2) = n</math> is also automatic.
}}
=== Epimorphic scales are CS ===

Latest revision as of 02:36, 31 January 2024

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