Epimorphic scale: Difference between revisions

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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.

Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is CS but not that it is epimorphic, there is a unique mapping ''v'' that witnesses that ''S'' is CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' is also linear.

An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI subgroup ''A'' is a temperament [[support]]ed by its epimorphism on ''A''. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales despite their relatively low accuracy:

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* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence|Tas series]].

Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is CS but not that it is epimorphic, there is a unique mapping ''v'' that witnesses that ''S'' is CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' is also linear.

== Example ==

Consider the Ptolemaic diatonic scale, {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}, which is nicetone with L = 9/8, M = 10/9, and s = 16/15. This scale is epimorphic because we can apply ⟨7 11 16], the [[7edo]] [[patent val]], to map the intervals into the number of scale steps:

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 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.
-Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is CS but not that it is epimorphic, there is a unique mapping ''v'' that witnesses that ''S'' is CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' is also linear.
 An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI subgroup ''A'' is a temperament [[support]]ed by its epimorphism on ''A''. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales despite their relatively low accuracy:
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 * The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence|Tas series]].
+Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is CS but not that it is epimorphic, there is a unique mapping ''v'' that witnesses that ''S'' is CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' is also linear.
 == Example ==
 Consider the Ptolemaic diatonic scale, {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}, which is nicetone with L = 9/8, M = 10/9, and s = 16/15. This scale is epimorphic because we can apply ⟨7 11 16], the [[7edo]] [[patent val]], to map the intervals into the number of scale steps: