Epimorphic scale: Difference between revisions

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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). When one assumes ''S'' is CS but not that it is epimorphic, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is CS and satisfies ''v''(''S''[i]) = ''i'' for all ''i''. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.

Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is CS but not that it is epimorphic, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is CS and satisfies ''v''(''S''[''i'']) = ''i'' for all ''i''. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.

== 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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 * 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). When one assumes ''S'' is CS but not that it is epimorphic, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is CS and satisfies ''v''(''S''[i]) = ''i'' for all ''i''. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.
+Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is CS but not that it is epimorphic, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is CS and satisfies ''v''(''S''[''i'']) = ''i'' for all ''i''. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.
 == 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: