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 ''v'': ''A'' → ℤ, called an '''epimorphism''', such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

~~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''.

Epimorphicity 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''.

This definition extends naturally to asking whether a higher-dimensional mapping <math>S:\mathbb{Z}^n \to P</math> for an arbitrary codomain <math>P</math> of relative pitches is epimorphic, in the same sense of there existing an abelian group <math>A</math> and a linear map <math>v : A \to \mathbb{Z}^n</math> such that <math>v(S(x)) = x.</math> This can be of practical interest: one might ask whether an isomorphic keyboard mapping <math>S : \mathbb{Z}^2 \to P</math> (for a theoretical infinite 2D isomorphic keyboard) is epimorphic.

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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 ''v'': ''A'' → ℤ, called an '''epimorphism''', such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.
-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''.
+Epimorphicity 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''.
 This definition extends naturally to asking whether a higher-dimensional mapping <math>S:\mathbb{Z}^n \to P</math> for an arbitrary codomain <math>P</math> of relative pitches is epimorphic, in the same sense of there existing an abelian group <math>A</math> and a linear map <math>v : A \to \mathbb{Z}^n</math> such that <math>v(S(x)) = x.</math> This can be of practical interest: one might ask whether an isomorphic keyboard mapping <math>S : \mathbb{Z}^2 \to P</math> (for a theoretical infinite 2D isomorphic keyboard) is epimorphic.