Epimorphic scale: Difference between revisions

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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.

Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be ~~used as epimorphic temperaments~~ for small epimorphic scales despite their relatively low accuracy:

Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small epimorphic scales despite their relatively low accuracy:

* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-note epimorphic scale) scale structures.

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

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 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.
-Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales despite their relatively low accuracy:
+Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small epimorphic scales despite their relatively low accuracy:
 * The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-note epimorphic scale) scale structures.
 * The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence|Tas series]].