Epimorphic scale: Difference between revisions
mNo edit summary |
|||
| Line 1: | Line 1: | ||
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. | 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 CS. One can 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 ''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: | 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: | ||