Detempering: Difference between revisions

Line 3:

Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.

== Epimorphic scales ==

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. Equivalently, it is a detempering of an [[equal ~~tenperament~~]] under some mapping where each note of the equal temperament is matched to exactly one note.

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. Equivalently, it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one note.

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''.

@@ Line 3: / Line 3: @@
 Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.
 == Epimorphic scales ==
-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. Equivalently, it is a detempering of an [[equal tenperament]] under some mapping where each note of the equal temperament is matched to exactly one note.
+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. Equivalently, it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one note.
 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''.