Detempering: Difference between revisions

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

== One-to-one detemperings ==

== One-to-one detemperings of equal temperaments ==

The following are two equivalent definitions for one-to-one detemperings:

The following are two equivalent definitions for one-to-one detemperings of an [[equal temperament]]:

# A JI scale is a ''one-to-one detempering'' ~~if it is a detempering~~ of an ~~[[equal temperament]] under some mapping where~~ each note of the equal temperament is matched to exactly one JI note which tempers to the note.

# A JI scale is a ''one-to-one detempering'' of an ET if each note of the equal temperament is matched to exactly one JI note which tempers to the 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 [[val]] ''v'': ''A'' → ℤ (which can be called an '''epimorphism''') such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

The two terms are equivalent because if a detempering of an ''n''-note equal temperament ''v'' is one-to-one, then second definition follows by the additivity of ''v'', and given the second definition, injectivity is immediate.

@@ Line 2: / Line 2: @@
 Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.
-== One-to-one detemperings ==
+== One-to-one detemperings of equal temperaments ==
-The following are two equivalent definitions for one-to-one detemperings:
+The following are two equivalent definitions for one-to-one detemperings of an [[equal temperament]]:
-# A JI scale is a ''one-to-one detempering'' if it is a detempering of an [[equal temperament]] under some mapping where each note of the equal temperament is matched to exactly one JI note which tempers to the note.
+# A JI scale is a ''one-to-one detempering'' of an ET if each note of the equal temperament is matched to exactly one JI note which tempers to the 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 [[val]] ''v'': ''A'' → ℤ (which can be called an '''epimorphism''') such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.
 The two terms are equivalent because if a detempering of an ''n''-note equal temperament ''v'' is one-to-one, then second definition follows by the additivity of ''v'', and given the second definition, injectivity is immediate.