Detempering: Difference between revisions

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

# 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 ''S'' is an ''~~epimorphism~~'' 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.

# 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.

The property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a one-to-one detempering, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is a CS and satisfies ''v''(''S''[''i'']) = ''i'' for all ''i''. Thus a CS scale ''S'' is a one-to-one detempering if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.

The second 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 ~~an epimorphism~~, 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 ~~a one-to-one epimorphism~~.

The second 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 epimmorphic.

Temperaments [[support]]ed by vals for one-to-one detemperings have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small one-to-one detemperings despite their relatively low accuracy:

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 The following are two equivalent definitions for one-to-one detemperings:
 # 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 ''S'' is an ''epimorphism'' 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.
+# 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.
 The property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a one-to-one detempering, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is a CS and satisfies ''v''(''S''[''i'']) = ''i'' for all ''i''. Thus a CS scale ''S'' is a one-to-one detempering if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.
-The second 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 an epimorphism, 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 a one-to-one epimorphism.
+The second 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 epimmorphic.
 Temperaments [[support]]ed by vals for one-to-one detemperings have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small one-to-one detemperings despite their relatively low accuracy: