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.

== ~~Strong CS scales~~ ==

== One-to-one detemperings ==

A JI scale ''S'' is a ''~~'strong constant structure'~~'' 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 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.

The following is one possible definition for one-to-one detemperings: A JI scale ''S'' is a ''one-to-one detempering'' 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 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 JI note which tempers to the note. The two definitions are equivalent because if a detempering of an n-note equal temperament ''v'' is one-to-one, then the first definition follows by the additivity of ''v'', and given the first definition the injectivity is immediate.

As suggested by its name, the property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a ~~strong CS~~, 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 ~~strong CS~~ if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.

As suggested by its name, 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''.

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 a ~~strong CS~~, 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.

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 a one-to-one detempering, 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.

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

Temperaments [[support]]ed by vals for one-to-one detempering scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small one-to-one detempering 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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Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written as ratios from ''S''(0) = 1 in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the [[interval class|set of ''k''-steps]] of ''S''. Then ''S'' ''is a [[constant structure]]'' (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math>

==== ~~Strong CS scales~~ are CSes ====

==== One-to-one detemperings are CSes ====

{{proof|contents=

Let ''v'': ''A'' → ℤ be the val associated with ''s''. Let <math>x \in C_j.</math> Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>

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}}

==== If the steps of a CS scale are linearly independent, then the scale is a ~~strong CS~~ ====

==== If the steps of a CS scale are linearly independent, then the scale is a one-to-one detempering ====

Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that <math>C_1</math> is a basis for the JI subgroup ''A'' generated by it. Then there exists an val <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).

(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is a CS but not a ~~strong CS~~. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].)

(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is a CS but not a one-to-one detempering. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].)

{{proof|contents=

Define the linear map <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have <math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i,</math> whence ''v'' is a ~~strong CS~~. That <math>v(2) = n</math> is also automatic.

Define the linear map <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have <math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i,</math> whence ''v'' is a one-to-one detempering. That <math>v(2) = n</math> is also automatic.

}}

=== Terminology ===

@@ Line 2: / Line 2: @@
 Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.
-== Strong CS scales ==
+== One-to-one detemperings ==
-A JI scale ''S'' is a '''strong constant structure''' 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 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.
+The following is one possible definition for one-to-one detemperings: A JI scale ''S'' is a ''one-to-one detempering'' 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 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 JI note which tempers to the note. The two definitions are equivalent because if a detempering of an n-note equal temperament ''v'' is one-to-one, then the first definition follows by the additivity of ''v'', and given the first definition the injectivity is immediate.
-As suggested by its name, the property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a strong CS, 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 strong CS if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.
+As suggested by its name, 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''.
-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 a strong CS, 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.
+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 a one-to-one detempering, 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.
-Temperaments [[support]]ed by vals for strong CS scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small strong CS scales despite their relatively low accuracy:
+Temperaments [[support]]ed by vals for one-to-one detempering scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small one-to-one detempering 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]].
@@ Line 36: / Line 36: @@
 Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written as ratios from ''S''(0) = 1 in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the [[interval class|set of ''k''-steps]] of ''S''. Then ''S'' ''is a [[constant structure]]'' (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math>
-==== Strong CS scales are CSes ====
+==== One-to-one detemperings are CSes ====
 {{proof|contents=
 Let ''v'': ''A'' → ℤ be the val associated with ''s''. Let <math>x \in C_j.</math> Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>
@@ Line 43: / Line 43: @@
 }}
-==== If the steps of a CS scale are linearly independent, then the scale is a strong CS ====
+==== If the steps of a CS scale are linearly independent, then the scale is a one-to-one detempering ====
 Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that <math>C_1</math> is a basis for the JI subgroup ''A'' generated by it. Then there exists an val <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).
-(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is a CS but not a strong CS. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].)
+(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is a CS but not a one-to-one detempering. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under [[5edo]]'s [[patent val]].)
 {{proof|contents=
-Define the linear map <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have <math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i,</math> whence ''v'' is a strong CS. That <math>v(2) = n</math> is also automatic.
+Define the linear map <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have <math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i,</math> whence ''v'' is a one-to-one detempering. That <math>v(2) = n</math> is also automatic.
 }}
 === Terminology ===