Detempering: Difference between revisions

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In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one interpretation is used for each degree of a scale, then the detempered scale is called a ''~~[[transversal]]~~'', ''~~'epimorphic scale'~~'' or ''~~one-to-one detempering~~''. Ideally the resultant detempered scale will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.

In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one interpretation is used for each degree of a scale, then the detempered scale is called a '''one-to-one detempering''' or a '''strong constant structure''' (strong CS). Ideally the resultant detempered scale will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.

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

== ~~Epimorphic~~ scales ==

== Strong CS 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 temperament]] under some mapping where each note of the equal temperament is matched to exactly one note.

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.

~~Epimorphicity~~ is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a 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 a 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''.

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''.

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 ~~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 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 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.

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

Temperament [[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:

* 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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</math>

where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is ~~an epimorphic temperament of~~ the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.

where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is a val associated with the the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.

=== Facts ===

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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>

==== ~~Epimorphic~~ scales are CSes ====

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

{{proof|contents=

Let ''v'': ''A'' → ℤ be the ~~epimorphism for~~ ''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>

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>

Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i + j - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.

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-In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one interpretation is used for each degree of a scale, then the detempered scale is called a ''[[transversal]]'', '''epimorphic scale''' or ''one-to-one detempering''. Ideally the resultant detempered scale will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.
+In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one interpretation is used for each degree of a scale, then the detempered scale is called a '''one-to-one detempering''' or a '''strong constant structure''' (strong CS). Ideally the resultant detempered scale will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.
 Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.
-== Epimorphic scales ==
+== Strong CS 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 temperament]] under some mapping where each note of the equal temperament is matched to exactly one note.
+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.
-Epimorphicity is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a 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 a 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''.
+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''.
-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 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 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 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.
-Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small epimorphic scales despite their relatively low accuracy:
+Temperament [[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:
 * 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 30: / Line 30: @@
 </math>
-where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.
+where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is a val associated with the the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.
 === Facts ===
@@ 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>
-==== Epimorphic scales are CSes ====
+==== Strong CS scales are CSes ====
 {{proof|contents=
-Let ''v'': ''A'' → ℤ be the epimorphism for  ''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>
+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>
 Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i + j - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.