Detempering - Revision history

ArrowHead294 at 11:37, 18 February 2026

2026-02-18T11:37:22Z

← Older revision		Revision as of 11:37, 18 February 2026
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	The following are two equivalent definitions for one-to-one detemperings of an [[equal temperament]]:		The following are two equivalent definitions for one-to-one detemperings of an [[equal temperament]]:
	# 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 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.		# 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]] {{nowrap\|''v'': ''A'' → ℤ}} (which can be called an '''epimorphism''') such that {{nowrap\|''v''(''S''[''i'']) {{=}} ''i''}} for all {{nowrap\|''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 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 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 {{nowrap\|''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 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> is epimorphic.		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> is epimorphic.

	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:		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:
	* 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]].		* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L 2m 2s), [[blackdye]] (5L 2m 3s) and superzarlino (a 17-note epimorphic scale) scale structures.
			* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L 3m 2s), [[diasem]] (5L 2m 2s), and other scales in the [[Generator sequence\|Tas series]].

	=== Example ===		=== Example ===
	Consider the Ptolemaic diatonic scale, {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}, which is nicetone with L = 9/8, M = 10/9, and s = 16/15. This scale is epimorphic because we can apply ⟨7 11 16], the [[7edo]] [[patent val]], to map the intervals into the number of scale steps:		Consider the Ptolemaic diatonic scale, {{nowrap\|{9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}<nowiki/>}}, which is nicetone with {{nowrap\|L {{=}} 9/8\|M {{=}} 10/9}}, and {{nowrap\|s {{=}} 16/15}}. This scale is epimorphic because we can apply {{val\| 7 11 16 }}, the [[7edo]] [[patent val]], to map the intervals into the number of scale steps:

	<math>		<math>
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	=== Facts ===		=== Facts ===
	==== Definition: constant structure (CS) ====		==== Definition: constant structure (CS) ====
	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>		Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written as ratios from {{nowrap\|''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>

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

	{{proof\|contents=		{{proof\|contents=
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	== Lifting ==		== Lifting ==

	The term ''lifting'' can be used as a [[JI-agnostic]] alternative to ''detempering''.		The term ''lifting'' can be used as a [[JI-agnostic]] alternative to ''detempering''.

Inthar: /* Deregularization */

2026-02-18T09:01:31Z

Deregularization

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	[[Category:Scale]]		[[Category:Scale]]

	== ~~Deregularization~~ ==		== Lifting ==

	The term ''~~deregularization~~'' can be used as a [[JI-agnostic]] alternative to ''detempering''.		The term ''lifting'' can be used as a [[JI-agnostic]] alternative to ''detempering''.

	In this sense, [[diasem]] (LMLSLMLSL) is a ~~deregularization~~ of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.		In this sense, [[diasem]] (LMLSLMLSL) is a lifting of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.

	== Examples ==		== Examples ==

BudjarnLambeth: /* Examples */ add link to orphan page

2025-12-28T00:39:54Z

Examples: add link to orphan page

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	== Examples ==		== Examples ==
	* [[Ringer scale]]s		* [[Ringer scale]]s
			* [[Fantasy detempers]]
	* [[87edo/13-limit detempering]]		* [[87edo/13-limit detempering]]
	* [[Diasem]]		* [[Diasem]]

Fredg999: Remove internal link that points to a redirect to this page itself

2025-08-05T19:51:47Z

Remove internal link that points to a redirect to this page itself

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	=== Terminology ===		=== Terminology ===
	As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''[[transversal]]'', ''epimorphic scale'', and ''strong CS''.		As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''transversal'', ''epimorphic scale'', and ''strong CS''.
	[[Category:Scale]]		[[Category:Scale]]

BudjarnLambeth: /* Examples */

2024-12-02T00:27:09Z

Examples

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	== Examples ==		== Examples ==
			* [[Ringer scale]]s
	* [[87edo/13-limit detempering]]		* [[87edo/13-limit detempering]]
	* [[Diasem]]		* [[Diasem]]

BudjarnLambeth: /* Examples */ add link to orphaned page

2024-12-02T00:26:36Z

Examples: add link to orphaned page

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	* [[87edo/13-limit detempering]]		* [[87edo/13-limit detempering]]
	* [[Diasem]]		* [[Diasem]]
			* [[Beautiful 27]]

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Inthar: /* One-to-one detemperings of equal temperaments */

2024-06-28T16:07:40Z

One-to-one detemperings of equal temperaments

← Older revision		Revision as of 16:07, 28 June 2024
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	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 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 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.		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> is epimorphic.

	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:		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:

Inthar: /* One-to-one detemperings of equal temperaments */

2024-06-26T19:58:07Z

One-to-one detemperings of equal temperaments

← Older revision		Revision as of 19:58, 26 June 2024
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	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 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 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~~.		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 epimorphic.

	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:		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:

Inthar: /* Terminology */

2024-06-26T11:35:27Z

Terminology

← Older revision		Revision as of 11:35, 26 June 2024
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	=== Terminology ===		=== Terminology ===
	As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''[[transversal]]'' ~~and~~ ''epimorphic scale'', and ''strong CS''.		As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''[[transversal]]'', ''epimorphic scale'', and ''strong CS''.
	[[Category:Scale]]		[[Category:Scale]]

Inthar: /* Terminology */

2024-06-26T11:30:29Z

Terminology

← Older revision		Revision as of 11:30, 26 June 2024
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	=== Terminology ===		=== Terminology ===
	As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''[[transversal]]'' and ''epimorphic scale''.		As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''[[transversal]]'' and ''epimorphic scale'', and ''strong CS''.
	[[Category:Scale]]		[[Category:Scale]]