Epimorphic scale - Revision history

Inthar: Redirected page to Detempering

2024-04-07T08:51:24Z

Redirected page to Detempering

← Older revision		Revision as of 08:51, 7 April 2024
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	~~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.		#redirect [[Detempering]]

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

	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.

	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:
	* 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]].
	~~== 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:

	~~<math>~~
	~~\left(\begin{array} {rrr}~~
	~~7 & 11 & 16~~
	~~\end{array} \right)~~
	~~\left(\begin{array}{rrrrrrr}~~
	~~-3 & -2 & 2 & -1 & 0 & -3 & 1 \\~~
	~~2 & 0 & -1 & 1 & -1 & 1 & 0 \\~~
	~~0 & 1 & 0 & 0 & 1 & 1 & 0~~
	~~\end{array}\right)~~
	=
	~~\left(\begin{array}{rrrrrrr}~~
	~~1 & 2 & 3 & 4 & 5 & 6 & 7~~
	~~\end{array}\right)~~
	~~</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.

	~~== Facts ==~~
	~~=== 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 [[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 CS ===~~
	~~{{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>

	~~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.~~
	}}

	~~=== If the steps of a CS scale are linearly independent, then the scale is epimorphic ===~~
	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 epimorphism <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 CS but not epimorphic. 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 an epimorphism. That <math>v(2) = n</math> is also automatic.
	}}

	~~[[Category:Scale~~]]

Inthar at 00:00, 7 April 2024

2024-04-07T00:00:55Z

← Older revision		Revision as of 00:00, 7 April 2024
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	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.		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.

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

	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 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.

Inthar at 23:56, 6 April 2024

2024-04-06T23:56:24Z

← Older revision		Revision as of 23:56, 6 April 2024
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	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 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.

	Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be ~~used as epimorphic temperaments~~ for small epimorphic scales despite their relatively low accuracy:		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:
	* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-note epimorphic scale) scale structures.		* 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 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

Inthar: Added an extension of the definition to arbitrary-dimension "scales".

2024-04-06T23:55:49Z

Added an extension of the definition to arbitrary-dimension "scales".

← Older revision		Revision as of 23:55, 6 April 2024
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	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.		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.

	An '''epimorphic ~~temperament~~''' ~~of an epimorphic~~ scale ''S'' on ~~a JI subgroup~~ ''A'' is a ~~temperament~~ [[support]]ed by ~~its epimorphism on ''A''~~. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales despite their relatively low accuracy:		Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.

			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.

			Temperament [[support]]ed by epimorphisms for epimorphic scales have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic 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 2.3.5 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 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

	Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.
	== 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, {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:

Inthar at 06:39, 20 February 2024

2024-02-20T06:39:15Z

← Older revision		Revision as of 06:39, 20 February 2024
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	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~~, called an~~ '''~~epimorphism~~''', ''v'': ''A'' ~~→ ℤ~~ 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 linear map ''v'': ''A'' → ℤ, called an '''epimorphism''', such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

	An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI subgroup ''A'' is a temperament [[support]]ed by its epimorphism on ''A''. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales despite their relatively low accuracy:		An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI subgroup ''A'' is a temperament [[support]]ed by its epimorphism on ''A''. Some [[temperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales despite their relatively low accuracy:

Inthar at 06:29, 20 February 2024

2024-02-20T06:29:41Z

← Older revision		Revision as of 06:29, 20 February 2024
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	* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

	Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.		Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.
	== 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, {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:

Inthar at 06:29, 20 February 2024

2024-02-20T06:29:10Z

← Older revision		Revision as of 06:29, 20 February 2024
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	* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

	Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.		Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.
	== 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, {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:

Inthar at 06:28, 20 February 2024

2024-02-20T06:28:51Z

← Older revision		Revision as of 06:28, 20 February 2024
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	* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

	Epimorphism is strictly stronger than [[constant structure]] (CS). ~~The reader should verify that when~~ one assumes ''S'' is 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 CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.		Epimorphism is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is 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 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''.
	== 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, {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:

Inthar at 06:21, 20 February 2024

2024-02-20T06:21:35Z

← Older revision		Revision as of 06:21, 20 February 2024
Line 5:		Line 5:
	* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

	Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is 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 CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.		Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is 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 CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.
	== 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, {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:

Inthar at 06:21, 20 February 2024

2024-02-20T06:21:11Z

← Older revision		Revision as of 06:21, 20 February 2024
Line 5:		Line 5:
	* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Generator sequence\|Tas series]].

	Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is CS but not that it is epimorphic, there is a unique set ~~mapping~~ <math>v : \{\text{intervals of $S$}\} -> \mathbb{Z}</math> that witnesses that ''S'' is CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' ~~is also~~ linear.		Epimorphism is strictly stronger than [[constant structure]] (CS). The reader should verify that when one assumes ''S'' is 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 CS. Thus a CS scale ''S'' is epimorphic if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.
	== 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, {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: