Epimorphism - Revision history

Fredg999: Link to main article, categories

2025-07-28T04:29:37Z

Link to main article, categories

← Older revision		Revision as of 04:29, 28 July 2025
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	#~~redirect~~ [[~~Periodic scale~~#~~Epimorphism~~]]		#REDIRECT [[Detempering#One-to-one detemperings of equal temperaments]]

			[[Category:Scale]]

Sintel: redirect to where the actual definition is

2025-04-24T21:34:48Z

redirect to where the actual definition is

← Older revision		Revision as of 21:34, 24 April 2025
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	#redirect [[~~Detempering~~]]		#redirect [[Periodic scale#Epimorphism]]

BudjarnLambeth: Fixed double redirect

2024-04-23T06:25:06Z

Fixed double redirect

← Older revision		Revision as of 06:25, 23 April 2024
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	#redirect [[~~epimorphic scale~~]]		#redirect [[Detempering]]

Inthar: Redirected page to Epimorphic scale

2024-01-31T02:39:29Z

Redirected page to Epimorphic scale

← Older revision		Revision as of 02:39, 31 January 2024
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	A JI scale is '''epimorphic''' if on the JI subgroup ''A'' generated by the scale's intervals, there exists a linear map, called an '''epimorphism''' or '''epimorphic val''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i''.		#redirect [[epimorphic scale]]

	~~An '''epimorphic temperament''' of an [[epimorphic]] scale ''S'' on a JI group ''A'' is a temperament supported by its epimorphic val on ''A''. Some~~ [[~~exotemperament]]s (including vals for small edos) can be used as~~ epimorphic ~~temperaments for small epimorphic scales scales:~~
	* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.
	* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].
	~~== Facts ==~~
	~~=== Definition: constant structure (CS) ===~~
	~~Given a periodic~~ scale ~~''S'', let <math>C_k</math> be the set of ''k''-steps of ''S''. Then ''S'' is ''constant structure'' (CS) if for any <math>i, j \in \mathbb{Z}</math> we have <math>C_i \cap C_j = \varnothing.</math>~~

	~~=== Epimorphic scales are CS ===~~
	~~{{proof\|contents=~~
	Let ''v'' be the val witnessing the epimorphicity of ''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 group ''A'' generated by it. Then there exists an epimorphic 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 CS but not epimorphic.~~

	~~{{proof\|contents=~~
	Define <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> That <math>v(2) = n</math> is also automatic.
	}}

	~~[[Category:Scale~~]]

Inthar: Removed redirect to Periodic scale#Epimorphism

2024-01-31T02:39:01Z

Removed redirect to Periodic scale#Epimorphism

← Older revision		Revision as of 02:39, 31 January 2024
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	~~#redirect~~ [[~~Periodic~~ scale~~#Epimorphism~~]]		A JI scale is '''epimorphic''' if on the JI subgroup ''A'' generated by the scale's intervals, there exists a linear map, called an '''epimorphism''' or '''epimorphic val''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i''.

			An '''epimorphic temperament''' of an [[epimorphic]] scale ''S'' on a JI group ''A'' is a temperament supported by its epimorphic val on ''A''. Some [[exotemperament]]s (including vals for small edos) can be used as epimorphic temperaments for small epimorphic scales scales:
			* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.
			* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].
			== Facts ==
			=== Definition: constant structure (CS) ===
			Given a periodic scale ''S'', let <math>C_k</math> be the set of ''k''-steps of ''S''. Then ''S'' is ''constant structure'' (CS) if for any <math>i, j \in \mathbb{Z}</math> we have <math>C_i \cap C_j = \varnothing.</math>

			=== Epimorphic scales are CS ===
			{{proof\|contents=
			Let ''v'' be the val witnessing the epimorphicity of ''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 group ''A'' generated by it. Then there exists an epimorphic 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 CS but not epimorphic.

			{{proof\|contents=
			Define <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> That <math>v(2) = n</math> is also automatic.
			}}

			[[Category:Scale]]

BudjarnLambeth: Create wanted redirect

2024-01-14T06:14:14Z

Create wanted redirect

New page

#redirect [[Periodic scale#Epimorphism]]