User:Inthar/Epimorphic temperament - Revision history

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2024-01-31T02:36:45Z

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← Older revision		Revision as of 02:36, 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''.

	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.
	}}

Inthar at 02:34, 31 January 2024

2024-01-31T02:34:47Z

← Older revision		Revision as of 02:34, 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''.		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 CS scales:		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 2.3.5 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]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].
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	=== Epimorphic scales are CS ===		=== Epimorphic scales are CS ===
	{{proof\|contents=		{{proof\|contents=
	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'' 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.		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.

Inthar at 02:33, 31 January 2024

2024-01-31T02:33:45Z

← Older revision		Revision as of 02:33, 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''.		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 CS scales:		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 CS scales:
	* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.		* 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]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].

Inthar at 02:32, 31 January 2024

2024-01-31T02:32:54Z

← Older revision		Revision as of 02:32, 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 ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i''.		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 CS scales:		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 CS scales:

Inthar: /* If the steps of a CS scale are linearly independent, then the scale is epimorphic */

2024-01-31T02:31:18Z

If the steps of a CS scale are linearly independent, then the scale is epimorphic

← Older revision		Revision as of 02:31, 31 January 2024
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	{{proof\|contents=		{{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		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.

	<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.
	}}		}}

Inthar at 02:30, 31 January 2024

2024-01-31T02:30:54Z

← Older revision		Revision as of 02:30, 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 ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'')		A JI scale is '''epimorphic''' if on the JI subgroup ''A'' generated by the scale's intervals, there exists a linear map ''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 CS scales:		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 CS scales:

Inthar: /* If the steps of a CS scale are linearly independent, then the scale is epimorphic */

2024-01-31T02:30:18Z

If the steps of a CS scale are linearly independent, then the scale is epimorphic

← Older revision		Revision as of 02:30, 31 January 2024
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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 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).		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 ~~otherwise~~ the scale {5/4, 32/25, 2/1} is ~~a counterexample~~.		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=		{{proof\|contents=

Inthar at 02:28, 31 January 2024

2024-01-31T02:28:40Z

← Older revision		Revision as of 02:28, 31 January 2024
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	~~An '''epimorphic temperament~~''' ~~of an [[~~epimorphic~~]] scale~~ ''S'' on a JI ~~group~~ ''A'' is a ~~temperament supported by its [[epimorphic val]] (~~linear map ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'') on ''G''. Some [[exotemperament]]s (including vals for small edos) can be used as epimorphic temperaments for small CS scales:		A JI scale is '''epimorphic''' if on the JI subgroup ''A'' generated by the scale's intervals, there exists a linear map ''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 CS scales:
	* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.		* 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]].		* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].

Inthar: /* If the steps of a CS scale are linearly independent, then the scale is epimorphic */

2024-01-31T02:21:40Z

If the steps of a CS scale are linearly independent, then the scale is epimorphic

← Older revision		Revision as of 02:21, 31 January 2024
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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 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).		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 ~~linear independence~~ cannot be omitted, since otherwise the scale {5/4, 32/25, 2/1} is a counterexample.		The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since otherwise the scale {5/4, 32/25, 2/1} is a counterexample.

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

Inthar: /* Epimorphic scales are CS */

2024-01-31T02:21:08Z

Epimorphic scales are CS

← Older revision		Revision as of 02:21, 31 January 2024
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	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 <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+k - 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.
	}}		}}