Generator complexity - Revision history

FloraC: Readers aren't obligated to see your poor-tasted operation name

2025-06-25T14:37:49Z

Readers aren't obligated to see your poor-tasted operation name

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	== Definition ==		== Definition ==
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	Suppose ''A'' = {{val\| 0 ''a''<sub>3</sub> ''a''<sub>5</sub> ''a''<sub>7</sub> … ''a''<sub>''p''</sub> }} is the generator mapping [[val]] for a [[rank-2 temperament]] with ''P'' [[period]]s to the [[octave]], and ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the same val in weighted coordinates. For instance, {{val\| 0 1 -2 -2 }} is the generator mapping val for seven limit [[pajara]], and {{val\| 0 1/log<sub>2</sub>(3) -2/log<sub>2</sub>(5) -2/log<sub>2</sub>(7) }} ≅ {{val\| 0 0.631 -0.831 -0.712 }} is the val in weighted coordinates. For any vector '''v''', let		Suppose ''A'' = {{val\| 0 ''a''<sub>3</sub> ''a''<sub>5</sub> ''a''<sub>7</sub> … ''a''<sub>''p''</sub> }} is the generator mapping [[val]] for a [[rank-2 temperament]] with ''P'' [[period]]s to the [[octave]], and ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the same val in weighted coordinates. For instance, {{val\| 0 1 -2 -2 }} is the generator mapping val for seven limit [[pajara]], and {{val\| 0 1/log<sub>2</sub>(3) -2/log<sub>2</sub>(5) -2/log<sub>2</sub>(7) }} ≅ {{val\| 0 0.631 -0.831 -0.712 }} is the val in weighted coordinates. For any vector '''v''', let

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VectorGraphics at 04:30, 25 June 2025

2025-06-25T04:30:42Z

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	== Definition ==		== Definition ==

Sintel: -legacy, todo intro

2025-03-29T18:00:32Z

-legacy, todo intro

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	== Definition ==		== Definition ==

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2025-03-17T20:18:03Z

Inaccessible

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Lériendil at 16:51, 17 February 2025

2025-02-17T16:51:15Z

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	== Definition ==		== Definition ==

Lériendil: deploying new cat

2025-02-17T16:16:03Z

deploying new cat

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	== Definition ==		== Definition ==

Lériendil: Changed protection level for "Generator complexity" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

2025-02-17T16:02:43Z

Changed protection level for "Generator complexity" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

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FloraC: Linking; style; recategorize; +todo

2023-11-27T11:00:56Z

Linking; style; recategorize; +todo

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	== Definition ==		== Definition ==
	Suppose ''A'' = {{val\| 0 ''a''<sub>3</sub> ''a''<sub>5</sub> ''a''<sub>7</sub> … ''a''<sub>''p''</sub> }} is the generator mapping val for a rank ~~two~~ temperament with ''P'' ~~periods~~ to the octave, and ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the same val in weighted coordinates. For instance, {{val\| 0 1 -2 -2 }} is the generator mapping val for seven limit [[pajara]], and {{val\| 0 1/log<sub>2</sub>(3) -2/log<sub>2</sub>(5) -2/log<sub>2</sub>(7) }} ≅ {{val\| 0 0.631 -0.831 -0.712 }} is the val in weighted coordinates. For any vector '''v''', let span(~~'''~~v~~'''~~) = max(~~'''~~v~~'''~~) - min(~~'''~~v~~'''~~). The '''generator complexity''' of the temperament is ''P~~''⋅span~~(''B''). In the case of pajara, which has two periods to the octave, this would be 2⋅(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie ''W'' of the temperament, as span(2∨''W''), which is the span of 0 followed by the first ''n'' - 1 elements of W, where ''n'' is the number of primes in the ''p''-limit.		Suppose ''A'' = {{val\| 0 ''a''<sub>3</sub> ''a''<sub>5</sub> ''a''<sub>7</sub> … ''a''<sub>''p''</sub> }} is the generator mapping [[val]] for a [[rank-2 temperament]] with ''P'' [[period]]s to the [[octave]], and ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the same val in weighted coordinates. For instance, {{val\| 0 1 -2 -2 }} is the generator mapping val for seven limit [[pajara]], and {{val\| 0 1/log<sub>2</sub>(3) -2/log<sub>2</sub>(5) -2/log<sub>2</sub>(7) }} ≅ {{val\| 0 0.631 -0.831 -0.712 }} is the val in weighted coordinates. For any vector '''v''', let

			<math>\displaystyle \operatorname {span}(\vec v) = \max(\vec v) - \min(\vec v)</math>

			The '''generator complexity''' of the temperament is

			<math>\displaystyle P \cdot \operatorname {span}(B)</math>

			In the case of pajara, which has two periods to the octave, this would be 2⋅(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie ''W'' of the temperament, as span(2∨''W''), which is the span of 0 followed by the first ''n'' - 1 elements of W, where ''n'' is the number of primes in the ''p''-limit.

	Generator complexity satisfies the inequality, for any ''p''-limit interval ''I'', G(''I'') ≤ ''C'' KE(''I''), where ''C'' is the generator complexity of the temperament, G(''I'') is the number of generator steps, times ''P'', required to reach the tempered version of ''I'', and KE(''I'') is the [[Kees semi-height\|Kees expressibility]] of ''I''. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log<sub>2</sub>(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)<sup>2</sup> ⋅ 45/64), and ''P'' = 2, so that G(5/4) = 2×2.		Generator complexity satisfies the inequality, for any ''p''-limit interval ''I'', G(''I'') ≤ ''C'' KE(''I''), where ''C'' is the generator complexity of the temperament, G(''I'') is the number of generator steps, times ''P'', required to reach the tempered version of ''I'', and KE(''I'') is the [[Kees semi-height\|Kees expressibility]] of ''I''. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log<sub>2</sub>(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)<sup>2</sup> ⋅ 45/64), and ''P'' = 2, so that G(5/4) = 2×2.
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	This inequality can be used to give an alternative definition of generator complexity: ''C'' = sup G(''I'')/KE(''I'') over non-octave intervals, where KE(''I'') > 0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(''I''), we can define a complexity measure for any rank of temperament by ''C'' = sup OETES(''I'')/KE(''I'').		This inequality can be used to give an alternative definition of generator complexity: ''C'' = sup G(''I'')/KE(''I'') over non-octave intervals, where KE(''I'') > 0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(''I''), we can define a complexity measure for any rank of temperament by ''C'' = sup OETES(''I'')/KE(''I'').

	Generator complexity has the nice property that for any mos of size ''N'', floor(''N''/(''C'' KE(''I''))) intervals with pitch class corresponding to ''I'' are guaranteed to exist in the mos. Generator complexity is also useful in making complete searches using the [[wedgie]] for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.		Generator complexity has the nice property that for any [[mos]] of size ''N'', floor(''N''/(''C'' KE(''I''))) intervals with pitch class corresponding to ''I'' are guaranteed to exist in the mos. Generator complexity is also useful in making complete searches using the [[wedgie]] for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.

	== Generator complexity and Kees expressibility ==		== Generator complexity and Kees expressibility ==
	The following proof is due to Mike Battaglia.		The following proof is due to [[Mike Battaglia]].

	If '''m''' = {{monzo\| ''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub> }} is a vector with weighted coordinates in interval space, then KE('''m'''), the Kees expressibility of '''m''', is (\|''m''<sub>3</sub> + ''m''<sub>5</sub> + … + ''m''<sub>''p''</sub>\| + \|''m''<sub>3</sub>\| + \|''m''<sub>5</sub>\| + … + \|''m''<sub>''p''</sub>\|)/2. The "2" coordinate, ''m''<sub>2</sub>, plays no role in Kees expressibility, so we may replace it with anything we choose. If we replace it with -''e''<sub>3</sub> - ''e''<sub>5</sub> - … - ''e''<sub>''p''</sub>, we may define expressibility in terms of the ''L''<sub>1</sub> norm, as ‖ {{monzo\| -''e''<sub>3</sub>-''e''<sub>5</sub>-…-''e''<sub>''p''</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }} ‖/2.		If '''m''' = {{monzo\| ''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub> }} is a vector with weighted coordinates in interval space, then KE('''m'''), the Kees expressibility of '''m''', is (\|''m''<sub>3</sub> + ''m''<sub>5</sub> + … + ''m''<sub>''p''</sub>\| + \|''m''<sub>3</sub>\| + \|''m''<sub>5</sub>\| + … + \|''m''<sub>''p''</sub>\|)/2. The "2" coordinate, ''m''<sub>2</sub>, plays no role in Kees expressibility, so we may replace it with anything we choose. If we replace it with -''e''<sub>3</sub> - ''e''<sub>5</sub> - … - ''e''<sub>''p''</sub>, we may define expressibility in terms of the ''L''<sub>1</sub> norm, as ‖ {{monzo\| -''e''<sub>3</sub>-''e''<sub>5</sub>-…-''e''<sub>''p''</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }} ‖/2.

	For any vector space X with a subspace A, we may define a quotient space X/A as the equivalence classes of vectors in X where two vectors are equivalent iff their difference lies in A. Then we have a short exact sequence 0 → A → X → X/A → 0. Taking the duals of this gives us 0 → (X/A)* → X* → A* → 0. The annihilator of A is the subspace A⁀ of X* consisting of those functionals ''f'' such that ⟨''f''\|A⟩ equals 0; that is, it is the subspace of all the functionals ''f'' such that ⟨''f''\|'''a'''⟩ equals 0 for every '''a''' in A. There is a natural isomorphism between the annihilator A⁀ of A and the dual of the quotient (X/A), and also between X/A⁀ and A*.		For any {{w\|vector space}} X with a {{w\|Linear subspace\|subspace}} A, we may define a {{w\|Quotient space (linear algebra)\|quotient space}} X/A as the equivalence classes of vectors in X where two vectors are equivalent iff their difference lies in A. Then we have a short exact sequence 0 → A → X → X/A → 0. Taking the duals of this gives us 0 → (X/A)* → X* → A* → 0. The {{w\|Dual space #Quotient spaces and annihilators\|annihilator}} of A is the subspace A⁀ of X* consisting of those functionals ''f'' such that ⟨''f''\|A⟩ equals 0; that is, it is the subspace of all the functionals ''f'' such that ⟨''f''\|'''a'''⟩ equals 0 for every '''a''' in A. There is a natural isomorphism between the annihilator A⁀ of A and the dual of the quotient (X/A), and also between X/A⁀ and A*.

	Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class ['''x'''], ‖['''x''']‖ equals inf {‖'''x''' + '''a'''‖, '''a''' ∈ A}. Algebraically X is (noncanonically) isomorphic to X, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X, defined by setting, over all nonzero '''x''' ∈ X, ‖''f''‖* = sup ⟨''f''\|'''x'''⟩/‖'''x'''‖. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A), and X/A⁀ is isometrically isomorphic to A*.		Now suppose X is a finite dimensional real {{w\|normed vector space}}. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class ['''x'''], ‖['''x''']‖ equals inf {‖'''x''' + '''a'''‖, '''a''' ∈ A}. Algebraically X is (noncanonically) isomorphic to X, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X, defined by setting, over all nonzero '''x''' ∈ X, ‖''f''‖* = sup ⟨''f''\|'''x'''⟩/‖'''x'''‖. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A), and X/A⁀ is isometrically isomorphic to A*.

	In the situation which concerns us, X is the ''p''-limit interval space of dimension ''n'' under a norm of one half times the ''L''<sub>1</sub> norm, A is a subspace of dimension ''n'' - 1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {''kJ''}, where ''J'' is the [[JIP]], as its annihilator. X has a norm of half the ''L''<sub>1</sub> norm, and hence X* has a norm of twice the ''L''<sub>∞</sub> norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {‖''f'' + ''kJ''‖} occurs for the value of ''k'' where the maximum of ''f'' + ''kJ'' and minus the minimum of ''f'' + ''kj'' are the same. In that case, 2‖''f'' + ''kJ''‖<sub>∞</sub> = span(''f''), which is the generator complexity of ''f''. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.		In the situation which concerns us, X is the ''p''-limit interval space of dimension ''n'' under a norm of one half times the ''L''<sub>1</sub> norm, A is a subspace of dimension ''n'' - 1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {''kJ''}, where ''J'' is the [[JIP]], as its annihilator. X has a norm of half the ''L''<sub>1</sub> norm, and hence X* has a norm of twice the ''L''<sub>∞</sub> norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {‖''f'' + ''kJ''‖} occurs for the value of ''k'' where the maximum of ''f'' + ''kJ'' and minus the minimum of ''f'' + ''kj'' are the same. In that case, 2‖''f'' + ''kJ''‖<sub>∞</sub> = span(''f''), which is the generator complexity of ''f''. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.

	== STD complexity ==		== STD complexity ==
	If ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the generator mapping val in weighted coordinates, and ''P'' is the period, then the '''STD complexity''' (a term due to Graham Breed) is ''P''⋅STD(''B''), where "STD" means the standard deviation. If ''μ''(''V'') is the mean of the components of the vector ''V'', and ''J'' is the [[JIP]] {{val\| 1 1 1 … 1 }}, then ''₱''(''V'') = ''V'' - ''μ''(''V'')''J'' is the projection of ''V'' onto the subspace of vectors with zero mean value. We have STD(''V'') = sqrt (''₱''(''V'')∙''₱''(''V'') / dim(''V'')), where dim(''V'') is the dimension of ''V'' and the "⋅" denotes the dot product. If ''M'' = [''M''<sub>0</sub>, ''M''<sub>1</sub>]<sup>T</sup> is the [[Temperament mapping matrices\|mapping matrix]] in weighted coordinates in the standard [[Normal lists #Normal val lists\|normal val list]] form, then we may express STD complexity as STDcom(''M'') = ''M''<sub>0</sub>[1]⋅STD(''M''<sub>1</sub>).		If ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the generator mapping val in weighted coordinates, and ''P'' is the period, then the '''STD complexity''' (a term due to [[Graham Breed]]) is ''P''⋅STD(''B''), where "STD" means the {{w\|standard deviation}}. If ''μ''(''V'') is the mean of the components of the vector ''V'', and ''J'' is the [[JIP]] {{val\| 1 1 1 … 1 }}, then ''₱''(''V'') = ''V'' - ''μ''(''V'')''J'' is the projection of ''V'' onto the subspace of vectors with zero mean value. We have STD(''V'') = sqrt (''₱''(''V'')∙''₱''(''V'') / dim(''V'')), where dim(''V'') is the dimension of ''V'' and the "⋅" denotes the dot product. If ''M'' = [''M''<sub>0</sub>, ''M''<sub>1</sub>]<sup>T</sup> is the [[Temperament mapping matrices\|mapping matrix]] in weighted coordinates in the standard [[Normal lists #Normal val lists\|normal val list]] form, then we may express STD complexity as STDcom(''M'') = ''M''<sub>0</sub>[1]⋅STD(''M''<sub>1</sub>).

	Associated to STD complexity is STD error. If ''S'' = ''₱''(''M''<sub>0</sub>) ∧ ₱(''M''<sub>1</sub>), then STDerr(''M'') = sqrt(''S''∙''S'' / dim(''M''<sub>1</sub>)⋅''₱''(''M''<sub>1</sub>)∙₱(''M''<sub>1</sub>)).		Associated to STD complexity is STD error. If ''S'' = ''₱''(''M''<sub>0</sub>) ∧ ₱(''M''<sub>1</sub>), then STDerr(''M'') = sqrt(''S''∙''S'' / dim(''M''<sub>1</sub>)⋅''₱''(''M''<sub>1</sub>)∙₱(''M''<sub>1</sub>)).

			[[Category:Math]]
			[[Category:Temperament complexity measures]]
	[[Category:Generator]]		[[Category:Generator]]
	~~[[Category:Math]]~~
	~~[[Category:Theory]]~~

	~~[[Category:todo:~~add examples]]		{{Todo\| add examples \| intro }}

FloraC: Improve wiki markup (2/2)

2023-11-27T10:48:17Z

Improve wiki markup (2/2)

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	== Definition ==		== Definition ==
	Suppose ''A'' = {{val\| 0 ~~A₃ A₅ A₇~~ … A<sub>''p''</sub> }} is the generator mapping val for a rank two temperament with ''P'' periods to the octave, and ''B'' = {{val\| 0 ~~B₃ B₅ B₇~~ … B<sub>''p''</sub> }} is the same val in weighted coordinates. For instance, {{val\| 0 1 -2 -2 }} is the generator mapping val for seven limit [[pajara]], and {{val\| 0 1/~~log2~~(3) -2/~~log2~~(5) -2/~~log2~~(7) }} ≅ {{val\| 0 0.631 -0.831 -0.712 }} is the val in weighted coordinates. For any vector V, let max(V) - min(~~V) = span(V~~). The '''generator complexity''' of the temperament is ''P'' ~~span~~(''B''). In the case of pajara, which has two periods to the octave, this would be 2·(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(~~2∨W~~), which is the span of 0 followed by the first ''n'' - 1 elements of W, where ''n'' is the number of primes in the ''p''-limit.		Suppose ''A'' = {{val\| 0 ''a''<sub>3</sub> ''a''<sub>5</sub> ''a''<sub>7</sub> … ''a''<sub>''p''</sub> }} is the generator mapping val for a rank two temperament with ''P'' periods to the octave, and ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the same val in weighted coordinates. For instance, {{val\| 0 1 -2 -2 }} is the generator mapping val for seven limit [[pajara]], and {{val\| 0 1/log<sub>2</sub>(3) -2/log<sub>2</sub>(5) -2/log<sub>2</sub>(7) }} ≅ {{val\| 0 0.631 -0.831 -0.712 }} is the val in weighted coordinates. For any vector '''v''', let span('''v''') = max('''v''') - min('''v'''). The '''generator complexity''' of the temperament is ''P''⋅span(''B''). In the case of pajara, which has two periods to the octave, this would be 2⋅(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie ''W'' of the temperament, as span(2∨''W''), which is the span of 0 followed by the first ''n'' - 1 elements of W, where ''n'' is the number of primes in the ''p''-limit.

	Generator complexity satisfies the inequality, for any ''p''-limit interval ''I'', G(''I'') ≤ ''C'' KE(''I''), where ''C'' is the generator complexity of the temperament, G(''I'') is the number of generator steps, times ''P'', required to reach the tempered version of ''I'', and KE(''I'') is the [[Kees semi-height\|Kees expressibility]] of ''I''. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = ~~log2~~(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)<sup>2</sup> · 45/64), and ''P'' = 2, so that G(5/4) = 2×2.		Generator complexity satisfies the inequality, for any ''p''-limit interval ''I'', G(''I'') ≤ ''C'' KE(''I''), where ''C'' is the generator complexity of the temperament, G(''I'') is the number of generator steps, times ''P'', required to reach the tempered version of ''I'', and KE(''I'') is the [[Kees semi-height\|Kees expressibility]] of ''I''. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log<sub>2</sub>(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)<sup>2</sup> ⋅ 45/64), and ''P'' = 2, so that G(5/4) = 2×2.

	This inequality can be used to give an alternative definition of generator complexity: ''C'' = sup G(''I'')/KE(''I'') over non-octave intervals, where KE(''I'') > 0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(''I''), we can define a complexity measure for any rank of temperament by ''C'' = sup OETES(''I'')/KE(''I'').		This inequality can be used to give an alternative definition of generator complexity: ''C'' = sup G(''I'')/KE(''I'') over non-octave intervals, where KE(''I'') > 0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(''I''), we can define a complexity measure for any rank of temperament by ''C'' = sup OETES(''I'')/KE(''I'').
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	The following proof is due to Mike Battaglia.		The following proof is due to Mike Battaglia.

	If '''m''' = {{monzo\| ~~m2 m3 m5 ... mp~~ }} is a vector with weighted coordinates in interval space, then KE('''m'''), the Kees expressibility of '''m''', is (\|m3 + m5 + … + mp\| + \|m3\| + \|m5\| + ~~...~~ + \|mp\|)/2. The "2" coordinate, m2, plays no role in Kees expressibility, so we may replace it with anything we choose. If we replace it with -e3 -e5-…-ep, we may define expressibility in terms of the ''L''<sub>1</sub> norm, as ‖ \|-e3-e5-…-ep e3 e5 … ~~ep>~~ ‖/2.		If '''m''' = {{monzo\| ''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>''p''</sub> }} is a vector with weighted coordinates in interval space, then KE('''m'''), the Kees expressibility of '''m''', is (\|''m''<sub>3</sub> + ''m''<sub>5</sub> + … + ''m''<sub>''p''</sub>\| + \|''m''<sub>3</sub>\| + \|''m''<sub>5</sub>\| + … + \|''m''<sub>''p''</sub>\|)/2. The "2" coordinate, ''m''<sub>2</sub>, plays no role in Kees expressibility, so we may replace it with anything we choose. If we replace it with -''e''<sub>3</sub> - ''e''<sub>5</sub> - … - ''e''<sub>''p''</sub>, we may define expressibility in terms of the ''L''<sub>1</sub> norm, as ‖ {{monzo\| -''e''<sub>3</sub>-''e''<sub>5</sub>-…-''e''<sub>''p''</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }} ‖/2.

	For any vector space X with a subspace A, we may define a quotient space X/A as the equivalence classes of vectors in X where two vectors are equivalent iff their difference lies in A. Then we have a short exact sequence 0 → A → X → X/A → 0. Taking the duals of this gives us 0 → (X/A)* → X* → A* → 0. The annihilator of A is the subspace A⁀ of X* consisting of those functionals ''f'' such that ~~<~~''f''\|~~A>~~ equals 0; that is, it is the subspace of all the functionals ''f'' such that ~~<~~''f''\|'''a'''~~>~~ equals 0 for every '''a''' in A. There is a natural isomorphism between the annihilator A⁀ of A and the dual of the quotient (X/A), and also between X/A⁀ and A*.		For any vector space X with a subspace A, we may define a quotient space X/A as the equivalence classes of vectors in X where two vectors are equivalent iff their difference lies in A. Then we have a short exact sequence 0 → A → X → X/A → 0. Taking the duals of this gives us 0 → (X/A)* → X* → A* → 0. The annihilator of A is the subspace A⁀ of X* consisting of those functionals ''f'' such that ⟨''f''\|A⟩ equals 0; that is, it is the subspace of all the functionals ''f'' such that ⟨''f''\|'''a'''⟩ equals 0 for every '''a''' in A. There is a natural isomorphism between the annihilator A⁀ of A and the dual of the quotient (X/A), and also between X/A⁀ and A*.

	Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class ['''x'''], ‖['''x''']‖ equals inf {‖'''x''' + '''a'''‖, '''a''' ∈ A}. Algebraically X is (noncanonically) isomorphic to X, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X, defined by setting, over all nonzero '''x''' ∈ X, ‖''f''‖* = sup ~~<~~''f''\|'''x'''~~>~~/‖'''x'''‖. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A), and X/A⁀ is isometrically isomorphic to A*.		Now suppose X is a finite dimensional real normed vector space. A is then also a finite dimensional real normed vector space, inheriting its norm from X, and X/A is a finite dimensional real normed vector space, with a norm given by, for an equivalence class ['''x'''], ‖['''x''']‖ equals inf {‖'''x''' + '''a'''‖, '''a''' ∈ A}. Algebraically X is (noncanonically) isomorphic to X, but in general they are no longer isomorphic as normed spaces. Instead, we have the [[dual norm]] on X, defined by setting, over all nonzero '''x''' ∈ X, ‖''f''‖* = sup ⟨''f''\|'''x'''⟩/‖'''x'''‖. Under the dual norm X* is also a finite dimensional normed vector space, A⁀ is isometrically isomorphic to (X/A), and X/A⁀ is isometrically isomorphic to A*.

	In the situation which concerns us, X is the ''p''-limit interval space of dimension ''n'' under a norm of one half times the ''L''<sub>1</sub> norm, A is a subspace of dimension ''n'' - 1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {''kJ''}, where ''J'' is the [[JIP]], as its annihilator. X has a norm of half the ''L''<sub>1</sub> norm, and hence X* has a norm of twice the ''L''<sub>∞</sub> norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {‖''f'' + ''kJ''‖} occurs for the value of ''k'' where the maximum of ''f'' + ''kJ'' and minus the minimum of ''f'' + ''kj'' are the same. In that case, 2‖''f'' + ''kJ''‖<sub>~~''L''~~∞</sub> = span(''f''), which is the generator complexity of ''f''. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.		In the situation which concerns us, X is the ''p''-limit interval space of dimension ''n'' under a norm of one half times the ''L''<sub>1</sub> norm, A is a subspace of dimension ''n'' - 1, whose coordinates sum to 0; hence A can be described as having the one-dimensional subspace A⁀ = {''kJ''}, where ''J'' is the [[JIP]], as its annihilator. X has a norm of half the ''L''<sub>1</sub> norm, and hence X* has a norm of twice the ''L''<sub>∞</sub> norm. The norm on A* is defined by its isomorphism with X*/A⁀; the minimum defining inf {‖''f'' + ''kJ''‖} occurs for the value of ''k'' where the maximum of ''f'' + ''kJ'' and minus the minimum of ''f'' + ''kj'' are the same. In that case, 2‖''f'' + ''kJ''‖<sub>∞</sub> = span(''f''), which is the generator complexity of ''f''. Hence generator complexity is the dual norm for Kees expressibility as a norm on pitch classes.

	== STD complexity ==		== STD complexity ==
	If ''B'' = {{val\| 0 ~~B₃ B₅ B₇ ... B~~<sub>''p''</sub> }} is the generator mapping val in weighted coordinates, and ''P'' is the period, then the '''STD complexity''' (a term due to Graham Breed) is ''P'' ~~STD~~(''B''), where "STD" means the standard deviation. If ''μ''(''V'') is the mean of the components of the vector ''V'', and ''J'' is the [[JIP]] {{val\| 1 1 1 … 1 }}, then ''₱''(''V'') = ''V'' - ''μ''(''V'')''J'' is the projection of ''V'' onto the subspace of vectors with zero mean value. We have STD(''V'') = √( ''₱''(''V'') ∙ ''₱''(''V'') / dim(''V'')), where dim(''V'') is the dimension of ''V'' and the " ∙ " denotes the dot product. If ''M'' = [''M''0, ''M''1] is the [[Temperament mapping matrices\|mapping matrix]] in weighted coordinates in the standard [[Normal lists #Normal val lists\|normal val list]] form, then we may express STD complexity as STDcom(''M'') = M0[1] ~~STD~~(M1).		If ''B'' = {{val\| 0 ''b''<sub>3</sub> ''b''<sub>5</sub> ''b''<sub>7</sub> … ''b''<sub>''p''</sub> }} is the generator mapping val in weighted coordinates, and ''P'' is the period, then the '''STD complexity''' (a term due to Graham Breed) is ''P''⋅STD(''B''), where "STD" means the standard deviation. If ''μ''(''V'') is the mean of the components of the vector ''V'', and ''J'' is the [[JIP]] {{val\| 1 1 1 … 1 }}, then ''₱''(''V'') = ''V'' - ''μ''(''V'')''J'' is the projection of ''V'' onto the subspace of vectors with zero mean value. We have STD(''V'') = sqrt (''₱''(''V'')∙''₱''(''V'') / dim(''V'')), where dim(''V'') is the dimension of ''V'' and the "⋅" denotes the dot product. If ''M'' = [''M''<sub>0</sub>, ''M''<sub>1</sub>]<sup>T</sup> is the [[Temperament mapping matrices\|mapping matrix]] in weighted coordinates in the standard [[Normal lists #Normal val lists\|normal val list]] form, then we may express STD complexity as STDcom(''M'') = ''M''<sub>0</sub>[1]⋅STD(''M''<sub>1</sub>).

	Associated to STD complexity is STD error. If ''S'' = ''₱''(''M''0) ∧ ₱(''M''1), then STDerr(''M'') = √(''S'' ∙ ''S'' / dim(''M''1)*₱(''M''1) ~~∙ ₱~~(''M''1)).		Associated to STD complexity is STD error. If ''S'' = ''₱''(''M''<sub>0</sub>) ∧ ₱(''M''<sub>1</sub>), then STDerr(''M'') = sqrt(''S''∙''S'' / dim(''M''<sub>1</sub>)⋅''₱''(''M''<sub>1</sub>)∙₱(''M''<sub>1</sub>)).

	[[Category:Generator]]		[[Category:Generator]]

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