7-limit symmetrical lattices - Revision history

Overthink: fix todo

2026-01-03T20:08:09Z

fix todo

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	== Norms and lattices ==		== Norms and lattices ==

Overthink: + todo

2025-12-27T12:13:38Z

+ todo

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	== Norms and lattices ==		== Norms and lattices ==

VectorGraphics at 01:48, 13 November 2025

2025-11-13T01:48:34Z

← Older revision		Revision as of 01:48, 13 November 2025
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	$$		$$

	If ''T'' is the ~~[[vals and tuning space\|val]]~~ {{nowrap\| ~~''T'' {{=}}~~ {{val\| 1 1 1 1 }} }} (~~note that this~~ is ~~''not'' the~~ [[~~just tuning map~~]]) then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all ~~vectors ''M'' in interval space such that~~ {{nowrap\| {{vmp\|''T''\|''M''}} {{=}} 0 }}, ~~which~~ has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.		If ''T'' is the equal temperament {{nowrap\|{{val\| 1 1 1 1 }} }} which maps 2, 3, 5, and 7 all to a single step (and is thus a val of [[1edo]], [[1edt]], etc), then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all intervals where {{nowrap\| {{vmp\|''T''\|''M''}} {{=}} 0 }} - or in other words, the number of primes (including repetition) in the numerator and denominator is the same. This has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space (because only one power of two in the numerator or denominator satisfies the criterion for any odd fraction), and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.

	If {{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }} is any element of symmetric interval class space, then by definition {{nowrap\|‖{{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }}‖}} = {{nowrap\|sqrt(2) sqrt(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''xy'' + ''yz'' + ''zx'')}} where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by {{monzo\| <nowiki>*</nowiki> ''x'' ''y'' ''z'' }}.		If {{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }} is any element of symmetric interval class space, then by definition {{nowrap\|‖{{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }}‖}} = {{nowrap\|sqrt(2) sqrt(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''xy'' + ''yz'' + ''zx'')}} where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by {{monzo\| <nowiki>*</nowiki> ''x'' ''y'' ''z'' }}.

FloraC: FloraC moved page The Seven Limit Symmetrical Lattices to 7-limit symmetrical lattices: For obvious reasons

2025-09-03T13:31:53Z

FloraC moved page The Seven Limit Symmetrical Lattices to 7-limit symmetrical lattices: For obvious reasons

← Older revision	Revision as of 13:31, 3 September 2025
(No difference)

Sintel: -legacy, todo intro

2025-03-29T18:12:39Z

-legacy, todo intro

← Older revision		Revision as of 18:12, 29 March 2025
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	{{~~Legacy~~}}		{{todo\|intro\|inline=1}}
	__FORCETOC__		__FORCETOC__
	== Norms and lattices ==		== Norms and lattices ==

ArrowHead294 at 13:45, 26 February 2025

2025-02-26T13:45:54Z

← Older revision		Revision as of 13:45, 26 February 2025
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	$$		$$

	If ''T'' is the [[vals and tuning space\|val]] {{nowrap\| ''T'' {{=}} {{val\| 1 1 1 1 }} }} (note that this is ''not'' the [[just tuning map]]) then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all vectors ''M'' in interval space such that {{nowrap\| {{~~vmprod~~\|''T''\|''M''}} {{=}} 0 }}, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.		If ''T'' is the [[vals and tuning space\|val]] {{nowrap\| ''T'' {{=}} {{val\| 1 1 1 1 }} }} (note that this is ''not'' the [[just tuning map]]) then we may define a subspace of 7-limit interval space, '''symmetric interval class space''', as the subspace of all vectors ''M'' in interval space such that {{nowrap\| {{vmp\|''T''\|''M''}} {{=}} 0 }}, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.

	If {{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }} is any element of symmetric interval class space, then by definition {{nowrap\|‖{{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }}‖}} = {{nowrap\|sqrt(2) sqrt(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''xy'' + ''yz'' + ''zx'')}} where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by {{monzo\| <nowiki>*</nowiki> ''x'' ''y'' ''z'' }}.		If {{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }} is any element of symmetric interval class space, then by definition {{nowrap\|‖{{monzo\| -''x''-''y''-''z'' ''x'' ''y'' ''z'' }}‖}} = {{nowrap\|sqrt(2) sqrt(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''xy'' + ''yz'' + ''zx'')}} where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by {{monzo\| <nowiki>*</nowiki> ''x'' ''y'' ''z'' }}.

FloraC: Cleanup

2025-02-18T10:43:26Z

Cleanup

Show changes

Lériendil at 16:46, 17 February 2025

2025-02-17T16:46:23Z

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	{{Legacy~~\|The_Seven_Limit_Symmetrical_Lattices~~}}		{{Legacy}}
	__FORCETOC__		__FORCETOC__
	=Norms and lattices=		=Norms and lattices=

Lériendil: deploying new cat

2025-02-17T16:17:44Z

deploying new cat

← Older revision		Revision as of 16:17, 17 February 2025
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			{{Legacy\|The_Seven_Limit_Symmetrical_Lattices}}
	__FORCETOC__		__FORCETOC__
	=Norms and lattices=		=Norms and lattices=

Lériendil: Changed protection level for "The Seven Limit Symmetrical Lattices" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

2025-02-17T16:03:39Z

Changed protection level for "The Seven Limit Symmetrical Lattices" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:03, 17 February 2025
(No difference)