Comma-based lattices - Revision history

Sintel: -legacy

2025-03-29T17:57:23Z

-legacy

← Older revision		Revision as of 17:57, 29 March 2025
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	~~{{Legacy}}~~
	When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.		When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

Lériendil at 16:45, 17 February 2025

2025-02-17T16:45:32Z

← Older revision		Revision as of 16:45, 17 February 2025
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	{{Legacy~~\|Comma-based_lattices~~}}		{{Legacy}}
	When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.		When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

Lériendil: deploying new cat

2025-02-17T16:19:12Z

deploying new cat

← Older revision		Revision as of 16:19, 17 February 2025
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			{{Legacy\|Comma-based_lattices}}
	When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.		When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

Lériendil: Changed protection level for "Comma-based lattices" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

2025-02-17T16:03:58Z

Changed protection level for "Comma-based 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)

Xenwolf: remove wiki cat

2021-12-04T12:21:09Z

remove wiki cat

← Older revision		Revision as of 12:21, 4 December 2021
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	[[Category:Lattice]]		[[Category:Lattice]]
	[[Category:What is]]		[[Category:What is]]
	~~[[Category:Wiki]]~~

Xenwolf: minor formal adjustments in wikitext

2021-12-04T12:20:14Z

minor formal adjustments in wikitext

← Older revision		Revision as of 12:20, 4 December 2021
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	The zero planes for ETs tempering out a particular comma form sheaves of planes radiating from that comma’s monzo vector. They appear as lines marking the intersection of their zero planes with the ''n''k = 1 plane, and fall into family groups including:		The zero planes for ETs tempering out a particular comma form sheaves of planes radiating from that comma’s monzo vector. They appear as lines marking the intersection of their zero planes with the ''n''k = 1 plane, and fall into family groups including:

	~~<ul><li>~~meantone temperaments: horizontal lines~~</li><li>~~schismic temperaments: vertical lines~~</li><li>~~diaschismic temperaments: leading diagonals~~</li><li>~~aristoxenean temperaments: trailing diagonals~~</li><li>~~misty temperaments: lines with gradient -2~~</li></ul>~~Other temperament families (such as kleismic) can be plotted as lines radiating from the tempered-out comma. Regular temperaments such as quarter-comma meantone can also be plotted, and the graphic has a number of other nice features.		* meantone temperaments: horizontal lines
			* schismic temperaments: vertical lines
			* diaschismic temperaments: leading diagonals
			* aristoxenean temperaments: trailing diagonals
			* misty temperaments: lines with gradient -2

			Other temperament families (such as kleismic) can be plotted as lines radiating from the tempered-out comma. Regular temperaments such as quarter-comma meantone can also be plotted, and the graphic has a number of other nice features.

	In a given prime limit there is an infinite number of comma basis sets to choose from, and any such set can be transformed in simple ways to generate others. For example, starting from the set described above, any comma in the ''n''k = 1 plane can substitute for the kleisma as the third basis comma.		In a given prime limit there is an infinite number of comma basis sets to choose from, and any such set can be transformed in simple ways to generate others. For example, starting from the set described above, any comma in the ''n''k = 1 plane can substitute for the kleisma as the third basis comma.

	Another basis set, [ \|monzisma> \|raider> \|atom> ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.		Another basis set, [ \|monzisma> \|raider> \|atom> ], turns up the magnification to focus on schismina-sized commas and the associated temperaments.

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:~~comma~~]]		[[Category:Comma]]
	[[Category:~~lattice~~]]		[[Category:Lattice]]
	[[Category:~~what_is~~]]		[[Category:What is]]
	[[Category:~~wiki~~]]		[[Category:Wiki]]

Mike Battaglia: Protected "Comma-based lattices" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:23:19Z

Protected "Comma-based lattices" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:23, 15 November 2021
(No difference)

Inthar at 22:25, 4 April 2021

2021-04-04T22:25:28Z

← Older revision		Revision as of 22:25, 4 April 2021
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	When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a lattice ~~of commas~~ is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.		When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a '''comma-based lattice''' is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

	The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma).xlsx\|spreadsheet ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png\|image]].		The theory behind this technique is set out below, illustrated for the 5-limit but extending in a straightforward way to any prime limit. An example of its application in the 5-limit can be viewed in this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma).xlsx\|spreadsheet ]]and this [[:File:Comma_lattice_(syntonic,_schisma,_kleisma)_3D.png\|image]].

Inthar: unnecessary header

2021-04-04T22:24:58Z

unnecessary header

← Older revision		Revision as of 22:24, 4 April 2021
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	~~= =~~
	When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a lattice of commas is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.		When plotted on the standard tonal lattice (in which the basis intervals have prime number frequency ratios up to some prime limit p) commas form a widely scattered cloud in which no obvious structure is discernible. But rebasing to a lattice in which the basis intervals are themselves of comma size has the effect of drawing a set of similar-sized commas into the region near the origin, where their interrelationships become apparent. The dual of such a lattice of commas is a lattice of equal temperaments (ETs), which provides a means of visualising the relationships between ETs and commas.

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

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