Trivial temperament - Revision history

FloraC: Use common terms rather than unattested/rare idiosyncratic terms (including "Om temperament"!)

2025-10-06T14:19:59Z

Use common terms rather than unattested/rare idiosyncratic terms (including "Om temperament"!)

← Older revision		Revision as of 14:19, 6 October 2025
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	A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two types of trivial temperaments: ~~identity temperaments (~~which ~~leave~~ all intervals untempered) and ~~Om temperaments (~~which ~~temper~~ out all intervals).		A '''trivial temperament''' is something that fits the mathematical definition of [[regular temperament]], but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two types of trivial temperaments: [[just intonation]], which leaves all intervals [[tempering\|untempered]], and [[single-pitch tuning]], which [[tempering out\|tempers out]] all intervals.

	== ~~Identity temperament~~ ==		== Just intonation ==
	The '''identity temperament''' for a subgroup of rank ''n'', so called because a possible mapping is an ''n''×''n'' identity matrix, transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are [[tempered out]] is {~~1/1~~}, but that's still a valid set, so identity temperaments are still valid regular temperaments. An identity temperament exists for each subgroup of JI, and there is an identity extension for any given temperament.		{{Main\| Just intonation }}

	The ~~2-limit version is the equal temperament~~ [[~~1edo~~]]~~. The [[3-limit]] version is~~ a ~~rank-2 temperament ("~~[[~~pythagorean~~]]~~"), which has all the properties~~ of ~~any other~~ rank~~-2 temperament except~~ that ~~it tempers no commas~~. The ~~5-limit identity temperament~~ is ~~rank-3 ("classical" - though note~~ that ~~this might be confused with [[meantone]]), the 7-limit identity temperament~~ is ~~rank-4 ("septimal")~~, ~~etc~~.		The [[mapping]] for a [[just intonation subgroup]] of rank ''n'' is an ''n''×''n'' {{w\|identity matrix}}, and transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are tempered out is {1/1}, but that is still a valid set, so just intonation still counts as valid regular temperaments.

	~~== Om temperament ==~~		There is an infinite family of these temperaments, one for each subgroup of JI. The 2-limit version is equivalent to [[1edo\|1et]]. The [[3-limit]] version, or [[pythagorean tuning]], is a rank-2 temperament, which has all the properties of any other rank-2 temperament except that it tempers out no commas. 5-limit JI is rank-3, 7-limit JI is rank-4, etc.
	~~'''Om''' temperament~~ is ~~the rank~~-~~0 temperament, in which every interval~~ is ~~a comma. Thus all notes are represented by the same note, leading~~ to [[~~single-pitch tuning~~]]~~. This is different from 1edo because not even octaves exist~~. The ~~mapping for this is the 0~~-~~val~~, ~~{{val\| 0 0 ... 0 }}~~, ~~and its multival~~ is a ~~single entry. It could also be called~~ the ~~''unison~~ temperament~~''<ref>http://www~~.~~robertinventor.com/tuning~~-~~math/s__12/msg_11050~~-~~11074.html</ref>~~, ~~as all intervals are equated to the unison~~.

	~~As with~~ identity ~~temperaments, there is technically an Om~~ temperament for ~~every subgroup~~.		[[User:VectorGraphics\|Vector]] proposes the name ''identity temperament''{{idio}} for this family of temperaments.

	The ~~name "Om"~~ is a ~~reference~~ to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices~~]]; [[Keenan Pepper]] gave it this name because~~ there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		== Single-pitch tuning ==
			{{Main\| Single-pitch tuning }}

			The single-pitch tuning is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, {{val\| 0 0 … 0 }}, and its [[wedgie]] is a single entry.

			As with JI, there is technically a temperament of a single pitch for every subgroup.

			[[Gene Ward Smith]] proposes the name ''unison temperament'' for this family of temperaments<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. [[Keenan Pepper]] proposes the name ''Om temperament''{{idio}}. [[Wikipedia:Om\|''Om'']] is a reference to that syllable's use in Hindu meditation practices, for there is only one temperament-distinct pitch in the whole system, in the same way that ''Om'' in the meditation sense is the only word you need to create the whole universe.

	== Notes and references ==		== Notes and references ==

VectorGraphics at 04:42, 6 October 2025

2025-10-06T04:42:26Z

← Older revision		Revision as of 04:42, 6 October 2025
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	== Om temperament ==		== Om temperament ==
	'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note, leading to [[single-pitch tuning]]. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}, and its multival is a single ~~zero~~. It could also be called the ''unison temperament''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison.		'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note, leading to [[single-pitch tuning]]. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}, and its multival is a single entry. It could also be called the ''unison temperament''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison.

	As with identity temperaments, there is technically an Om temperament for every subgroup.		As with identity temperaments, there is technically an Om temperament for every subgroup.

VectorGraphics at 04:41, 6 October 2025

2025-10-06T04:41:21Z

← Older revision		Revision as of 04:41, 6 October 2025
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	A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two ~~kinds~~ of trivial temperaments ~~- [[JI]], in~~ which ~~nothing is tempered,~~ and ~~'''~~Om~~''' temperament, in~~ which ~~everything is tempered~~.		A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two types of trivial temperaments: identity temperaments (which leave all intervals untempered) and Om temperaments (which temper out all intervals).

	~~Just intonation~~ is ~~a codimension-0 "temperament"~~, ~~which~~ means nothing is tempered. The set of commas that are ~~made to~~ [[~~vanish~~]] is ~~the set~~ {1/1}, but that's still a set, so ~~JI is~~ still a regular temperament~~. There is an infinite family of these "temperaments", one~~ for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is a rank-2 temperament ("[[pythagorean]]"), which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3 ("classical"), and ~~so on. The mapping for this temperament~~ is an ~~''n''×''n''~~ identity ~~matrix~~.		== Identity temperament ==
			The '''identity temperament''' for a subgroup of rank ''n'', so called because a possible mapping is an ''n''×''n'' identity matrix, transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are [[tempered out]] is {1/1}, but that's still a valid set, so identity temperaments are still valid regular temperaments. An identity temperament exists for each subgroup of JI, and there is an identity extension for any given temperament.

	[[~~Single~~-~~pitch tuning\|~~'''Om''' temperament]] is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist~~; it could be described as 0edo~~. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		The 2-limit version is the equal temperament [[1edo]]. The [[3-limit]] version is a rank-2 temperament ("[[pythagorean]]"), which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit identity temperament is rank-3 ("classical" - though note that this might be confused with [[meantone]]), the 7-limit identity temperament is rank-4 ("septimal"), etc.

			== Om temperament ==
			'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note, leading to [[single-pitch tuning]]. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}, and its multival is a single zero. It could also be called the ''unison temperament''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison.

			As with identity temperaments, there is technically an Om temperament for every subgroup.

			The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

	== Notes and references ==		== Notes and references ==

VectorGraphics at 04:38, 12 June 2025

2025-06-12T04:38:33Z

← Older revision		Revision as of 04:38, 12 June 2025
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	A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.		A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.

	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is a rank-2 temperament ("[[pythagorean]]"), which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3 ("classical"), and so on. The mapping for this temperament is an ''n''×''n'' identity matrix~~, with wedgies of {{val\| 1 }}, {{multival\|rank=2\| 1 }}, {{multival\|rank=3\| 1 }}, etc~~.		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is a rank-2 temperament ("[[pythagorean]]"), which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3 ("classical"), and so on. The mapping for this temperament is an ''n''×''n'' identity matrix.

	[[Single-pitch tuning\|'''Om''' temperament]] is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as 0edo. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		[[Single-pitch tuning\|'''Om''' temperament]] is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as 0edo. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

VectorGraphics at 20:45, 14 April 2025

2025-04-14T20:45:23Z

← Older revision		Revision as of 20:45, 14 April 2025
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	A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.		A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.

	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is ~~the~~ rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an ''n''×''n'' identity matrix, with wedgies of {{val\| 1 }}, {{multival\|rank=2\| 1 }}, {{multival\|rank=3\| 1 }}, etc.		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is a rank-2 temperament ("[[pythagorean]]"), which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3 ("classical"), and so on. The mapping for this temperament is an ''n''×''n'' identity matrix, with wedgies of {{val\| 1 }}, {{multival\|rank=2\| 1 }}, {{multival\|rank=3\| 1 }}, etc.

	[[Single-pitch tuning\|'''Om''' temperament]] is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as 0edo. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		[[Single-pitch tuning\|'''Om''' temperament]] is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as 0edo. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

VectorGraphics at 06:15, 13 February 2025

2025-02-13T06:15:55Z

← Older revision		Revision as of 06:15, 13 February 2025
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	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an ''n''×''n'' identity matrix, with wedgies of {{val\| 1 }}, {{multival\|rank=2\| 1 }}, {{multival\|rank=3\| 1 }}, etc.		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an ''n''×''n'' identity matrix, with wedgies of {{val\| 1 }}, {{multival\|rank=2\| 1 }}, {{multival\|rank=3\| 1 }}, etc.

	'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		[[Single-pitch tuning\|'''Om''' temperament]] is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as 0edo. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

	== Notes and references ==		== Notes and references ==

ArrowHead294 at 14:56, 4 October 2024

2024-10-04T14:56:53Z

← Older revision		Revision as of 14:56, 4 October 2024
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	A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.		A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.

	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an ~~nxn~~ identity matrix, with wedgies of ~~<~~1\|, ~~<<1~~\|\|, ~~<<<1~~\|\|~~\|...~~ .		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an ''n''×''n'' identity matrix, with wedgies of {{val\| 1 }}, {{multival\|rank=2\| 1 }}, {{multival\|rank=3\| 1 }}, etc.

	'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, ~~<~~0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, {{val\| 0 0 ... 0 }}. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

			== Notes and references ==
			<references />

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Jerdle: Removed false information about tempering out the unison - every EDO does that

2023-05-30T19:23:45Z

Removed false information about tempering out the unison - every EDO does that

← Older revision		Revision as of 19:23, 30 May 2023
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	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1\|, <<1\|\|, <<<1\|\|\|... .		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1\|, <<1\|\|, <<<1\|\|\|... .

	'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, <0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, ~~following the common pattern whereby temperaments~~ are ~~named after the intervals they [[temper out]], where in this case the interval made~~ to ~~vanish is~~ the ~~[[unison]] (and therefore all intervals are brought together in~~ unison). The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, <0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Cmloegcmluin: this temperament is already named "om temperament"; it does not need another arbitrary name (the additional "unison temperament" name suggested here is simply an acknowledgment of the well-established naming pattern whereby temperaments can be named after any existing well-established names for the comma/interval they temper out, e.g. such that "meantone temperament" might also be called "syntonic temperament")

2023-05-23T17:05:03Z

this temperament is already named "om temperament"; it does not need another arbitrary name (the additional "unison temperament" name suggested here is simply an acknowledgment of the well-established naming pattern whereby temperaments can be named after any existing well-established names for the comma/interval they temper out, e.g. such that "meantone temperament" might also be called "syntonic temperament")

← Older revision		Revision as of 17:05, 23 May 2023
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	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1\|, <<1\|\|, <<<1\|\|\|... .		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1\|, <<1\|\|, <<<1\|\|\|... .

	'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, <0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval made to vanish is the [[unison]] (and therefore all intervals are brought together in unison)~~. Because the question of whether the unison can vanish is like a Zen koan, it It could also be called the '''zen temperament'''~~. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, <0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval made to vanish is the [[unison]] (and therefore all intervals are brought together in unison). The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Moremajorthanmajor at 02:39, 20 May 2023

2023-05-20T02:39:16Z

← Older revision		Revision as of 02:39, 20 May 2023
Line 3:		Line 3:
	Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1\|, <<1\|\|, <<<1\|\|\|... .		Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of <1\|, <<1\|\|, <<<1\|\|\|... .

	'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, <0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval made to vanish is the [[unison]] (and therefore all intervals are brought together in unison). The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.		'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist; it could be described as [[0edo]]. The mapping for this is the 0-val, <0 0 ... 0\|. It could also be called the '''unison temperament'''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval made to vanish is the [[unison]] (and therefore all intervals are brought together in unison). Because the question of whether the unison can vanish is like a Zen koan, it It could also be called the '''zen temperament'''. The name "Om" is a reference to [[Wikipedia:Om\|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]