Rank-3 temperament - Revision history

FloraC: Improve stuff

2026-06-08T11:29:57Z

Improve stuff

← Older revision		Revision as of 11:29, 8 June 2026
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	{{See also\| Table of rank-3 temperament lattice bases }}		{{See also\| Table of rank-3 temperament lattice bases }}

	The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the [[comma]]s of the temperament. To do this we need a Euclidean metric on the space in which ''p''-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[~~Euclidean~~ interval space]].		The most elegant way to put a Euclidean metric, and hence a normed lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the [[comma]]s of the temperament. To do this we need a Euclidean metric on the space in which ''p''-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Tenney–Euclidean metrics\|Tenney–Euclidean interval space]].

	For example, 7-limit [[marvel]] temperament is defined by [[tempering out]] a single comma, [[225/224]]. If we convert that to a weighted [[monzo]] '''m''' = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "'''t'''", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix {{nowrap\| ''U'' {{=}} ['''t''', '''m'''] }}. If ''U''<sup>+</sup> denotes the [[pseudoinverse]] of ''U'', then letting {{nowrap\| ''Q'' {{=}} ''U''<sup>+</sup>''U'' }} take {{nowrap\| ''P'' {{=}} ''I'' - ''Q'' }}, where ''I'' is the identity matrix. ''P'' is the [[projection matrix]] that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that '''m'''''P'' and '''t'''''P'' are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the [[secor]], 16/15 and 15/14, and the second closest independent point the [[3/2\|fifth]] (or alternatively, fourth). The secor and the fifth give a [[Minkowski basis]] for the lattice, but we could also use the [[5/4\|major third]] and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.		For example, 7-limit [[marvel]] temperament is defined by [[tempering out]] a single comma, [[225/224]]. If we convert that to a weighted [[monzo]] '''m''' = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "'''t'''", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix {{nowrap\| ''U'' {{=}} ['''t''', '''m'''] }}. If ''U''<sup>+</sup> denotes the [[pseudoinverse]] of ''U'', then letting {{nowrap\| ''Q'' {{=}} ''U''<sup>+</sup>''U'' }} take {{nowrap\| ''P'' {{=}} ''I'' - ''Q'' }}, where ''I'' is the identity matrix. ''P'' is the [[projection matrix]] that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that '''m'''⋅''P'' and '''t'''⋅''P'' are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the [[secor]], 16/15 and 15/14, and the second closest independent point the [[3/2\|perfect fifth]] (or alternatively, perfect fourth). The secor and the perfect fifth give a [[Minkowski basis]] for the lattice, but we could also use the [[5/4\|classical major third]] and perfect fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.

	If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.		If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.

	=== ~~Projection pair~~ ===		=== Other tools ===
	~~{{Main\|~~ Projection pair }}		* [[Projection pair]]
	~~{{Todo\|inline=1\|complete section}}~~		* [[Complexity spectrum]]

	~~=== Complexity spectrum ===~~
	~~{{Main\|~~ Complexity spectrum }}
	~~{{Todo\|inline=1\|complete section}}~~

	== See also ==		== See also ==

FloraC: + projection pair and complexity spectrum

2026-06-07T07:11:00Z

+ projection pair and complexity spectrum

← Older revision		Revision as of 07:11, 7 June 2026
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	See [[Tour of regular temperaments #Rank-3 temperaments]] for a list of families, clans, and collections of rank-3 temperaments.		See [[Tour of regular temperaments #Rank-3 temperaments]] for a list of families, clans, and collections of rank-3 temperaments.

	== Euclidean metric on the lattice ==		== Analytical tools for rank-3 temperaments ==
			=== Euclidean metric on the lattice ===
			{{See also\| Table of rank-3 temperament lattice bases }}

	The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the [[comma]]s of the temperament. To do this we need a Euclidean metric on the space in which ''p''-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].		The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the [[comma]]s of the temperament. To do this we need a Euclidean metric on the space in which ''p''-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].

	~~=== Example ===~~		For example, 7-limit [[marvel]] temperament is defined by [[tempering out]] a single comma, [[225/224]]. If we convert that to a weighted [[monzo]] '''m''' = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "'''t'''", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix {{nowrap\| ''U'' {{=}} ['''t''', '''m'''] }}. If ''U''<sup>+</sup> denotes the [[pseudoinverse]] of ''U'', then letting {{nowrap\| ''Q'' {{=}} ''U''<sup>+</sup>''U'' }} take {{nowrap\| ''P'' {{=}} ''I'' - ''Q'' }}, where ''I'' is the identity matrix. ''P'' is the [[projection matrix]] that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that '''m'''''P'' and '''t'''''P'' are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the [[secor]], 16/15 and 15/14, and the second closest independent point the [[3/2\|fifth]] (or alternatively, fourth). The secor and the fifth give a [[Minkowski basis]] for the lattice, but we could also use the [[5/4\|major third]] and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.
	7-limit [[marvel]] temperament is defined by [[tempering out]] a single comma, [[225/224]]. If we convert that to a weighted [[monzo]] '''m''' = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "'''t'''", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix {{nowrap\| ''U'' {{=}} ['''t''', '''m'''] }}. If ''U''<sup>+</sup> denotes the [[pseudoinverse]] of ''U'', then letting {{nowrap\| ''Q'' {{=}} ''U''<sup>+</sup>''U'' }} take {{nowrap\| ''P'' {{=}} ''I'' - ''Q'' }}, where ''I'' is the identity matrix. ''P'' is the [[projection matrix]] that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that '''m'''''P'' and '''t'''''P'' are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the [[secor]], 16/15 and 15/14, and the second closest independent point the [[3/2\|fifth]] (or alternatively, fourth). The secor and the fifth give a [[Minkowski basis]] for the lattice, but we could also use the [[5/4\|major third]] and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.

	If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.		If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.

			=== Projection pair ===
			{{Main\| Projection pair }}
			{{Todo\|inline=1\|complete section}}

			=== Complexity spectrum ===
			{{Main\| Complexity spectrum }}
			{{Todo\|inline=1\|complete section}}

	== See also ==		== See also ==

FloraC: Replace the outdated list with links to more comprehensive lists

2025-11-25T14:10:43Z

Replace the outdated list with links to more comprehensive lists

@@ Line 1: / Line 1: @@
 A '''rank-3 temperament''' is a [[regular temperament]] with three [[generator]]s. If one of the generators can be an [[2/1|octave]], it is called a '''planar temperament''', though the word is sometimes applied to any rank-3 temperament. There are two interpretations for the name ''planar temperament'': first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of three or more dimensions.
 == Euclidean metric on the lattice ==
@@ Line 9: / Line 11: @@
 If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.
-== List of rank-3 temperament families and clans ==
+== See also ==
-=== Planar temperaments ===
+* [[:Category: Rank-3 temperaments]]
 == External links ==

FloraC: Fix linking

2025-08-27T13:34:05Z

Fix linking

← Older revision		Revision as of 13:34, 27 August 2025
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	* [[Hemimean family]]		* [[Hemimean family]]
	* [[Archytas family]]		* [[Archytas family]]
	* [[Kleismic rank ~~three~~ family]]		* [[Kleismic rank-3 family]]

	=== Rank-3 but not planar ===		=== Rank-3 but not planar ===

FloraC: Cleanup

2025-06-12T10:01:57Z

Cleanup

← Older revision		Revision as of 10:01, 12 June 2025
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	A '''rank ~~three~~ temperament''' is a [[regular temperament]] with three ~~generators~~. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank ~~three~~ temperament. There are two interpretations for the name "planar temperament": first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[~~Wikipedia:Lattice\|~~lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of 3 or ~~higher~~ dimensions. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].		A '''rank-3 temperament''' is a [[regular temperament]] with three [[generator]]s. If one of the generators can be an [[2/1\|octave]], it is called a '''planar temperament''', though the word is sometimes applied to any rank-3 temperament. There are two interpretations for the name ''planar temperament'': first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of three or more dimensions.

	== ~~Example~~ ==		== Euclidean metric on the lattice ==
			The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the [[comma]]s of the temperament. To do this we need a Euclidean metric on the space in which ''p''-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].

	7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted [[monzo]] m = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "t", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix U = [t, m]. If U<sup>+</sup> denotes the [[~~Moore-Penrose~~ pseudoinverse]] of U, then letting Q = U<sup>+</sup>U take P = I - Q, where I is the identity matrix. P is the projection matrix that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.		=== Example ===
			7-limit [[marvel]] temperament is defined by [[tempering out]] a single comma, [[225/224]]. If we convert that to a weighted [[monzo]] '''m''' = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "'''t'''", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix {{nowrap\| ''U'' {{=}} ['''t''', '''m'''] }}. If ''U''<sup>+</sup> denotes the [[pseudoinverse]] of ''U'', then letting {{nowrap\| ''Q'' {{=}} ''U''<sup>+</sup>''U'' }} take {{nowrap\| ''P'' {{=}} ''I'' - ''Q'' }}, where ''I'' is the identity matrix. ''P'' is the [[projection matrix]] that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that '''m'''''P'' and '''t'''''P'' are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the [[secor]], 16/15 and 15/14, and the second closest independent point the [[3/2\|fifth]] (or alternatively, fourth). The secor and the fifth give a [[Minkowski basis]] for the lattice, but we could also use the [[5/4\|major third]] and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.

	If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.		If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.

	~~== Planar temperaments ==~~

			== List of rank-3 temperament families and clans ==
			=== Planar temperaments ===
	* [[Marvel family]]		* [[Marvel family]]
	* [[Starling family]]		* [[Starling family]]
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	* [[Kleismic rank three family]]		* [[Kleismic rank three family]]

	== Rank ~~three~~ but not planar ==		=== Rank-3 but not planar ===

	* [[Jubilismic temperament]]		* [[Jubilismic temperament]]

	~~'' An earlier version of this article may be found at~~ [http://lumma.org/tuning/gws/planar.htm ~~http://lumma.org/tuning/gws/planar.htm]~~ ''		== External links ==
			* [http://lumma.org/tuning/gws/planar.htm Xenharmony \| ''Planar Temperaments'']

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Rank 3\| ]] <!-- main article -->		[[Category:Rank 3\| ]] <!-- main article -->
	[[Category:Math]]		[[Category:Math]]

Akselai at 15:57, 12 February 2024

2024-02-12T15:57:53Z

← Older revision		Revision as of 15:57, 12 February 2024
Line 1:		Line 1:
	A '''rank three temperament''' is a [[regular temperament]] with three generators. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank three temperament. ~~The~~ octave classes of notes of a planar temperament can be embedded in a plane as a [[Wikipedia:Lattice\|lattice]], ~~hence~~ the ~~name~~. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].		A '''rank three temperament''' is a [[regular temperament]] with three generators. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank three temperament. There are two interpretations for the name "planar temperament": first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[Wikipedia:Lattice\|lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of 3 or higher dimensions. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].

	== Example ==		== Example ==

Fredg999 category edits: Moving from Category:Theory to Category:Regular temperament theory using Cat-a-lot

2023-03-05T05:34:04Z

Moving from Category:Theory to Category:Regular temperament theory using Cat-a-lot

← Older revision		Revision as of 05:34, 5 March 2023
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	'' An earlier version of this article may be found at [http://lumma.org/tuning/gws/planar.htm http://lumma.org/tuning/gws/planar.htm] ''		'' An earlier version of this article may be found at [http://lumma.org/tuning/gws/planar.htm http://lumma.org/tuning/gws/planar.htm] ''

	[[Category:~~Theory~~]]		[[Category:Regular temperament theory]]
	[[Category:Rank 3\| ]] <!-- main article -->		[[Category:Rank 3\| ]] <!-- main article -->
	[[Category:Math]]		[[Category:Math]]

Cmloegcmluin: projection map → projection matrix

2021-11-06T20:55:16Z

projection map → projection matrix

← Older revision		Revision as of 20:55, 6 November 2021
Line 3:		Line 3:
	== Example ==		== Example ==

	7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted [[monzo]] m = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "t", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix U = [t, m]. If U<sup>+</sup> denotes the [[Moore-Penrose pseudoinverse]] of U, then letting Q = U<sup>+</sup>U take P = I - Q, where I is the identity matrix. P is the projection ~~map~~ from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.		7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted [[monzo]] m = {{monzo\| -5 3.17 4.64 -2.81 }} and call the weighted monzo {{monzo\| 1 0 0 0 }} for 2 "t", then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2×4 matrix U = [t, m]. If U<sup>+</sup> denotes the [[Moore-Penrose pseudoinverse]] of U, then letting Q = U<sup>+</sup>U take P = I - Q, where I is the identity matrix. P is the projection matrix that maps from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector {{monzo\| 0 0 0 0 }} representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96, and the major third is angled 129.84 to the fifth.

	If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.		If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates.

FloraC: FloraC moved page Planar temperament to Rank-3 temperament: Adopt a more inclusive page title

2021-07-14T11:27:00Z

FloraC moved page Planar temperament to Rank-3 temperament: Adopt a more inclusive page title

← Older revision	Revision as of 11:27, 14 July 2021
(No difference)

Xenllium: /* Planar temperaments */

2021-04-04T11:45:16Z

Planar temperaments

← Older revision		Revision as of 11:45, 4 April 2021
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	* [[Horwell family]]		* [[Horwell family]]
	* [[Sensamagic family]]		* [[Sensamagic family]]
			* [[Sengic family]]
	* [[Keemic family]]		* [[Keemic family]]
	* [[Hemimage family]]		* [[Hemimage family]]