Projection - Revision history

Inthar: Remove contentious "beginner" template

2025-03-04T15:40:17Z

Remove contentious "beginner" template

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	~~{{Beginner\|Projection matrices}}~~

	A '''projection matrix''', or '''projection''' for short<ref group="note">If an alternative word to "mapping" is sought that does not generically refer to the mathematical structure in the way that "matrix" does, the noun "operator" could be used, according to this: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12787#12787</ref>, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament.<ref group="note">All temperaments discussed on this page hereafter will be assumed to be "regular".</ref>		A '''projection matrix''', or '''projection''' for short<ref group="note">If an alternative word to "mapping" is sought that does not generically refer to the mathematical structure in the way that "matrix" does, the noun "operator" could be used, according to this: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12787#12787</ref>, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament.<ref group="note">All temperaments discussed on this page hereafter will be assumed to be "regular".</ref>

Cmloegcmluin: /* The unchanged-interval basisv */ fix broken links introduced by Arrowhead

2025-02-20T00:33:54Z

The unchanged-interval basisv: fix broken links introduced by Arrowhead

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	An alternative method for finding <math>\textrm{C}</math> is discussed in the "Alternative method for the comma and unchanged-interval bases" section below.		An alternative method for finding <math>\textrm{C}</math> is discussed in the "Alternative method for the comma and unchanged-interval bases" section below.

	=== The unchanged-interval ~~basisv~~===		=== The unchanged-interval basis ===
	The '''unchanged-interval basis''' of a tuning is the [[basis]] for all of its [[unchanged-interval]]s.		The '''unchanged-interval basis''' of a tuning is the [[basis]] for all of its [[unchanged-interval]]s.

VectorGraphics at 21:02, 19 February 2025

2025-02-19T21:02:43Z

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			{{Beginner\|Projection matrices}}

	A '''projection matrix''', or '''projection''' for short<ref group="note">If an alternative word to "mapping" is sought that does not generically refer to the mathematical structure in the way that "matrix" does, the noun "operator" could be used, according to this: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12787#12787</ref>, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament.<ref group="note">All temperaments discussed on this page hereafter will be assumed to be "regular".</ref>		A '''projection matrix''', or '''projection''' for short<ref group="note">If an alternative word to "mapping" is sought that does not generically refer to the mathematical structure in the way that "matrix" does, the noun "operator" could be used, according to this: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12787#12787</ref>, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament.<ref group="note">All temperaments discussed on this page hereafter will be assumed to be "regular".</ref>

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	== See also ==		== See also ==
	* [[~~Fractional monzo~~]]: a more mathematical discussion of these ideas		* [[Projection matrices]]: a more mathematical discussion of these ideas
	* [[Eigenmonzo basis]]: an alternative conceptualization for the unchanged-interval basis		* [[Eigenmonzo basis]]: an alternative conceptualization for the unchanged-interval basis

ArrowHead294: Update links

2024-11-02T01:36:00Z

Update links

ArrowHead294: Markup formatting

2024-11-02T01:32:22Z

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Cmloegcmluin: /* Generator information types */

2024-06-24T19:43:33Z

Generator information types

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	Mappings combine types (1) and (3). Generator embeddings combine types (2) and (3). Projections combine types (1) and (2). So each possible subset of two of these pieces of information is accounted for by these three objects.		Mappings combine types (1) and (3). Generator embeddings combine types (2) and (3). Projections combine types (1) and (2). So each possible subset of two of these pieces of information is accounted for by these three objects.

	One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see: [[~~https://en.xen.wiki/w/Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT~~#~~Pure_representation_of_temperament_information~~]]		One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Pure representation of temperament information]].

	In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the diagram to the right, with type (1) red, type (2) blue, type (3) green.		In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the diagram to the right, with type (1) red, type (2) blue, type (3) green.

Cmloegcmluin: /* The unrotated vectors and scaling factors */

2024-06-24T19:42:31Z

The unrotated vectors and scaling factors

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	===The unrotated vectors and scaling factors===		===The unrotated vectors and scaling factors===

	To obtain the unrotated vectors and scaling factors, we can find these in matrix form via a mathematical process known as "eigendecomposition", which can be handled by most math software. For example, in Wolfram Language, we can perform an eigendecomposition using the function `Eigensystem[]`. Here's an example of quarter-comma meantone:		To obtain the unrotated vectors and scaling factors, we can find these in matrix form via a mathematical process known as "eigendecomposition", which can be handled by most math software. For example, in Wolfram Language, we can perform an eigendecomposition using the function <code>Eigensystem[]</code>. Here's an example of quarter-comma meantone:
	<nowiki>In: P = {{1,1,0},{0,0,0},{0,1/4,1}}		<nowiki>In: P = {{1,1,0},{0,0,0},{0,1/4,1}}
	In: Eigensystem[P] // MatrixForm		In: Eigensystem[P] // MatrixForm

Cmloegcmluin: /* Form matrix */

2024-06-24T19:38:28Z

Form matrix

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	M_{\text{8ave,5th}} \\		M_{\text{8ave,5th}} \\
	\left[ \begin{array} {c}		\left[ \begin{array} {c}
	1 & 0 ~~& -4~~ \\		1 & 1 & 0 \\
	0 & 1 & 4 \\		0 & 1 & 4 \\
	\end{array} \right]		\end{array} \right]
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	F_{\text{8ave,5th}} \\		F_{\text{8ave,5th}} \\
	\left[ \begin{array} {c}		\left[ \begin{array} {c}
	1 & -1 \\		1 & 1 \\
	0 & 1 \\		0 & 1 \\
	\end{array} \right]		\end{array} \right]
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	M \\		M \\
	\left[ \begin{array} {c}		\left[ \begin{array} {c}
	1 & 1 & 0 \\		1 & 0 & -4 \\
	0 & 1 & 4 \\		0 & 1 & 4 \\
	\end{array} \right]		\end{array} \right]
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	\end{array}		\end{array}
	</math>		</math>


	==Units==		==Units==

Cmloegcmluin: /* With respect to the generator tuning map */

2024-06-24T19:34:27Z

With respect to the generator tuning map

← Older revision		Revision as of 19:34, 24 June 2024
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	=== With respect to the generator tuning map===		=== With respect to the generator tuning map===

	A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map\|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map\|1.000 1.585 2.232}}·{{ket\|{{map\|1 1 0}} {{map\|0 0 0}} {{map\|0 1/4 1}}}} = {{map\|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map\|log₂2 log₂3 log₂5}} ≈ {{map\|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map\|1.000 1.585 2.232}}·{{rbra\|{{vector\|1 0 0}} {{vector\|0 0 1/4}}}} = {{map\|1.000 0.580}}, or in cents instead of octaves, that's {{~~map~~\|1200.000 696.578}}.		A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map\|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map\|1.000 1.585 2.232}}·{{ket\|{{map\|1 1 0}} {{map\|0 0 0}} {{map\|0 1/4 1}}}} = {{map\|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map\|log₂2 log₂3 log₂5}} ≈ {{map\|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map\|1.000 1.585 2.232}}·{{rbra\|{{vector\|1 0 0}} {{vector\|0 0 1/4}}}} = {{map\|1.000 0.580}}, or in cents instead of octaves, that's {{rbra\|1200.000 696.578}}.

	Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.		Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.

Cmloegcmluin at 19:31, 24 June 2024

2024-06-24T19:31:03Z

← Older revision		Revision as of 19:31, 24 June 2024
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	Thus <math>\frac65</math> is an unchanged-interval of third-comma meantone, instead of <math>\frac54</math>. The reader can verify that <math>\frac21</math> is still an unchanged-interval.		Thus <math>\frac65</math> is an unchanged-interval of third-comma meantone, instead of <math>\frac54</math>. The reader can verify that <math>\frac21</math> is still an unchanged-interval.

	Nearly two-dozen further examples of generator embeddings may be found throughout the article [[~~Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation~~]], including some with tempered octaves.		Nearly two-dozen further examples of generator embeddings may be found throughout the article [[Generator embedding optimization]], including some with tempered octaves.

	==Generator embedding==		==Generator embedding==

← Older revision		Revision as of 01:36, 2 November 2024
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	=== With respect to the generator tuning map ===		=== With respect to the generator tuning map ===
	A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map\|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref group="note">Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map\|1.000 1.585 2.232}}·{{ket\|{{map\|1 1 0}} {{map\|0 0 0}} {{map\|0 1/4 1}}}} = {{map\|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map\|log₂2 log₂3 log₂5}} ≈ {{map\|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map\|1.000 1.585 2.232}}·{{rbra\|{{vector\|1 0 0}} {{vector\|0 0 1/4}}}} = {{map\|1.000 0.580}}, or in cents instead of octaves, that's {{rbra\|1200.000 696.578}}.		A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map\|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref group="note">Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map\|1.000 1.585 2.232}}·{{ket\|{{map\|1 1 0}} {{map\|0 0 0}} {{map\|0 1/4 1}}}} = {{map\|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map\|log₂2 log₂3 log₂5}} ≈ {{map\|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map\|1.000 1.585 2.232}}·{{rbra\|{{vector\|1 0 0}} {{vector\|0 0 1/4}}}} = {{map\|1.000 0.580}}, or in cents instead of octaves, that's {{rbra\|1200.000 696.578}}.

	Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.		Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.
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	The subscripts indicate which primes and which generators are related. So the columns, as previously stated, correspond to the two generators of the temperament, g₁ and g₂, while the rows correspond to the three primes for this temperament, p₁, p₂, and p₃, which are primes 2, 3, and 5, respectively.		The subscripts indicate which primes and which generators are related. So the columns, as previously stated, correspond to the two generators of the temperament, g₁ and g₂, while the rows correspond to the three primes for this temperament, p₁, p₂, and p₃, which are primes 2, 3, and 5, respectively.

	See also [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis]], and/or the [[#Units\|Units]] section later in this article for more details.		See also [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]], and/or the [[#Units\|Units]] section later in this article for more details.

	== Uniqueness ==		== Uniqueness ==
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	=== Keeping the mapping and generator embedding in sync ===		=== Keeping the mapping and generator embedding in sync ===
	The way to transform from one mapping form <math>M_1</math> to another equivalent mapping form <math>M_2</math> is to perform elementary row operations, the most common of which is to add some multiple of one row to another (or subtract some multiple of one row from another). For more information on this, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: mappings~~#Adding and subtracting rows\|the detailed explanation here]]. Similarly, we can transform from one generator embedding <math>G_1</math> to another equivalent generator embedding <math>G_2</math> by performing elementary ''column'' operations.		The way to transform from one mapping form <math>M_1</math> to another equivalent mapping form <math>M_2</math> is to perform elementary row operations, the most common of which is to add some multiple of one row to another (or subtract some multiple of one row from another). For more information on this, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Mappings#Adding and subtracting rows\|the detailed explanation here]]. Similarly, we can transform from one generator embedding <math>G_1</math> to another equivalent generator embedding <math>G_2</math> by performing elementary ''column'' operations.

	Supposing one desires to transform from a pair of <math>M_1</math> and <math>G_1</math> to another pair of <math>M_2</math> and <math>G_2</math> where both pairs multiply to the same <math>P</math>, or—said another way—you wish to keep your <math>M</math> and <math>G</math> ''in sync'', the simplest approach would be to—for each elementary row operation you apply to <math>M</math> you must apply the opposite elementary column operation to <math>G</math>, e.g. if you add three times the second row to the first row of <math>M</math>, then you must ''subtract'' three times the second ''column'' from the first ''column'' of <math>G</math>. This is along the same lines as the explanations provided for manipulating generator form by changing forms of <math>M</math>, which you can find here: [[Generator form manipulation]].		Supposing one desires to transform from a pair of <math>M_1</math> and <math>G_1</math> to another pair of <math>M_2</math> and <math>G_2</math> where both pairs multiply to the same <math>P</math>, or—said another way—you wish to keep your <math>M</math> and <math>G</math> ''in sync'', the simplest approach would be to—for each elementary row operation you apply to <math>M</math> you must apply the opposite elementary column operation to <math>G</math>, e.g. if you add three times the second row to the first row of <math>M</math>, then you must ''subtract'' three times the second ''column'' from the first ''column'' of <math>G</math>. This is along the same lines as the explanations provided for manipulating generator form by changing forms of <math>M</math>, which you can find here: [[Generator form manipulation]].
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	In the first several of the following subsections, we examine units-only analyses of some RTT objects; for simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis#Units-only analyses]]. Then the last few sections are more general units analyses.		In the first several of the following subsections, we examine units-only analyses of some RTT objects; for simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Units-only analyses]]. Then the last few sections are more general units analyses.

	=== Generator embedding, mapping: Projection matrix ===		=== Generator embedding, mapping: Projection matrix ===
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	=== The JI mapping times the JI generator embedding ===		=== The JI mapping times the JI generator embedding ===
	This situation is a variation upon the situation [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis#Cancelation\|described here]], where the projection matrix <math>P</math> is derived as the matrix product of the generator embedding matrix <math>G</math> and the temperament mapping matrix <math>M</math>. What we're going to do here is work through the variation where it's the ''JI'' embedding matrix <math>G_{\text{j}}</math> and the ''JI'' mapping matrix <math>M_{\text{j}}</math>.		This situation is a variation upon the situation [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Cancelation\|described here]], where the projection matrix <math>P</math> is derived as the matrix product of the generator embedding matrix <math>G</math> and the temperament mapping matrix <math>M</math>. What we're going to do here is work through the variation where it's the ''JI'' embedding matrix <math>G_{\text{j}}</math> and the ''JI'' mapping matrix <math>M_{\text{j}}</math>.

	The key difference here is that both of these matrices are [[identity matrix\|identity matrices]]. Thus, upon multiplying them, the result is ''also'' an identity matrix.		The key difference here is that both of these matrices are [[identity matrix\|identity matrices]]. Thus, upon multiplying them, the result is ''also'' an identity matrix.
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	=== The comma basis ===		=== The comma basis ===
	To obtain the comma basis from <math>P</math>, simply take the nullspace as you would take it of the mapping (see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: exploring~~ temperaments#Nullspace]] for more information).		To obtain the comma basis from <math>P</math>, simply take the nullspace as you would take it of the mapping (see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace]] for more information).

	Remember, the mapping represents the temperament, and the projection represents a particular tuning of this temperament, so no matter which projection we use, while they will each have their own unchanged-intervals, they will share the same commas: the commas of the temperament.		Remember, the mapping represents the temperament, and the projection represents a particular tuning of this temperament, so no matter which projection we use, while they will each have their own unchanged-intervals, they will share the same commas: the commas of the temperament.
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	=== The mapping ===		=== The mapping ===
	To obtain (some form of) the mapping from a projection, find its comma basis per the above, then take the nullspace of that comma basis to get the mapping. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: exploring~~ temperaments#Nullspace]].		To obtain (some form of) the mapping from a projection, find its comma basis per the above, then take the nullspace of that comma basis to get the mapping. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace]].


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	For a simpler take on this idea which doesn't involve the projection matrix, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: tuning~~ computation#Solving for the generators]].		For a simpler take on this idea which doesn't involve the projection matrix, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Solving for the generators]].

	=== The unrotated vectors and scaling factors ===		=== The unrotated vectors and scaling factors ===
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	The basic idea is that any two commas' projections are the zero vector, so an indefinite number of these may be combined with each other. And any unchanged-interval's projection is equal to itself, so an indefinite number of these may be combined as well. But any interval that is a combination of some number of unchanged-intervals and some number of commas will have the comma part projected to zero but the unchanged-interval part left alone, and thus be rotated by the projection. In other words, only intervals with the same scaling factor can be combined and still remain unrotated, which is to say that the commas form one type of unrotated vector basis and the unchanged-intervals form another type of unrotated vector basis, but these bases do not combine into one basis together.		The basic idea is that any two commas' projections are the zero vector, so an indefinite number of these may be combined with each other. And any unchanged-interval's projection is equal to itself, so an indefinite number of these may be combined as well. But any interval that is a combination of some number of unchanged-intervals and some number of commas will have the comma part projected to zero but the unchanged-interval part left alone, and thus be rotated by the projection. In other words, only intervals with the same scaling factor can be combined and still remain unrotated, which is to say that the commas form one type of unrotated vector basis and the unchanged-intervals form another type of unrotated vector basis, but these bases do not combine into one basis together.

	That said, <math>V</math> must still be full-rank, meaning there is some relationship between the comma vectors and the unchanged-interval vectors. Specifically, it means that no unchanged-interval can be a linear combination of other unchanged-intervals and commas. If any were, we'd find an impossible situation, such as two unchanged-intervals off by a comma, so neither interval can change, yet they must both project to the same thing. For another take on this idea, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: tuning~~ computation#Edge cases]].		That said, <math>V</math> must still be full-rank, meaning there is some relationship between the comma vectors and the unchanged-interval vectors. Specifically, it means that no unchanged-interval can be a linear combination of other unchanged-intervals and commas. If any were, we'd find an impossible situation, such as two unchanged-intervals off by a comma, so neither interval can change, yet they must both project to the same thing. For another take on this idea, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Edge cases]].

	== Generator information types ==		== Generator information types ==