Defactoring algorithms - Revision history

FloraC: Undo revision 203567 by VectorGraphics (talk). Readers aren't obligated to see your poor-tasted operation name

2025-06-25T14:48:47Z

Undo revision 203567 by VectorGraphics (talk). Readers aren't obligated to see your poor-tasted operation name

← Older revision		Revision as of 14:48, 25 June 2025
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	{{Todo~~\|rewrite~~\|inline=1\|text=~~Remove all references to~~ wedgies~~, as per [[User:VectorGraphics/Operation_Loosen_Underpants]]~~}}		{{Todo\|inline=1\| rework \|text=Try explaining without wedgies outside historical notes.}}


	This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring\|defactor]] it.		This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring\|defactor]] it.

VectorGraphics at 04:30, 25 June 2025

2025-06-25T04:30:15Z

← Older revision		Revision as of 04:30, 25 June 2025
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			{{Todo\|rewrite\|inline=1\|text=Remove all references to wedgies, as per [[User:VectorGraphics/Operation_Loosen_Underpants]]}}


	This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring\|defactor]] it.		This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring\|defactor]] it.

FloraC: Recategorize

2025-03-31T07:58:30Z

Recategorize

← Older revision		Revision as of 07:58, 31 March 2025
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
	[[Category:~~Algorithm~~]]		[[Category:Algorithms]]

Dave Keenan: /* Putting it all together */ Fix typo &inus; introduced by Arrowhead294

2025-03-30T08:46:52Z

Putting it all together: Fix typo &inus; introduced by Arrowhead294

← Older revision		Revision as of 08:46, 30 March 2025
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	\end{matrix} \right]</math>		\end{matrix} \right]</math>

	The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket\|{{map\|17 16 -4}} {{map\|4 -4 1}}}}, i.e. with a common factor of {{nowrap\|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &~~inus~~;3 of the 2nd row, i.e. 2{{map\|6 5 -4}} + (−3){{map\|4 -4 1}} {{=}} {{map\|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket\|{{map\|6 5 -4}} {{map\|4 -4 1}}}} mapping the other direction like {{rbra\|{{vector\|6 4}} {{vector\|5 -4}} {{vector\|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector\|6 4}} + 2{{vector\|5 -4}} + 1{{vector\|-4 1}} {{=}} {{vector\|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.		The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket\|{{map\|17 16 -4}} {{map\|4 -4 1}}}}, i.e. with a common factor of {{nowrap\|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with −3 of the 2nd row, i.e. 2{{map\|6 5 -4}} + (−3){{map\|4 -4 1}} {{=}} {{map\|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket\|{{map\|6 5 -4}} {{map\|4 -4 1}}}} mapping the other direction like {{rbra\|{{vector\|6 4}} {{vector\|5 -4}} {{vector\|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector\|6 4}} + 2{{vector\|5 -4}} + 1{{vector\|-4 1}} {{=}} {{vector\|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.
	}}		}}

Sintel: Category:Algorithm

2025-03-29T18:35:25Z

Category:Algorithm

← Older revision		Revision as of 18:35, 29 March 2025
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Pages with proofs]]		[[Category:Pages with proofs]]
			[[Category:Algorithm]]

Zhenlige: /* Hermite decomposition by hand */fix typo

2025-01-15T08:24:52Z

Hermite decomposition by hand: fix typo

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	<math>		<math>
	\left[ \begin{array} {ccc\|cc}		\left[ \begin{array} {ccc\|cc}
	0 & -11 & 4 & 13 & -6		0 & -11 & 4 & 13 & -6 \\
	2 & 5 & 4 & -2 & 1 \\		2 & 5 & 4 & -2 & 1 \\
	\end{array} \right]		\end{array} \right]

ArrowHead294: /* Development */

2024-11-09T16:53:59Z

Development

← Older revision		Revision as of 16:53, 9 November 2024
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	<blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group \| ''Standardizing our wedge product'']</ref></blockquote>		<blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group \| ''Standardizing our wedge product'']</ref></blockquote>

	Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: ~~Tempered~~ lattice fractions generated by prime combinations\|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.		Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: tempered lattice fractions generated by prime combinations\|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.

	Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.		Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.

ArrowHead294: /* Development */ Databox isn't a good idea

2024-11-09T16:52:03Z

Development: Databox isn't a good idea

← Older revision		Revision as of 16:52, 9 November 2024
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	Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.		Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.

	~~{{Databox\|~~Failed defactoring methods\|		=== Failed defactoring methods ===
	When in development on an ideal defactoring method—the effort which culminated in column Hermite defactoring—Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated).		When in development on an ideal defactoring method—the effort which culminated in column Hermite defactoring—Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated).

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	==== Addabilization defactoring ====		==== Addabilization defactoring ====
	This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found in [[Temperament addition #3. Addabiliziation defactoring]].		This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found in [[Temperament addition #3. Addabiliziation defactoring]].
	}}

	== Finding the greatest factor ==		== Finding the greatest factor ==

ArrowHead294: /* Development */ I feel like this could go in a databox, but feel free to undo

2024-11-09T16:50:28Z

Development: I feel like this could go in a databox, but feel free to undo

← Older revision		Revision as of 16:50, 9 November 2024
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	<blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group \| ''Standardizing our wedge product'']</ref></blockquote>		<blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group \| ''Standardizing our wedge product'']</ref></blockquote>

	Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[~~Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_EA_for_RTT~~#~~Multicomma_entries~~:~~_tempered_lattice_fractions_generated_by_prime_combinations~~\|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.		Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: Tempered lattice fractions generated by prime combinations\|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.

	Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.		Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.

	~~===~~ Failed defactoring methods ~~===~~		{{Databox\|Failed defactoring methods\|
	When in development on an ideal defactoring method—the effort which culminated in column Hermite defactoring—Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated).		When in development on an ideal defactoring method—the effort which culminated in column Hermite defactoring—Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated).

Line 888:		Line 888:
	==== Addabilization defactoring ====		==== Addabilization defactoring ====
	This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found in [[Temperament addition #3. Addabiliziation defactoring]].		This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found in [[Temperament addition #3. Addabiliziation defactoring]].
			}}

	== Finding the greatest factor ==		== Finding the greatest factor ==

ArrowHead294 at 16:45, 9 November 2024

2024-11-09T16:45:37Z

← Older revision		Revision as of 16:45, 9 November 2024
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	This is the furthest progress that was made on SAD defactor before efforts were shifted to column Hermite defactor. This implementation still exhibits problems 2 and 3; only problems 1 and 4 were solved here. Problem 3 started to be solved, but was not quite quashed.		This is the furthest progress that was made on SAD defactor before efforts were shifted to column Hermite defactor. This implementation still exhibits problems 2 and 3; only problems 1 and 4 were solved here. Problem 3 started to be solved, but was not quite quashed.

	{{Databox\|~~Code~~\|		{{Databox\|Wolfram Language code for SAD defactoring\|
	<pre>		<pre>
	confirmEnfactoredRowReplaced[m_] := Module[{i, enfactoredRowReplaced},		confirmEnfactoredRowReplaced[m_] := Module[{i, enfactoredRowReplaced},