Talk:Mathematical theory of saturation - Revision history

Cmloegcmluin: updates

2021-11-13T19:13:04Z

updates

← Older revision		Revision as of 19:13, 13 November 2021
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	I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)		I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)

	: If anyone comes to this later, Dave and I managed to answer our own question. Our findings are compiled here: [[~~Canonical_form~~#~~defactoring_methods~~]] --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 06:15, 27 September 2021 (UTC)		: If anyone comes to this later, Dave and I managed to answer our own question. Our findings are compiled here: [[Defactoring_algorithms#Precedent:_Smith_defactoring]] --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 06:15, 27 September 2021 (UTC)

	== how to relate this page to the new page for canonical form ==		== how to relate this page to the new page for canonical form ==
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	I don't want to be hasty here so I don't plan to take any action until I hear plenty of other opinions. Thanks for your attention to this. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 17:39, 27 September 2021 (UTC)		I don't want to be hasty here so I don't plan to take any action until I hear plenty of other opinions. Thanks for your attention to this. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 17:39, 27 September 2021 (UTC)

	Okay, I decided that at least a "See also" at the bottom seemed reasonable enough. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 19:03, 29 September 2021 (UTC)		: Okay, I decided that at least a "See also" at the bottom seemed reasonable enough. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 19:03, 29 September 2021 (UTC)

			:: Alright, after discussing with Mike Battaglia, this page has been moved to be the mathematically-inclined page, and a new general audience page has been created here: [[Saturation,_torsion,_contorsion,_and_defactoring]] --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 19:13, 13 November 2021 (UTC)

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Cmloegcmluin: Cmloegcmluin moved page Talk:Saturation to Talk:Mathematical theory of saturation: this page targets a more mathematically-inclined audience; a new general audience page has just been created

2021-11-12T22:53:06Z

Cmloegcmluin moved page Talk:Saturation to Talk:Mathematical theory of saturation: this page targets a more mathematically-inclined audience; a new general audience page has just been created

← Older revision	Revision as of 22:53, 12 November 2021
(No difference)

Cmloegcmluin: /* how to relate this page to the new page for canonical form */

2021-09-29T19:03:53Z

how to relate this page to the new page for canonical form

← Older revision		Revision as of 19:03, 29 September 2021
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	I don't want to be hasty here so I don't plan to take any action until I hear plenty of other opinions. Thanks for your attention to this. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 17:39, 27 September 2021 (UTC)		I don't want to be hasty here so I don't plan to take any action until I hear plenty of other opinions. Thanks for your attention to this. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 17:39, 27 September 2021 (UTC)

			Okay, I decided that at least a "See also" at the bottom seemed reasonable enough. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 19:03, 29 September 2021 (UTC)

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Cmloegcmluin: question about how best to relate this page to the new page for canonical form

2021-09-27T17:39:03Z

question about how best to relate this page to the new page for canonical form

← Older revision		Revision as of 17:39, 27 September 2021
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	: If anyone comes to this later, Dave and I managed to answer our own question. Our findings are compiled here: [[Canonical_form#defactoring_methods]] --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 06:15, 27 September 2021 (UTC)		: If anyone comes to this later, Dave and I managed to answer our own question. Our findings are compiled here: [[Canonical_form#defactoring_methods]] --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 06:15, 27 September 2021 (UTC)

			== how to relate this page to the new page for canonical form ==

			I recently published [[Canonical form\|a page on the xen wiki proposing a canonical form for RTT mappings (or comma-bases)]], a key component of which is "defactoring" — removing common factors from their rows (or columns).

			This defactoring process is in fact the same mathematical thing as "saturation". And the opposite situation, being "enfactored", is identical to the situation of being "contorted", a term which redirects to this page on the wiki as well.

			Of the several methods for eliminating enfactoring/contortion by achieving defactoring/saturation which are presented on my new page, one of them is the exact algorithm described here, and another of them was derived from it.

			However, I chose not to add any new thoughts about this topic to this existing page. Instead I added only to my new page. My reasons were:

			# I have been asked not to significantly alter the works of Gene Ward Smith, and I respect the intent to protect his original work here. I have made a few changes to this page recently, but only changes that I thought should help preserve Gene's genius thinking as well as help it reach people even better: I cleaned up the mathematical formatting using recently developed Mediwiki templates and what I've learned about LaTeX, I added some parenthetical clarifications that my colleagues have found helpful when discussing this page, and I also included a code implementation of Gene's original algorithm that we developed.
			# I do think that it is a proper home for the topic of defactoring/saturation to be as a subtopic of canonical form. I think this especially because if the RTT community leverages this canonical form, then defactoring/saturation ceases to be a problem. At least, the average practitioner should barely need to understand it or be aware of it.
			# Gene's page has a different target audience than mine, at least insofar as the high level of mathematical rigor and technicality it uses. My page is written as a non-mathematician, for non-mathematicians. And my understanding is that most xen wiki users and contributors (including myself) are amenable to the idea of achieving whatever structure or organization is practical such that those who are interested in such mathematical theory and proof can find it, but those who aren't don't get overwhelming levels of exposure to it and get turned off from xen music-making. So, I think both pages have good reason to stick around.
			# I have serious terminological concerns about both the words "saturation" and "contorsion", which I go into in detail on my new page.

			My question is, to what extent do people think is appropriate for me to recognize/associate the new canonical form page here on this page? It could be as slight as a "See also" at the bottom. It could be a new section at the bottom where I briefly summarize the relationship between this page and the new canonical form page. Or it could be some footnotes interspersed with the original text. Or something else entirely that I haven't thought of yet.

			I don't want to be hasty here so I don't plan to take any action until I hear plenty of other opinions. Thanks for your attention to this. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 17:39, 27 September 2021 (UTC)



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Cmloegcmluin at 06:15, 27 September 2021

2021-09-27T06:15:48Z

← Older revision		Revision as of 06:15, 27 September 2021
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	I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)		I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)

			: If anyone comes to this later, Dave and I managed to answer our own question. Our findings are compiled here: [[Canonical_form#defactoring_methods]] --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 06:15, 27 September 2021 (UTC)

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Cmloegcmluin: /* Edge case? */ never mind: just needed to use MatrixRank instead of length, and the description of the algorithm works fine as it's written, because it says to use rank

2021-08-02T21:59:57Z

Edge case?: never mind: just needed to use MatrixRank instead of length, and the description of the algorithm works fine as it's written, because it says to use rank

← Older revision		Revision as of 21:59, 2 August 2021
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	I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)		I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)

	~~== Edge case? ==~~

	I think that if the saturation algorithm begins with a matrix with a row of all zeroes, it fails as it is currently implemented, e.g. [⟨12 19 28] ⟨0 0 0]⟩ → [⟨12 19 28] ⟨-9 -14 -21]⟩. Probably we need to insert a step at the beginning where all-zero rows are eliminated. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 21:34, 2 August 2021 (UTC)

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Cmloegcmluin: edge case?

2021-08-02T21:34:22Z

edge case?

← Older revision		Revision as of 21:34, 2 August 2021
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	I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)		I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)

			== Edge case? ==

			I think that if the saturation algorithm begins with a matrix with a row of all zeroes, it fails as it is currently implemented, e.g. [⟨12 19 28] ⟨0 0 0]⟩ → [⟨12 19 28] ⟨-9 -14 -21]⟩. Probably we need to insert a step at the beginning where all-zero rows are eliminated. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 21:34, 2 August 2021 (UTC)

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Cmloegcmluin at 22:53, 1 August 2021

2021-08-01T22:53:40Z

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			== Origin of desaturating algorithm ==

			I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices ''works''. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 22:53, 1 August 2021 (UTC)

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Mike Battaglia: 1 revision imported: Moving archived Wikispaces discussion to subpage

2018-10-01T18:02:42Z

1 revision imported: Moving archived Wikispaces discussion to subpage

← Older revision	Revision as of 18:02, 1 October 2018
(No difference)

Wikispaces>FREEZE at 18:01, 1 October 2018

2018-10-01T18:01:16Z

← Older revision		Revision as of 18:01, 1 October 2018
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	~~= ARCHIVED WIKISPACES DISCUSSION BELOW =~~		{{WSArchiveLink}}
	~~'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''~~
	~~----~~

	~~== Saturation Problem ==~~
	~~"Because unsaturated subgroups of Z^n are problematic…"~~

	~~It seems like the article took an unexpected turn as soon as I read this sentence. Can somebody explain to me why this is problematic?~~

	~~- '''Sarzadoce''' October 27, 2012, 11:52:11 AM UTC-0700~~
	~~----~~
	~~The previous two paragraphs consist of an explanation for this claim.~~

	~~- '''genewardsmith''' October 27, 2012, 08:06:19 PM UTC-0700~~
	~~----~~
	~~I assume you are talking about this:~~

	~~"… if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals."~~

	~~So I'll reiterate my question: Why is this problematic?~~

	~~- '''Sarzadoce''' October 27, 2012, 11:59:00 PM UTC-0700~~
	~~----~~
	~~This doesn't seem correct:~~

	"If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals."

	The whole concept of having notes which cannot be reached from the unison by tempered rational intervals would seem to apply only if V is unsaturated, not C. If C isn't saturated, then if G is the group of JI intervals, the problem is that G/C has torsion, not that it has unmapped intervals.

	~~- '''mbattaglia1''' October 28, 2012, 12:09:30 AM UTC-0700~~
	~~----~~

	~~== examples ==~~
	Yes, there is one example, but it is rarely applicable for ordinary musicians, composers or mikrotonalists who have no mathematical background. Can you show us how this concept can be used in microtonal / xenharmonic music?

	~~:) Thanks in advance~~

	~~- '''xenwolf''' June 16, 2011, 11:59:40 PM UTC-0700~~
	~~----~~