Complexity spectrum - Revision history

FloraC: +archive. Rework the intro for a more encyclopedic style

2024-08-25T10:55:32Z

+archive. Rework the intro for a more encyclopedic style

← Older revision		Revision as of 10:55, 25 August 2024
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	~~One of the things one can look at when analyzing a temperament is its~~ '''complexity spectrum'''~~. This may be defined as the result~~ of ~~sorting the~~ [[~~complexity~~]] of ~~the intervals in the~~ ''q''-[[odd-limit]] [[~~tonality diamond~~]] between the unison and half an octave, where ''q'' is two less than the next prime after the prime limit of the temperament in question. In the rank ~~two case~~, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|~~OE complexity~~]], which is proportional to Graham complexity in the rank ~~two~~ case, but is also valid for higher limits.		The '''complexity spectrum''' of a [[regular temperament\|temperament]] is a sequence of [[odd limit\|''q''-odd-limit]] [[interval]]s between the [[unison]] and half an [[octave]] sorted by their [[Tenney-Euclidean metrics\|temperamental complexity]], where ''q'' is two less than the next [[prime]] after the [[prime limit]] of the temperament in question. In the case of rank-2 temperaments, the complexity is [[Graham complexity]], but for higher limits we can use the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|octave-equivalent TE seminorm]], which is proportional to Graham complexity in the rank-2 case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same.		Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same.

			== External links ==
			* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18933.html Yahoo! Tuning Group \| ''Spectrum of a temperament''] – [[Gene Ward Smith]]'s original post

	[[Category:Complexity]]		[[Category:Complexity]]
	[[Category:Terms]]		[[Category:Terms]]

Fredg999: Internal link, categories

2023-03-10T22:00:28Z

Internal link, categories

← Older revision		Revision as of 22:00, 10 March 2023
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	One of the things one can look at when analyzing a temperament is its '''complexity spectrum'''. This may be defined as the result of sorting the complexity of the intervals in the ''q''-[[odd-limit]] [[tonality diamond]] between the unison and half an octave, where ''q'' is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.		One of the things one can look at when analyzing a temperament is its '''complexity spectrum'''. This may be defined as the result of sorting the [[complexity]] of the intervals in the ''q''-[[odd-limit]] [[tonality diamond]] between the unison and half an octave, where ''q'' is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same.		Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same.

	[[Category:~~Regular temperament theory~~]]		[[Category:Complexity]]
	[[Category:Terms]]		[[Category:Terms]]

FloraC: Improve categories; some wording

2021-04-03T09:38:11Z

Improve categories; some wording

← Older revision		Revision as of 09:38, 3 April 2021
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	One of the things one can look at when analyzing a temperament is its '''complexity spectrum'''. This may be defined as the result of sorting the complexity of the intervals in the ''q'' [[odd limit]] [[tonality diamond]] between the unison and half an octave, where ''q'' is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.		One of the things one can look at when analyzing a temperament is its '''complexity spectrum'''. This may be defined as the result of sorting the complexity of the intervals in the ''q''-[[odd-limit]] [[tonality diamond]] between the unison and half an octave, where ''q'' is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11		: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11

	You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.		You can see it favors 5 over 7 and 7 over 11; for how much we could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.

	Here's the spectrum for 13-limit [[Werckismic temperaments #History\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[the Archipelago]]:		Here's the spectrum for 13-limit [[Werckismic temperaments #History\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[the Archipelago]]:
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	: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7		: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

	Even leaving aside the somewhat greater complexity and accuracy, it just ~~won't~~ taste the same.		Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same.

	[[Category:~~Theory~~]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]

	~~{{todo\|discuss title}}~~

FloraC: FloraC moved page Spectrum of a temperament to Complexity spectrum: In accordance with the content

2021-02-21T11:01:13Z

FloraC moved page Spectrum of a temperament to Complexity spectrum: In accordance with the content

← Older revision	Revision as of 11:01, 21 February 2021
(No difference)

FloraC: +category

2020-12-12T16:12:25Z

+category

← Older revision		Revision as of 16:12, 12 December 2020
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	[[Category:Theory]]		[[Category:Theory]]
			[[Category:Terms]]

	{{todo\|discuss title}}		{{todo\|discuss title}}

FloraC: Bold lemma; improve links and readability

2020-12-12T16:04:58Z

Bold lemma; improve links and readability

← Older revision		Revision as of 16:04, 12 December 2020
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	One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[odd limit]] [[tonality diamond]] between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-~~Euclidean_metrics~~\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.		One of the things one can look at when analyzing a temperament is its '''complexity spectrum'''. This may be defined as the result of sorting the complexity of the intervals in the ''q'' [[odd limit]] [[tonality diamond]] between the unison and half an octave, where ''q'' is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics #Octave equivalent TE seminorm\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.		You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.

	Here's the spectrum for 13-limit [[~~Werckismic_temperaments~~\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[~~The_Archipelago\|~~the Archipelago]]:		Here's the spectrum for 13-limit [[Werckismic temperaments #History\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[the Archipelago]]:

	: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7		: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
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	Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.		Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.

	[[Category:~~theory~~]]		[[Category:Theory]]

			{{todo\|discuss title}}

Spt3125: category maintenance, clean up formatting

2018-12-09T04:08:51Z

category maintenance, clean up formatting

← Older revision		Revision as of 04:08, 9 December 2018
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	One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[~~Odd_limit\|~~odd limit]] [[~~Tonality_diamond\|~~tonality diamond]] between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[~~Graham_complexity\|~~Graham complexity]], but for higher limits we can use [[Tenney-Euclidean_metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.		One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[odd limit]] [[tonality diamond]] between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean_metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	Here's the spectrum for 11-limit marvel:		Here's the spectrum for 11-limit marvel:

	5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11		: 5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11

	You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.		You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.
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	Here's the spectrum for 13-limit [[Werckismic_temperaments\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[The_Archipelago\|the Archipelago]]:		Here's the spectrum for 13-limit [[Werckismic_temperaments\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[The_Archipelago\|the Archipelago]]:

	11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7		: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

	Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.		Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.

	[[Category:theory]]		[[Category:theory]]
	~~[[Category:todo:clear_some_link_texts]]~~

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[Odd_limit\|odd limit]] [[Tonality_diamond\|tonality diamond]] between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham_complexity\|Graham complexity]], but for higher limits we can use [[Tenney-Euclidean_metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.
	~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-11-13 13:02:18 UTC</tt>.<br>~~
	~~: The original revision id was <tt>274836228</tt>.<br>~~
	~~: The revision comment was: <tt></tt><br>~~
	~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~
	~~<h4>Original Wikitext content:</h4>~~
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[odd limit]] [[tonality diamond]] between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-~~Euclidean metrics~~\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.		You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.

	Here's the spectrum for 13-limit [[~~Werckismic temperaments~~\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[the Archipelago]]:		Here's the spectrum for 13-limit [[Werckismic_temperaments\|history]], the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of [[The_Archipelago\|the Archipelago]]:

	11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7		11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

	Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.~~</pre></div>~~		Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:theory]]
	~~<div style="width~~:~~100%; max-height~~:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Spectrum of a temperament</title></head><body>One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q <a class="wiki_link" href="/odd%20limit">odd limit</a> <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a>, but for higher limits we can use <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE complexity</a>, which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.<br />		[[Category:todo:clear_some_link_texts]]
	~~<br />~~
	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.<br />
	~~<br />~~
	~~Here's the spectrum for 11-limit marvel:<br />~~
	~~<br />~~
	~~5/4, 4/3, 7/6, 8/7, 7/5, 6/5, 9/7, 12/11, 9/8, 11/8, 11/9, 10/9, 11/10, 14/11<br />~~
	~~<br />~~
	You can see it favors 5 over 7 and 7 over 11; for how much I could stick in the actual numerical complexities, but you can see that 9/8 and 10/9 are more complex than some 7 and 11 limit intervals just from the above.<br />
	~~<br />~~
	Here's the spectrum for 13-limit <a class="wiki_link" href="/Werckismic%20temperaments">history</a>, the temperament tempering out 364/363, 441/440 and 1001/1000 which is part of <a class="wiki_link" href="/the%20Archipelago">the Archipelago</a>:<br />
	~~<br />~~
	~~11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7<br />~~
	~~<br />~~
	~~Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.</body></html></pre></div>~~

Wikispaces>genewardsmith: Imported revision 274836228 - Original comment:

2011-11-13T13:02:18Z

**Imported revision 274836228 - Original comment: **

Wikispaces>xenwolf: Imported revision 239087885 - Original comment:

2011-06-28T03:08:53Z

**Imported revision 239087885 - Original comment: **

← Older revision		Revision as of 13:02, 13 November 2011
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~xenwolf~~\|~~xenwolf~~]] and made on <tt>2011-06-~~28 03~~:08:53 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-11-13 13:02:18 UTC</tt>.<br>
	: The original revision id was <tt>~~239087885~~</tt>.<br>		: The original revision id was <tt>274836228</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[odd limit]] [[tonality diamond]] between the unison and half an octave, where q is two less than the next prime after p. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[odd limit]] [[tonality diamond]] between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.</pre></div>		Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Spectrum of a temperament</title></head><body>One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q <a class="wiki_link" href="/odd%20limit">odd limit</a> <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> between the unison and half an octave, where q is two less than the next prime after p. In the rank two case, the complexity is <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a>, but for higher limits we can use <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE complexity</a>, which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.<br />		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Spectrum of a temperament</title></head><body>One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q <a class="wiki_link" href="/odd%20limit">odd limit</a> <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> between the unison and half an octave, where q is two less than the next prime after the prime limit of the temperament in question. In the rank two case, the complexity is <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a>, but for higher limits we can use <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE complexity</a>, which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.<br />
	<br />		<br />
	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.<br />		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.<br />

← Older revision		Revision as of 03:08, 28 June 2011
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~genewardsmith~~\|~~genewardsmith~~]] and made on <tt>2011-02-28 17:55:24 UTC</tt>.<br>		: This revision was by author [[User:xenwolf\|xenwolf]] and made on <tt>2011-06-28 03:08:53 UTC</tt>.<br>
	: The original revision id was <tt>~~205859688~~</tt>.<br>		: The original revision id was <tt>239087885</tt>.<br>
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	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q odd limit tonality diamond between the unison and half an octave, where q is two less than the next prime after p. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q [[odd limit]] [[tonality diamond]] between the unison and half an octave, where q is two less than the next prime after p. In the rank two case, the complexity is [[Graham complexity]], but for higher limits we can use [[Tenney-Euclidean metrics\|OE complexity]], which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.

	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.
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	Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.</pre></div>		Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Spectrum of a temperament</title></head><body>One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q odd limit tonality diamond between the unison and half an octave, where q is two less than the next prime after p. In the rank two case, the complexity is <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a>, but for higher limits we can use <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE complexity</a>, which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.<br />		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Spectrum of a temperament</title></head><body>One of the things one can look at when analyzing a temperament is its complexity spectrum. This may be defined as the result of sorting the complexity of the intervals in the q <a class="wiki_link" href="/odd%20limit">odd limit</a> <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> between the unison and half an octave, where q is two less than the next prime after p. In the rank two case, the complexity is <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a>, but for higher limits we can use <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE complexity</a>, which is proportional to Graham complexity in the rank two case, but is also valid for higher limits.<br />
	<br />		<br />
	The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.<br />		The different flavors of a temperament, so to speak, are shown in its spectrum. A temperament like meantone, which favors 3 over 5, and 5 over 7, has quite a different flavor than miracle, which favors 7, 11/9 and 7/5.<br />