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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | 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. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-28 17:55:24 UTC</tt>.<br>
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| : The original revision id was <tt>205859688</tt>.<br>
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| : The revision comment was: <tt></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>
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| <h4>Original Wikitext content:</h4>
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| <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.
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| 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: |
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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 |
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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 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. |
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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]]: | | 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 |
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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 will not taste the same. |
| <h4>Original HTML content:</h4>
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| <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 />
| | == External links == |
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| | * [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18933.html Yahoo! Tuning Group | ''Spectrum of a temperament''] – [[Gene Ward Smith]]'s original post |
| 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 />
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| | [[Category:Complexity]] |
| Here's the spectrum for 11-limit marvel:<br />
| | [[Category:Terms]] |
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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<br />
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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.<br />
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| 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 />
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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<br />
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| Even leaving aside the somewhat greater complexity and accuracy, it just won't taste the same.</body></html></pre></div>
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