Complexity spectrum: Difference between revisions
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Wikispaces>xenwolf **Imported revision 239087885 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 274836228 - Original comment: ** |
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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: | : 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> | : 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 | <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 | <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 /> | ||
Revision as of 13:02, 13 November 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-11-13 13:02:18 UTC.
- The original revision id was 274836228.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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. 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 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]]: 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.
Original HTML content:
<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 /> 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>