Complexity spectrum
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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 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. 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 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 /> <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>