Complexity spectrum: Difference between revisions

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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.

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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:Terms]]

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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.
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 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:Terms]]