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. | 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 | 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 | Even leaving aside the somewhat greater complexity and accuracy, it just will not taste the same. | ||
[[Category: | [[Category:Regular temperament theory]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
Revision as of 09:38, 3 April 2021
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 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 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 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 will not taste the same.