Catakleismic: Difference between revisions
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The '''catakleismic''' temperament is one of the best extensions of hanson, the 5-limit temperament tempering out the [[kleisma]]. | The '''catakleismic''' temperament is one of the best [[7-limit]] extensions of [[hanson]], the 5-limit temperament tempering out the [[15625/15552|kleisma]] (15625/15552), though it is naturally viewed as a 2.3.5.7.13 temperament, first extending hanson to include the harmonic 13 (called '''cata'''), and then to include 7. | ||
In addition to the kleisma, catakleismic tempers the [[marvel comma]] (225/224), equating the interval of [[25/24]] (which is already equated to [[26/25]] and [[27/26]] in the 2.3.5.13 subgroup interpretation of kleismic) to [[28/27]]. This forces a flatter interpretation of 25/24, which is found 4 [[6/5]] generators up, and therefore a flatter interpretation of the generator, which confines reasonable catakleismic tunings to the portion of the kleismic tuning spectrum. between [[19edo]] and [[34edo]]; in fact, catakleismic is the 19 & 34d temperament in the 7-limit. It can additionally be defined by tempering out the marvel comma and the [[ragisma]] (4375/4374), which finds [[7/6]] at the square of [[27/25]], which is found at the square of 25/24. Therefore the 7th harmonic appears 22 generators up the chain. | |||
Various reasonable extensions exist for harmonic 11. These are ''undecimal catakleismic'', mapping 11 to -21 generator steps, ''cataclysmic'', to +32 steps, ''catalytic'', to +51 steps, and cataleptic, to -2 steps. | |||
See [[Kleismic family #Catakleismic]] for technical data. | See [[Kleismic family #Catakleismic]] for technical data. | ||
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== Tuning spectrum == | == Tuning spectrum == | ||
The tuning spectrum, presumably, assumes undecimal catakleismic, while neglecting to specify the details of that specific extension. | |||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
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| | | | ||
|- | |- | ||
| 5\19 | | [[19edo|5\19]] | ||
| | | | ||
| 315.789 | | 315.789 | ||
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| | | | ||
|- | |- | ||
| 19\72 | | [[72edo|19\72]] | ||
| | | | ||
| 316.667 | | 316.667 | ||
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| 11-odd-limit minimax | | 11-odd-limit minimax | ||
|- | |- | ||
| 52\197 | | [[197edo|52\197]] | ||
| | | | ||
| 316.751 | | 316.751 | ||
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| 10/9 | | 10/9 | ||
| 316.799 | | 316.799 | ||
| | | 1/7-kleisma | ||
|- | |- | ||
| 33\125 | | [[125edo|33\125]] | ||
| | | | ||
| 316.800 | | 316.800 | ||
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|- | |- | ||
| 14\53 | | [[53edo|14\53]] | ||
| | | | ||
| 316.981 | | 316.981 | ||
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| 4/3 | | 4/3 | ||
| 316.993 | | 316.993 | ||
| 5-odd-limit minimax | | 5-odd-limit minimax, 1/6-kleisma | ||
|- | |- | ||
| | | | ||
| 16/15 | | 16/15 | ||
| 317.115 | | 317.115 | ||
| | | 2/11-kleisma | ||
|- | |- | ||
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|- | |- | ||
| 23\87 | | [[87edo|23\87]] | ||
| | | | ||
| 317.241 | | 317.241 | ||
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| 5/4 | | 5/4 | ||
| 317.263 | | 317.263 | ||
| | | 1/5-kleisma | ||
|- | |- | ||
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|- | |- | ||
| 9\34 | | [[34edo|9\34]] | ||
| | | | ||
| 317.647 | | 317.647 | ||