Catakleismic: Difference between revisions
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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. | 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 | 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. | 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. | ||
Revision as of 22:44, 21 October 2024
The catakleismic temperament is one of the best 7-limit extensions of hanson, the 5-limit temperament tempering out the 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.
Interval chain
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 316.7 | 6/5 |
| 2 | 633.5 | 13/9 |
| 3 | 950.2 | 26/15 |
| 4 | 67.0 | 25/24, 26/25, 27/26, 28/27 |
| 5 | 383.7 | 5/4 |
| 6 | 700.4 | 3/2 |
| 7 | 1017.2 | 9/5 |
| 8 | 133.9 | 13/12, 14/13, 27/25 |
| 9 | 450.7 | 13/10 |
| 10 | 767.4 | 14/9 |
| 11 | 1084.1 | 15/8, 28/15 |
| 12 | 200.9 | 9/8 |
| 13 | 517.6 | 27/20 |
| 14 | 834.4 | 13/8, 21/13 |
| 15 | 1151.1 | 35/18 |
| 16 | 267.9 | 7/6 |
| 17 | 584.6 | 7/5 |
| 18 | 901.3 | 27/16 |
| 19 | 18.1 | 81/80 |
* in 2.3.5.7.13 POTE tuning
Chords
- Main article: Chords of catakleismic and Chords of tridecimal catakleismic
Scales
Tuning spectrum
The tuning spectrum, presumably, assumes undecimal catakleismic, while neglecting to specify the details of that specific extension.
| EDO generator |
eigenmonzo (unchanged-interval) |
generator (¢) |
comments |
|---|---|---|---|
| 6/5 | 315.641 | ||
| 5\19 | 315.789 | Lower bound of 9-odd-limit diamond monotone | |
| 14/13 | 316.037 | ||
| 15/14 | 316.414 | ||
| 9/7 | 316.492 | ||
| 11/8 | 316.604 | ||
| 7/5 | 316.618 | ||
| 19\72 | 316.667 | ||
| 7/6 | 316.679 | ||
| 14/11 | 316.686 | ||
| 12/11 | 316.690 | ||
| 11/10 | 316.731 | ||
| 11/9 | 316.745 | 11-odd-limit minimax | |
| 52\197 | 316.751 | ||
| 8/7 | 316.765 | 7-, 9-, 13- and 15-odd-limit minimax | |
| 15/11 | 316.780 | ||
| 10/9 | 316.799 | 1/7-kleisma | |
| 33\125 | 316.800 | ||
| 13/11 | 316.835 | ||
| 14\53 | 316.981 | ||
| 4/3 | 316.993 | 5-odd-limit minimax, 1/6-kleisma | |
| 16/15 | 317.115 | 2/11-kleisma | |
| 13/10 | 317.135 | ||
| 16/13 | 317.181 | ||
| 23\87 | 317.241 | ||
| 5/4 | 317.263 | 1/5-kleisma | |
| 13/12 | 317.322 | ||
| 15/13 | 317.420 | ||
| 9\34 | 317.647 | Upper bound of 9-odd-limit diamond monotone | |
| 18/13 | 318.309 |