Catakleismic: Difference between revisions
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The '''catakleismic''' [[regular temperament|temperament]] is one of the best [[7-limit]] [[extension]]s 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-[[subgroup]] temperament, first extending hanson to include the [[harmonic]] [[13/1|13]] (called [[cata]]), and then to include [[7/1|7]]. | The '''catakleismic''' [[regular temperament|temperament]] is one of the best [[7-limit]] [[extension]]s 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-[[subgroup]] temperament, first extending hanson to include the [[harmonic]] [[13/1|13]] (called [[cata]]), and then to include [[7/1|7]]. | ||
In addition to the kleisma, catakleismic tempers out 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 four [[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 addition to the kleisma, catakleismic tempers out 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 four [[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]]—or further, between [[19edo]] and [[53edo]], as beyond 53, the [[countercata]] mapping of 7 is more reasonable, with the two meeting at 53edo. 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 | 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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== Interval chain == | == Interval chain == | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
! # | ! # | ||
! Cents* | ! Cents* | ||
! Approximate ratios | ! Approximate ratios | ||
| Line 93: | Line 93: | ||
| 81/80 | | 81/80 | ||
|} | |} | ||
<nowiki>* | <nowiki />* In 2.3.5.7.13 POTE tuning | ||
== Chords == | == Chords == | ||
| Line 110: | Line 110: | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! Edo<br>generator | ! Edo<br />generator | ||
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]* | ! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
| Line 265: | Line 265: | ||
| | | | ||
|} | |} | ||
<nowiki>* | <nowiki />* Besides the octave | ||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
Revision as of 23:25, 28 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-subgroup temperament, first extending hanson to include the harmonic 13 (called cata), and then to include 7.
In addition to the kleisma, catakleismic tempers out 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 four 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—or further, between 19edo and 53edo, as beyond 53, the countercata mapping of 7 is more reasonable, with the two meeting at 53edo. 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
Tunings
Tuning spectrum
This tuning spectrum assumes undecimal catakleismic.
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 5/3 | 315.641 | ||
| 5\19 | 315.789 | Lower bound of 9-odd-limit diamond monotone | |
| 13/7 | 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 | ||
| 11/7 | 316.686 | ||
| 11/6 | 316.690 | ||
| 11/10 | 316.731 | ||
| 11/9 | 316.745 | 11-odd-limit minimax | |
| 52\197 | 316.751 | 197ef val | |
| 7/4 | 316.765 | 7-, 9-, 13- and 15-odd-limit minimax | |
| 15/11 | 316.780 | ||
| 9/5 | 316.799 | 1/7-kleisma | |
| 33\125 | 316.800 | 125f val | |
| 13/11 | 316.835 | ||
| 14\53 | 316.981 | ||
| 3/2 | 316.993 | 5-odd-limit minimax, 1/6-kleisma | |
| 15/8 | 317.115 | 2/11-kleisma | |
| 13/10 | 317.135 | ||
| 13/8 | 317.181 | ||
| 23\87 | 317.241 | 87de val | |
| 5/4 | 317.263 | 1/5-kleisma | |
| 13/12 | 317.322 | ||
| 15/13 | 317.420 | ||
| 9\34 | 317.647 | 34de val, upper bound of 9-odd-limit diamond monotone | |
| 13/9 | 318.309 |
* Besides the octave