Catakleismic: Difference between revisions
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Cmloegcmluin (talk | contribs) →Spectrum of catakleismic tunings by eigenmonzos: improve and standardize tuning spectra tables |
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* [[Catakleismic34]] | * [[Catakleismic34]] | ||
== | == Tuning spectrum == | ||
{| class="wikitable center- | {| class="wikitable center-all" | ||
|- | |- | ||
! | ! ET<br>generator | ||
! | ! [[eigenmonzo|eigenmonzo<br>(unchanged interval]]) | ||
! | ! minor<br>third (¢) | ||
! comments | |||
|- | |- | ||
| | |||
| 6/5 | | 6/5 | ||
| 315.641 | | 315.641 | ||
| | | | ||
|- | |- | ||
| | | 5\19 | ||
| | |||
| 315.789 | | 315.789 | ||
| | | | ||
|- | |- | ||
| | |||
| 14/13 | | 14/13 | ||
| 316.037 | | 316.037 | ||
| | | | ||
|- | |- | ||
| | |||
| 15/14 | | 15/14 | ||
| 316.414 | | 316.414 | ||
| | | | ||
|- | |- | ||
| | |||
| 9/7 | | 9/7 | ||
| 316.492 | | 316.492 | ||
| | | | ||
|- | |- | ||
| | |||
| 11/8 | | 11/8 | ||
| 316.604 | | 316.604 | ||
| | | | ||
|- | |- | ||
| | |||
| 7/5 | | 7/5 | ||
| 316.618 | | 316.618 | ||
| | | | ||
|- | |- | ||
| | | 19\72 | ||
| | |||
| 316.667 | | 316.667 | ||
| | | | ||
|- | |- | ||
| | |||
| 7/6 | | 7/6 | ||
| 316.679 | | 316.679 | ||
| | | | ||
|- | |- | ||
| | |||
| 14/11 | | 14/11 | ||
| 316.686 | | 316.686 | ||
| | | | ||
|- | |- | ||
| | |||
| 12/11 | | 12/11 | ||
| 316.690 | | 316.690 | ||
| | | | ||
|- | |- | ||
| | |||
| 11/10 | | 11/10 | ||
| 316.731 | | 316.731 | ||
| | | | ||
|- | |- | ||
| | |||
| 11/9 | | 11/9 | ||
| 316.745 | | 316.745 | ||
| 11-odd-limit minimax | | 11-odd-limit minimax | ||
|- | |- | ||
| | | 52\197 | ||
| | |||
| 316.751 | | 316.751 | ||
| | | | ||
|- | |- | ||
| | |||
| 8/7 | | 8/7 | ||
| 316.765 | | 316.765 | ||
| 7-, 9-, 13- and 15-odd-limit minimax | | 7-, 9-, 13- and 15-odd-limit minimax | ||
|- | |- | ||
| | |||
| 15/11 | | 15/11 | ||
| 316.780 | | 316.780 | ||
| | | | ||
|- | |- | ||
| | |||
| 10/9 | | 10/9 | ||
| 316.799 | | 316.799 | ||
| | | | ||
|- | |- | ||
| | | 33\125 | ||
| | |||
| 316.800 | | 316.800 | ||
| | | | ||
|- | |- | ||
| | |||
| 13/11 | | 13/11 | ||
| 316.835 | | 316.835 | ||
| | | | ||
|- | |- | ||
| | | 14\53 | ||
| | |||
| 316.981 | | 316.981 | ||
| | | | ||
|- | |- | ||
| | |||
| 4/3 | | 4/3 | ||
| 316.993 | | 316.993 | ||
| 5-odd-limit minimax | | 5-odd-limit minimax | ||
|- | |- | ||
| | |||
| 16/15 | | 16/15 | ||
| 317.115 | | 317.115 | ||
| | | | ||
|- | |- | ||
| | |||
| 13/10 | | 13/10 | ||
| 317.135 | | 317.135 | ||
| | | | ||
|- | |- | ||
| | |||
| 16/13 | | 16/13 | ||
| 317.181 | | 317.181 | ||
| | | | ||
|- | |- | ||
| | | 23\87 | ||
| | |||
| 317.241 | | 317.241 | ||
| | | | ||
|- | |- | ||
| | |||
| 5/4 | | 5/4 | ||
| 317.263 | | 317.263 | ||
| | | | ||
|- | |- | ||
| | |||
| 13/12 | | 13/12 | ||
| 317.322 | | 317.322 | ||
| | | | ||
|- | |- | ||
| | |||
| 15/13 | | 15/13 | ||
| 317.420 | | 317.420 | ||
| | | | ||
|- | |- | ||
| | | 9\34 | ||
| | |||
| 317.647 | | 317.647 | ||
| | | | ||
|- | |- | ||
| | |||
| 18/13 | | 18/13 | ||
| 318.309 | | 318.309 | ||
Revision as of 00:27, 30 September 2021
The catakleismic temperament is one of the best extensions of hanson, the 5-limit temperament tempering out the kleisma.
Catakleismic 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. 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
| ET generator |
eigenmonzo (unchanged interval) |
minor third (¢) |
comments |
|---|---|---|---|
| 6/5 | 315.641 | ||
| 5\19 | 315.789 | ||
| 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 | ||
| 33\125 | 316.800 | ||
| 13/11 | 316.835 | ||
| 14\53 | 316.981 | ||
| 4/3 | 316.993 | 5-odd-limit minimax | |
| 16/15 | 317.115 | ||
| 13/10 | 317.135 | ||
| 16/13 | 317.181 | ||
| 23\87 | 317.241 | ||
| 5/4 | 317.263 | ||
| 13/12 | 317.322 | ||
| 15/13 | 317.420 | ||
| 9\34 | 317.647 | ||
| 18/13 | 318.309 |