Catakleismic
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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
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 | ||
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 | Upper bound of 9-odd-limit diamond monotone | |
18/13 | 318.309 |