Catakleismic
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. Undecimal catakleismic is shown in the tables below; additionally, tempering out 286/285 gives us an extension to prime 19 at -18 generator steps.
See Kleismic family #Catakleismic for technical data.
Interval chain
In the following table, harmonics 1–21 and their inverses are in bold.
| # | Cents* | Approximate ratios | |
|---|---|---|---|
| 2.3.5.7.13 subgroup | add-11 add-19 extension | ||
| 0 | 0.0 | 1/1 | |
| 1 | 316.8 | 6/5 | |
| 2 | 633.6 | 13/9 | |
| 3 | 950.4 | 26/15 | 19/11 |
| 4 | 67.2 | 25/24, 26/25, 27/26, 28/27 | |
| 5 | 384.0 | 5/4 | |
| 6 | 700.8 | 3/2 | |
| 7 | 1017.6 | 9/5 | |
| 8 | 134.4 | 13/12, 14/13, 27/25 | |
| 9 | 451.1 | 13/10 | |
| 10 | 767.9 | 14/9 | |
| 11 | 1084.7 | 15/8, 28/15 | |
| 12 | 201.5 | 9/8 | |
| 13 | 518.3 | 27/20 | |
| 14 | 835.1 | 13/8, 21/13 | |
| 15 | 1151.9 | 35/18, 39/20 | 64/33 |
| 16 | 268.7 | 7/6 | |
| 17 | 585.5 | 7/5 | |
| 18 | 902.3 | 27/16 | 32/19 |
| 19 | 19.1 | 81/80, 91/90, 105/104 | 77/76, 78/77, 96/95, 100/99, 133/132, 144/143 |
| 20 | 335.9 | 39/32 | 40/33 |
| 21 | 652.7 | 35/24 | 16/11 |
| 22 | 969.5 | 7/4 | |
| 23 | 86.3 | 21/20 | 20/19 |
| 24 | 403.1 | 63/50 | 24/19 |
| 25 | 719.8 | 91/60 | 50/33 |
| 26 | 1036.6 | 91/50 | 20/11 |
| 27 | 153.4 | 35/32 | 12/11 |
| 28 | 470.2 | 21/16 | |
| 29 | 787.0 | 63/40 | 30/19 |
| 30 | 1103.8 | 91/48 | 36/19 |
| 31 | 220.6 | 91/80 | 25/22 |
| 32 | 537.4 | 117/80 | 15/11, 26/19 |
| 33 | 854.2 | 49/30 | 18/11 |
| 34 | 1171.0 | 63/32 | 49/25, 65/33 |
* In 2.3.5.7.13-subgroup CWE tuning
As a detemperament of 19et
Catakleismic is naturally considered as a detemperament of the 19 equal temperament. The diagram on the right shows a 72-tone detempered scale, with a generator range of -35 to +36. 72 is the largest number of tones for a mos where intervals in the 19 categories do not overlap. Each category is divided into three or four qualities separated by 19 generator steps, which represent the syntonic comma. Combining this division with the minor and major diatonic qualities of the 19 equal temperament, catakleismic gives us seven or eight qualities for each diatonic category in addition to the four qualities for the categories corresponding to interseptimal intervals.
Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 53 generator steps, so it vanishes in 53edo, but is tuned to the same size as the syntonic comma in 72edo. 125edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.
Chords
- Main article: Chords of catakleismic and Chords of tridecimal catakleismic
Scales
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~6/5 = 316.9026 ¢ | CSEE: ~6/5 = 316.8354 ¢ | POEE: ~6/5 = 316.5718 ¢ |
| Tenney | CTE: ~6/5 = 316.8865 ¢ | CWE: ~6/5 = 316.7939 ¢ | POTE: ~6/5 = 316.7410 ¢ |
| Benedetti, Wilson |
CBE: ~6/5 = 316.8827 ¢ | CSBE: ~6/5 = 316.7927 ¢ | POBE: ~6/5 = 316.7673 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~6/5 = 316.7941 ¢ | CSEE: ~6/5 = 316.7860 ¢ | POEE: ~6/5 = 316.8002 ¢ |
| Tenney | CTE: ~6/5 = 316.8070 ¢ | CWE: ~6/5 = 316.7816 ¢ | POTE: ~6/5 = 316.7778 ¢ |
| Benedetti, Wilson |
CBE: ~6/5 = 316.8299 ¢ | CSBE: ~6/5 = 316.7884 ¢ | POBE: ~6/5 = 316.7625 ¢ |
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