Kleismic
| Kleismic |
325/324, 625/624 (2.3.5.13)
2.3.5.13 15-odd-limit: 2.35 ¢
2.3.5.13 15-odd-limit: 15 notes
Kleismic, alternatively called hanson in the 5-limit, is a rank-2 temperament and parent of the kleismic family, generated by a classical minor third (6/5), six of which stacked are equated to the perfect twelfth (3/1), and thereby characterized by the vanishing of the kleisma (ratio: 15625/15552, monzo: [-6 -5 6⟩).
Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, 25/24, into a third-tone by equating three of this interval to 9/8. As 9/8 = (27/26)⋅(26/25)⋅(25/24), it is natural to equate 25/24 to 26/25 and 27/26 as well, thereby tempering out the tunbarsma 625/624 (S25) and the marveltwin comma 325/324 (S25⋅S26) respectively, and resulting in a low-complexity but high-accuracy extension to the 2.3.5.13-subgroup sometimes known as cata. From there we can see that 676/675 (S26) is also tempered out, meaning 4/3 is split into two 15/13's and that 3/1 is split into two 26/15's. From 325/324 = (13/9)/(6/5)2 we can see that 13/9 is split into two 6/5's, so that it is equated with 36/25 (giving rise to the other S-expression of 325/324, S10/S12); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.
Extensions with prime 7 include catakleismic (which adds 225/224, finding 7 at 22 generators up), countercata (which adds 5120/5103, finding 7 at 31 generators down), metakleismic (which adds 179200/177147, finding 7 at 56 generators up), keemun (which adds 49/48, finding 7 at 3 generators up), anakleismic (which adds 2240/2187, finding 7 at 37 generators up), and catalan (which adds 64/63, finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical extension, as it makes an intuitive further equivalence of 25/24~26/25~27/26 to 28/27 (by tempering out the square superparticular comma 729/728 (S27) in addition to 625/624 and 676/675), and can be defined independently in the 7-limit by tempering out 225/224 and 4375/4374. However, countercata is well-tuned closer to the optimal range of kleismic (between 53edo and 87edo), especially that of 2.3.5.13 cata, and naturally emerges in that context, identifying 64/63 with 65/64 by tempering out 4096/4095. Catakleismic and countercata merge in 53edo, as the former finds 7 at 22 generators up while the latter finds it at 31 generators down (22 + 31 = 53).
Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out 385/384, equating the ~6/5 generator to 77/64. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between just 6/5 (just flat of 19edo) and 77/64 (just sharp of 15edo).
For technical data, see Kleismic family #Kleismic a.k.a. hanson.
Interval chain
In the following table, odd harmonics 1–15 are labeled in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 317.1 | 6/5 |
| 2 | 634.2 | 13/9, 36/25 |
| 3 | 951.3 | 26/15 |
| 4 | 68.4 | 25/24, 26/25, 27/26 |
| 5 | 385.5 | 5/4 |
| 6 | 702.6 | 3/2 |
| 7 | 1019.6 | 9/5 |
| 8 | 136.7 | 13/12, 27/25 |
| 9 | 453.8 | 13/10 |
| 10 | 770.9 | 25/16, 39/25 |
| 11 | 1088.0 | 15/8 |
| 12 | 205.1 | 9/8 |
| 13 | 522.2 | 27/20 |
| 14 | 839.3 | 13/8 |
| 15 | 1156.4 | 39/20 |
| 16 | 273.5 | 75/64 |
| 17 | 590.6 | 45/32 |
| 18 | 907.7 | 27/16 |
| 19 | 24.7 | 65/64, 81/80 |
* In 2.3.5.13-subgroup CWE tuning, octave reduced
Tunings
Optimized tunings
| Euclidean | ||
|---|---|---|
| Constrained | Destretched | |
| Tenney | CTE: ~6/5 = 317.0308 ¢ | POTE: ~6/5 = 317.007 ¢ |
| Equilateral | CEE: ~6/5 = 317.1033 ¢ (11/61-kleisma) | |
| Euclidean | ||
|---|---|---|
| Constrained | Destretched | |
| Tenney | CTE: ~6/5 = 317.1110 ¢ | POTE: ~6/5 = 317.0756 ¢ |
| Optimized chord | Generator value | Polynomial | Further notes |
|---|---|---|---|
| 3:4:5 (+1 +1) | ~6/5 = 317.1496 | g6 + 2g5 − 8 = 0 | 1 – 3 – 5 equal-beating tuning, close to 8/43-kleisma |
| 4:5:6 (+1 +1) | ~6/5 = 317.9593 | g6 − 2g5 + 2 = 0 | 1 – 3 – 5 equal-beating tuning, close to 2/7-kleisma |
| 10:12:15 (+2 +3) | ~6/5 = 317.6675 | g6 − 5g + 3 = 0 | Close to 1/4-kleisma |
| 9:13:15 (+2 +1) | ~6/5 = 317.5679 | 3g3 + 4g − 10 = 0 | Close to 13/36-marveltwin comma |
| 13:15:18 (+2 +3) | ~6/5 = 317.0010 | 3g3 − g − 4 = 0 | Close to 13/51-marveltwin comma |
Other tunings
- DKW (2.3.5): ~2 = 1200.0000 ¢, ~6/5 = 317.1983 ¢
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 6/5 | 315.6413 | Untempered tuning, lower bound of 5-odd-limit diamond tradeoff | |
| 5\19 | 315.7895 | Lower bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone | |
| 27/26 | 316.3343 | 1/4-tunbarsma | |
| 29\110 | 316.3636 | 110ff val | |
| 24\91 | 316.4835 | 91f val | |
| 27/25 | 316.6547 | 1/8-kleisma | |
| 19\72 | 316.6667 | ||
| 9/5 | 316.7995 | 1/7-kleisma | |
| 33\125 | 316.8000 | 125f val | |
| 26/25 | 316.9750 | 1/4-marveltwin comma | |
| 14\53 | 316.9811 | ||
| 3/2 | 316.9925 | 1/6-kleisma; 5- and 9-odd-limit minimax tuning | |
| 65\246 | 317.0732 | ||
| 51\193 | 317.0984 | ||
| 15/8 | 317.1153 | 2/11-kleisma | |
| 13/10 | 317.1349 | 13- and 15-odd-limit minimax tuning | |
| 37\140 | 317.1429 | ||
| 13/8 | 317.1805 | ||
| 60\227 | 317.1807 | ||
| 23\87 | 317.2414 | ||
| 5/4 | 317.2627 | 1/5-kleisma, upper bound of 5-odd-limit diamond tradeoff | |
| 13/12 | 317.3216 | ||
| 32\121 | 317.3554 | ||
| 41\155 | 317.4194 | ||
| 15/13 | 317.4197 | 1/3-marveltwin comma | |
| 9\34 | 317.6471 | ||
| 25/24 | 317.6681 | 1/4-kleisma, virtually DR 10:12:15 | |
| 22\83 | 318.0723 | 83f val | |
| 13/9 | 318.3088 | 1/2-marveltwin comma, upper bound of 2.3.5.13-subgroup 15-odd-limit diamond tradeoff | |
| 125/72 | 318.3437 | 1/3-kleisma | |
| 13\49 | 318.3673 | 49f val | |
| 625/432 | 319.6949 | 1/2-kleisma | |
| 4\15 | 320.0000 | Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone |
* Besides the octave