Keemun
| Keemun |
49/48, 56/55, 100/99 (11-limit);
49/48, 56/55, 91/90, 100/99 (13-limit)
13-odd-limit: 27.3 ¢
13-odd-limit: 19 notes
Keemun is an extension of the hanson temperament. It identifies the hemitwelfth as 7/4~12/7 and the two-generator step as 13/9~16/11, tempering out 49/48, 56/55, 91/90 and 100/99 in the 13-limit. This means it uses the same simple mappings for the 7th and 11th harmonics as orgone. Unfortunately, the optimal tuning for the 3rd and 5th harmonics is substantially flatter than that for the 7th & 11th, requiring you to compromise one set for the other. Edos that support keemun in their patent vals are 4edo, 15edo, 19edo, and 34edo, with 49d and 64bde coming closer to balancing the errors equally.
This temperament was originally discovered by Dave Keenan and named by Herman Miller in 2006 after the Chinese black tea.[1][2]
See Kleismic family #Keemun for technical data. See Keemun extensions for a discussion on alternative 11- and 13-limit extensions.
Interval chain
In the following table, odd harmonics 1–13 and their inverses are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 317.4 | 6/5 |
| 2 | 634.8 | 10/7, 13/9, 16/11 |
| 3 | 952.2 | 7/4, 12/7, 15/13 |
| 4 | 69.6 | 21/20, 25/24, 27/26, 33/32, 36/35 |
| 5 | 387.0 | 5/4, 14/11 |
| 6 | 704.4 | 3/2 |
| 7 | 1021.8 | 9/5, 20/11 |
| 8 | 139.3 | 12/11, 13/12, 15/14 |
| 9 | 456.7 | 9/7, 13/10, 21/16 |
| 10 | 774.1 | 25/16 |
| 11 | 1091.5 | 13/7, 15/8 |
| 12 | 208.9 | 9/8 |
| 13 | 526.3 | 15/11, 27/20 |
| 14 | 843.7 | 13/8, 18/11 |
| 15 | 1161.1 | 27/14 |
* In 13-limit CWE tuning, octave reduced
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~6/5 = 317.3927 ¢ | CWE: ~6/5 = 316.8293 ¢ | POTE: ~6/5 = 316.4727 ¢ |
Tuning spectrum
This tuning spectrum assumes septimal keemun; for the tuning spectrum of tridecimal keemun, see Keemun extensions #Tuning spectra.
| Edo generator |
Eigenmonzo (unchanged interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 1\4 | 300.000 | Lower bound of 5- and 7-odd-limit diamond monotone | |
| 7/5 | 308.744 | ||
| 7/6 | 311.043 | ||
| 6\23 | 313.043 | 23d val | |
| 15/14 | 314.930 | ||
| 9/7 | 315.009 | ||
| 5/3 | 315.641 | ||
| 5\19 | 315.789 | Lower bound of 9-odd-limit diamond monotone | |
| 9/5 | 316.799 | 1/7-kleisma | |
| 14\53 | 316.981 | 53d val | |
| 3/2 | 316.993 | 5-, 7- and 9-odd-limit minimax, 1/6-kleisma | |
| 15/8 | 317.115 | 2/11-kleisma | |
| 5/4 | 317.263 | 1/5-kleisma | |
| 9\34 | 317.647 | ||
| 13\49 | 318.367 | 49d val | |
| 4\15 | 320.000 | Upper bound of 9-odd-limit diamond monotone | |
| 7/4 | 322.942 | ||
| 3\11 | 327.273 | 11b val, upper bound of 5- and 7-odd-limit diamond monotone |
* Besides the octave