Unidec
Unidec is a temperament generated by a wide major third representing 80/63 which can be taken as 14/11 from the 11-limit onwards. Two generators minus a semi-octave period make a ~8/7, and three ~8/7's make a perfect fifth of 3/2, tempering out 1029/1024, the gamelisma, and 4375/4374, the ragisma, with the 11-limit extension adding 385/384 or 441/440 to the comma list.
| Unidec |
385/384, 441/440, 4375/4374 (11-limit)
11-limit 21-odd-limit: 2.95 ¢
11-limit 21-odd-limit: 46 notes
Unidec has two viable 13-limit extensions: hendec and ekadash, discussed in Unidec extensions.
See Gamelismic clan #Unidec for technical data.
Interval chain
In the following table, odd harmonics 1–21 and their inverses are in bold.
| # | Period 0 | Period 1 | ||
|---|---|---|---|---|
| Cents* | Approximate ratios | Cents* | Approximate ratios | |
| 0 | 0.00 | 1/1 | 600.00 | 99/70, 140/99 |
| 1 | 416.85 | 14/11 | 1016.80 | 9/5 |
| 2 | 833.71 | 81/50, 160/99 | 233.71 | 8/7 |
| 3 | 50.56 | 33/32, 36/35 | 650.56 | 16/11 |
| 4 | 467.42 | 21/16 | 1067.42 | 50/27 |
| 5 | 884.27 | 5/3 | 284.27 | 33/28 |
| 6 | 101.13 | 35/33 | 701.13 | 3/2 |
| 7 | 517.98 | 27/20 | 1117.98 | 21/11, 40/21 |
| 8 | 934.83 | 12/7 | 334.83 | 40/33 |
| 9 | 151.69 | 12/11 | 751.69 | 54/35 |
| 10 | 568.54 | 25/18 | 1168.54 | 55/28, 63/32, 96/49, 108/55 |
| 11 | 985.40 | 99/56 | 385.40 | 5/4 |
| 12 | 202.25 | 9/8 | 802.25 | 35/22 |
| 13 | 619.11 | 10/7 | 19.11 | 81/80, 99/98, 100/99 |
* In 11-limit CWE tuning, octave reduced
Chords and harmony
Scales
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~80/63 = 416.9403 ¢ | CWE: ~80/63 = 416.8688 ¢ | POTE: ~80/63 = 416.8385 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~14/11 = 416.9262 ¢ | CWE: ~14/11 = 416.8543 ¢ | POTE: ~14/11 = 416.8350 ¢ |
Tuning spectrum
| Eigenmonzo (unchanged interval) |
Generator (¢) | Comments |
|---|---|---|
| 13/12 | 415.357 | |
| 7/4 | 415.587 | |
| 13/10 | 416.198 | |
| 11/8 | 416.227 | |
| 13/9 | 416.338 | |
| 15/13 | 416.516 | |
| 7/6 | 416.641 | |
| 7/5 | 416.730 | 7-odd-limit minimax |
| 11/6 | 416.737 | |
| 11/10 | 416.785 | |
| 9/7 | 416.792 | 9- and 11-odd-limit minimax |
| 15/14 | 416.813 | |
| 11/9 | 416.839 | |
| 15/11 | 416.848 | |
| 5/3 | 416.872 | |
| 5/4 | 416.938 | 5-, 13- and 15-odd-limit minimax |
| 15/8 | 416.957 | |
| 3/2 | 416.993 | |
| 11/7 | 417.508 | |
| 9/5 | 417.596 | |
| 13/11 | 417.842 | |
| 13/7 | 417.925 | |
| 13/8 | 420.264 |
Music
- "Hypnocloudsmack 2" from Newbeams