Modus
The modus temperament is one of the 7-limit extensions of tetracot, the 5-limit temperament tempering out the tetracot comma (20000/19683), and is naturally a full 13-limit temperament.
| Modus |
64/63, 100/99, 243/242 (11-limit)
64/63, 78/77, 100/99, 144/143
(13-limit)
13-odd-limit: 16.7 ¢
13-odd-limit: 20 notes
In addition to the tetracot comma, modus tempers out 64/63, making it a member of the archytas clan. As such, septimal intervals are tempered together with Pythagorean intervals; in particular, a stack of two perfect fifths octave reduced represents 8/7~9/8 at 8 generator steps. Modus also tempers out 4375/4374, making it a ragismic temperament. In the 11- and 13-limit it can be viewed as a weak extension of suhajira as well.
Additionally, the generator can be taken to represent 21/19, which gives us an extension for prime 19 at -5 generator steps.
Modus was named by Mike Battaglia in 2012 for its fantastic modmos structures[1].
See Tetracot family #Modus for technical data.
Interval chain
In the following tables, odd harmonics 1–13 and their inverses are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 176.9 | 10/9, 11/10 |
| 2 | 353.7 | 11/9, 16/13 |
| 3 | 530.6 | 15/11 |
| 4 | 707.5 | 3/2 |
| 5 | 884.4 | 5/3 |
| 6 | 1061.2 | 11/6, 13/7, 24/13 |
| 7 | 38.1 | 36/35, 40/39, 45/44, 55/54 |
| 8 | 215.0 | 8/7, 9/8 |
| 9 | 391.9 | 5/4 |
| 10 | 568.7 | 11/8, 18/13 |
| 11 | 745.6 | 20/13 |
| 12 | 922.5 | 12/7, 22/13 |
| 13 | 1099.4 | 15/8, 40/21 |
| 14 | 76.2 | 22/21, 25/24, 27/26 |
| 15 | 253.1 | 15/13 |
| 16 | 430.0 | 9/7 |
| 17 | 606.8 | 10/7 |
| 18 | 783.7 | 11/7 |
| 19 | 960.6 | 45/26 |
| 20 | 1137.5 | 27/14 |
* in 13-limit CWE tuning
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 176.8176 ¢ | CWE: ~10/9 = 177.1188 ¢ | POTE: ~10/9 = 177.2035 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 176.4456 ¢ | CWE: ~10/9 = 176.9286 ¢ | POTE: ~10/9 = 177.0530 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 176.4708 ¢ | CWE: ~10/9 = 176.8735 ¢ | POTE: ~10/9 = 176.9532 ¢ |
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 11/10 | 165.004 | ||
| 1\7 | 171.429 | ||
| 11/9 | 173.704 | ||
| 11/6 | 174.894 | ||
| 11/8 | 175.132 | ||
| 3/2 | 175.489 | ||
| 13/11 | 175.899 | ||
| 15/8 | 176.021 | ||
| 5/4 | 176.257 | 5-odd-limit minimax | |
| 13/9 | 176.338 | ||
| 5\34 | 176.471 | 34d val, lower bound of 7- to 15-odd-limit diamond monotone | |
| 15/13 | 176.516 | ||
| 11/7 | 176.805 | 11-, 13- and 15-odd-limit minimax | |
| 5/3 | 176.872 | ||
| 13/10 | 176.890 | ||
| 13/12 | 176.905 | ||
| 9\61 | 177.049 | 61de val | |
| 15/14 | 177.116 | ||
| 9/7 | 177.193 | 9-odd-limit minimax | |
| 7/5 | 177.499 | 7-odd-limit minimax | |
| 7/6 | 177.761 | ||
| 4\27 | 177.778 | 27e val, upper bound of 11- to 15-odd-limit diamond monotone | |
| 13/7 | 178.617 | ||
| 7/4 | 178.897 | ||
| 15/11 | 178.984 | ||
| 13/8 | 179.736 | ||
| 3\20 | 180.000 | 20ce val, upper bound of 7- and 9-odd-limit diamond monotone | |
| 9/5 | 182.404 |
* Besides the octave
Music
See Tetracot #Music.