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.
See Tetracot family #Modus for technical data.
Interval chain
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.00 | 1/1 |
| 1 | 176.95 | 10/9, 11/10 |
| 2 | 353.91 | 11/9, 16/13 |
| 3 | 530.86 | 15/11 |
| 4 | 707.81 | 3/2 |
| 5 | 884.77 | 5/3 |
| 6 | 1061.72 | 11/6, 13/7, 24/13 |
| 7 | 38.67 | 36/35, 40/39, 45/44, 55/54 |
| 8 | 215.63 | 8/7, 9/8 |
| 9 | 392.58 | 5/4 |
| 10 | 569.53 | 11/8, 18/13 |
| 11 | 746.49 | 20/13 |
| 12 | 923.44 | 12/7, 22/13 |
| 13 | 1100.39 | 15/8, 40/21 |
| 14 | 77.35 | 22/21, 25/24, 27/26 |
| 15 | 254.30 | 15/13 |
| 16 | 431.25 | 9/7 |
| 17 | 608.20 | 10/7 |
| 18 | 785.16 | 11/7 |
| 19 | 962.11 | |
| 20 | 1139.06 | 27/14 |
| 21 | 116.02 | 15/14 |
* in 13-limit POTE tuning
Tunings
Tuning spectrum
| Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|
| 11/10 | 165.004 | |
| 11/9 | 173.704 | |
| 12/11 | 174.894 | |
| 11/8 | 175.132 | |
| 4/3 | 175.489 | |
| 13/11 | 175.899 | |
| 16/15 | 176.021 | |
| 5/4 | 176.257 | 5-odd-limit minimax |
| 18/13 | 176.338 | |
| 15/13 | 176.516 | |
| 14/11 | 176.805 | 11, 13 and 15-odd-limit minimax |
| 6/5 | 176.872 | |
| 13/10 | 176.890 | |
| 13/12 | 176.905 | |
| 15/14 | 177.116 | |
| 9/7 | 177.193 | 9-odd-limit minimax |
| 7/5 | 177.499 | 7-odd-limit minimax |
| 7/6 | 177.761 | |
| 14/13 | 178.617 | |
| 8/7 | 178.897 | |
| 15/11 | 178.984 | |
| 16/13 | 179.736 | |
| 10/9 | 182.404 |
Music
See Tetracot #Music.