Trisected
| Trisected |
56/55, 128/125, 1029/1000 (11-limit);
56/55, 91/90, 128/125, 1029/1000 (13-limit)
13-limit 21-odd-limit: 17.5 ¢
13-limit 21-odd-limit: 36 notes
Trisected is the rank-2 temperament tempering out 128/125, 1029/1000, and 1029/1024 in the 7-limit, making it a member of the augmented family, keegic temperaments, and gamelismic clan. Since it tempers out 128/125, the octave is split into 3 ~5/4's, each tuned to 400 ¢ if the octave is pure. Since it tempers out 1029/1024, the perfect fifth is split into three intervals of ~8/7.Since it tempers out 1029/1000, the tritave is split into three intervals of 10/7. This means that every Pythagorean interval is split into three equal parts.
In the 11-limit, the perfect fourth is split into three ~11/10's, thus tempering out 4000/3993. Additionally, the 1/3-octave period represents 14/11, tempering out 56/55 and 176/175. The 13-limit extension equates the ~10/7 with 13/9, tempering out 91/90 and 2197/2187.
The 2.3.7.11/5 subgroup restriction, known as trisect, removes the individual mappings for 5 and 11 while still tempering out 1029/1024 and 4000/3993, and is much more accurate.
For technical data, see Augmented family #Trisected.
Intervals
In the following table, odd harmonics 1–21 are in bold.
| # | Period 0 | Period 1 | Period 2 | |||
|---|---|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | 400.0 | 5/4, 14/11 | 800.0 | 8/5, 11/7 |
| 1 | 235.0 | 8/7 | 635.0 | 10/7, 16/11, 13/9 | 1035.0 | 20/11 |
| 2 | 470.0 | 21/16 | 870.0 | 33/20 | 70.0 | 21/20, 33/32 |
| 3 | 705.0 | 3/2 | 1105.0 | 15/8, 21/11 | 305.0 | 6/5 |
| 4 | 940.0 | 12/7, 26/15 | 140.0 | 12/11, 13/12, 15/14 | 540.0 | 15/11 |
| 5 | 1175.0 | 63/32 | 375.0 | 26/21 | 775.0 | 52/33 |
| 6 | 209.9 | 9/8 | 609.9 | 45/32, 63/44 | 1009.9 | 9/5 |
| 7 | 444.9 | 9/7, 13/10 | 844.9 | 18/11, 13/8 | 44.9 | 36/35 |
| 8 | 679.9 | 52/35, 72/49 | 1079.9 | 13/7 | 279.9 | 13/11 |
* In 13-limit CWE tuning, octave reduced
Tunings
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/7 = 633.889 ¢ | CWE: ~10/7 = 634.339 ¢ | POTE: ~10/7 = 634.476 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/7 = 634.215 ¢ | CWE: ~10/7 = 634.769 ¢ | POTE: ~10/7 = 634.893 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/7 = 634.286 ¢ | CWE: ~10/7 = 634.991 ¢ | POTE: ~10/7 = 635.144 ¢ |
Target tunings
| Target | Minimax | |
|---|---|---|
| Generator | Eigenmonzo* | |
| 7-odd-limit | ~10/7 = 633.282 ¢ | 7/6 |
| 9-odd-limit | ~10/7 = 633.583 ¢ | 9/7 |
| 11-odd-limit | ~10/7 = 633.760 ¢ | 77/45 |
| 13-odd-limit | ~10/7 = 633.962 ¢ | 13/7 |
| 15-odd-limit | ~10/7 = 633.962 ¢ | 13/7 |
| 13-limit 21-odd-limit | ~10/7 = 634.129 ¢ | 45/44 |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 7/5 | 617.488 | ||
| 11\21 | 628.571 | Lower bound of 7-odd-limit diamond monotone 21f val | |
| 15/8 | 629.423 | ||
| 15/14 | 629.861 | ||
| 7/4 | 631.174 | ||
| 7/6 | 633.282 | 7-odd-limit minimax | |
| 19\36 | Lower bound of 9- through 13-odd-limit diamond monotone 15-odd-limit diamond monotone (singleton) | ||
| 9/7 | 633.583 | 9-odd-limit minimax | |
| 21/13 | 633.949 | ||
| 13/7 | 633.962 | 13- and 15-odd-limit minimax | |
| 3/2 | 633.985 | ||
| 15/11 | 634.238 | ||
| 46\87 | 634.483 | 87cee val | |
| 13/8 | 634.361 | ||
| 13/12 | 634.643 | ||
| 11/10 | 634.996 | ||
| 27\51 | 635.294 | 51ce val | |
| 21/16 | 635.390 | ||
| 11/9 | 636.085 | ||
| 13/11 | 636.151 | ||
| 9/5 | 636.266 | ||
| 13/10 | 636.316 | ||
| 35\66 | 636.364 | 66cef val | |
| 13/9 | 636.618 | ||
| 11/6 | 637.659 | ||
| 15/13 | 638.065 | ||
| 5/3 | 638.547 | ||
| 21/11 | 639.821 | ||
| 8\15 | 640.000 | Upper bound of 7- through 13-odd-limit diamond monotone | |
| 21/20 | 642.234 |
* Besides the octave