Juggernaut
Juggernaut is a 5.7.11 nonoctave regular temperament, first documented by User:CompactStar, tempering out 125/121. It's subgroup does not contain harmonics 2 and 3 and so it uses the pentave (5/1) as its equivalence instead of the more common octave or even tritave. It has a period of 1\2ed5 (1393 cents) representing 11/5, and a generator representing 7/5 (in fact, in the CTE tuning it is exactly 7/5). This gives juggernaut an extremely low complexity with 5th, 7th, and 11th harmonics all reachable within just 1 generator, while still having only a moderately high error. It is one of the lowest-badness 5/1-equivalent or "no-twos-or-threes" temperaments, similar to meantone and BPS/lambda in their respective spheres. 14ed5 (practically the same as 6edo) is the first ed5 offering a workable tuning of juggernaut with the generator as 3\14ed5, while 24ed5 offers a more accurate tuning with the generator of 5\24ed5.
The best extension of juggernaut to the no-twos-or-threes 13-limit, named "cuthbernaut", splits the 7/5 generator into two 13/11 by tempering out 847/845. The next best extension has been named "tridecimal juggernaut" since it preserves the original 7/5 generator, mapping 13/5 to -2 generators by tempering out 637/625. Tridecimal juggernaut favors a flatter 7/5 (in the vicinity of 570 cents) for the least error.
Juggernaut contains multi-MOS scales of the families 4L 2s, 4L 6s, 10L 4s, 14L 10s, and 24L 14s. The 6-note MOS is rendered unusable because it has very large melodic steps (it corresponds to to 6*log(2)/log(5) ≈ 2.6 note octave-repeating scale) and contains too little 5:7:11 chords for the usage in no-twos-or-threes harmony.
Intervals
| Generator | Cents* | Ratios | Ratios (tridecimal juggernaut) |
|---|---|---|---|
| -5 | 1266.911 | 715/343 | |
| -4 | 456.266 | 3025/2401 | 65/49 |
| -3 | 1038.778 | 605/343, 625/343 | 13/7, 1625/847 |
| -2 | 228.133 | 55/49, 625/539 | 13/11, 143/125 |
| -1 | 810.645 | 11/7, 125/77 | 91/55, 1001/625 |
| 0 | 0.000 | 1/1 | |
| 1 | 582.512 | 7/5, 847/625 | 121/91, 125/91 |
| 2 | 1165.024 | 49/25, 5929/3125 | 25/13, 121/65 |
| 3 | 354.379 | 343/275, 3773/3125 | 77/65, 9317/8125 |
| 4 | 936.891 | 2401/1375 | 539/325 |
| 5 | 126.246 | 343/325 |
*In no-twos-or-threes 11-limit CTE tuning
Tuning spectrum
This assume tridecimal juggernaut mapping for no-twos-or-threes 13-limit intervals.
| ED5 generator | Eigenmonzo (unchanged-interval) | Cents |
|---|---|---|
| 91/25 | 549.588 | |
| 13/11 | 551.974 | |
| 77/25 | 554.360 | |
| 2\10ed5 | 557.263 | |
| 13/5 | 566.050 | |
| 539/125 | 568.436 | |
| 13/7 | 571.538 | |
| 7\34ed5 | 573.653 | |
| 49/13 | 574.281 | |
| 5\24ed5 | 580.482 | |
| 143/125 | 580.126 | |
| 7/5 | 582.512 | |
| 8\38ed5 | 586.592 | |
| 11\52ed5 | 589.413 | |
| 49/11 | 596.589 | |
| 3\14ed5 | 597.067 | |
| 343/121 | 601.281 | |
| 10\46ed5 | 605.720 | |
| 7\32ed5 | 609.506 | |
| 11/7 | 610.665 | |
| 4\18ed5 | 619.181 | |
| 5\22ed5 | 633.253 | |
| 121/35 | 638.818 |