Ripple family
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The ripple family of temperaments tempers out the ripple comma (ratio: 6561/6250, monzo: [-1 8 -5⟩), which equates a stack of five 27/25's with 4/3.
Ripple
The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered fourth reached at 5 semitones. Interestingly, in optimal tunings, the major third of ~5/4 does not tend to be damaged much sharpwards as one might expect from the equivalence, and is in practice often even flat, so that prime 3 takes on practically the whole damage of the 5-limit equivalence, for which it has the advantage of being the simplest so still having a good chance at psychoacoustic viability. As a result though, the mapping of ~9/8 is often inconsistent, so that ripple can in practice be thought of as a dual-fifth temperament unless you use tunings close to 12edo.
Reasonable patent val tunings not appearing in the optimal ET sequence are 35edo and 47edo.
Subgroup: 2.3.5
Comma list: 6561/6250
Mapping: [⟨1 2 3], ⟨0 -5 -8]]
- mapping generators: ~2, ~27/25
Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 100.838
- 5-odd-limit diamond monotone: [92.308, 109.091] (1\13 to 1\11)
- 5-odd-limit diamond tradeoff: [99.609, 105.214]
Optimal ET sequence: 11c, 12, 71b, 83b, 95b, 107bc, 119bc
Badness (Smith): 0.138948
Badness (Dirichlet): 3.259
Septimal ripple
Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~(21/16)⋅(4/3) = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.
If you are looking for the former canonical extension, see: #Rip.
Subgroup: 2.3.5.7
Mapping: [⟨1 2 3 4], ⟨0 -5 -8 -14]]
- CTE: ~15/14 = 101.538
- error map: ⟨0 -9.643 1.385 9.647]
- CE: ~15/14 = 101.881
- error map: ⟨0 -11.361 -1.364 4.837]
Optimal ET sequence: 11cd, 12, 35, 47
Badness (Dirichlet): 1.521
11-limit
A notable patent val tuning of 11-limit ripple not appearing in the optimal ET sequence is 47edo.
Subgroup: 2.3.5.7.11
Comma list: 126/125, 99/98, 45/44
Mapping: [⟨1 2 3 4 5], ⟨0 -5 -8 -14 -18]]
- CTE: ~15/14 = 101.538
- error map: ⟨0 -11.785, -2.041, 3.651, 13.296]
- CE: ~15/14 = 102.319
- error map: ⟨0 -13.551 -4.868 -1.296 6.935]
Optimal ET sequence: 11cdee, 12, 23de, 35
Badness (Dirichlet): 1.334
Rip
Formerly known as septimal ripple, but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that #Ripple admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.
Subgroup: 2.3.5.7
Comma list: 36/35, 2560/2401
Mapping: [⟨1 2 3 3], ⟨0 -5 -8 -2]]
Wedgie: ⟨⟨ 5 8 2 1 -11 -18 ]]
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 99.483
Badness (Smith): 0.059735
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 80/77, 126/121
Mapping: [⟨1 2 3 3 4], ⟨0 -5 -8 -2 -6]]
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 99.385
Optimal ET sequence: 12
Badness (Smith): 0.038811
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 40/39, 66/65, 147/143
Mapping: [⟨1 2 3 3 4 4], ⟨0 -5 -8 -2 -6 -3]]
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 98.572
Optimal ET sequence: 12f
Badness (Smith): 0.031639
Hemiripple
Subgroup: 2.3.5.7
Comma list: 49/48, 6561/6250
Mapping: [⟨1 2 3 3], ⟨0 -10 -16 -5]]
Wedgie: ⟨⟨ 10 16 5 2 -20 -33 ]]
Optimal tuning (POTE): ~2 = 1200.000, ~36/35 = 50.826
Optimal ET sequence: 23d, 24, 47d, 71bdd
Badness (Smith): 0.175113
11-limit
Subgroup: 2.3.5.7.11
Comma list: 49/48, 121/120, 567/550
Mapping: [⟨1 2 3 3 4], ⟨0 -10 -16 -5 -13]]
Optimal tuning (POTE): ~2 = 1200.000, ~36/35 = 50.826
Optimal ET sequence: 23de, 24, 47de, 71bdde
Badness (Smith): 0.066834
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 66/65, 121/120, 351/350
Mapping: [⟨1 2 3 3 4 4], ⟨0 -10 -16 -5 -13 -7]]
Optimal tuning (POTE): ~2 = 1200.000, ~36/35 = 50.635
Optimal ET sequence: 23de, 24, 47de, 71bdde
Badness (Smith): 0.046588
Cohemiripple
Subgroup: 2.3.5.7
Comma list: 245/243, 1323/1250
Mapping: [⟨1 7 11 12], ⟨0 -10 -16 -17]]
Wedgie: ⟨⟨ 10 16 17 2 -1 -5 ]]
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 549.944
Optimal ET sequence: 11cd, 13cd, 24
Badness (Smith): 0.190208
11-limit
Subgroup: 2.3.5.7.11
Comma list: 77/75, 243/242, 245/242
Mapping: [⟨1 7 11 12 17], ⟨0 -10 -16 -17 -25]]
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 549.945
Optimal ET sequence: 11cdee, 13cdee, 24
Badness (Smith): 0.082716
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 77/75, 147/143, 243/242
Mapping: [⟨1 7 11 12 17 14], ⟨0 -10 -16 -17 -25 -19]]
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 549.958
Optimal ET sequence: 11cdeef, 13cdeef, 24
Badness (Smith): 0.049933