Ripple family

The ripple family of temperaments tempers out the ripple comma, 6561/6250 = [-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

Tuning ranges:

• 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

See also: Dual-fifth temperaments

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.

Subgroup: 2.3.5.7

Comma list: 126/125, 405/392

Mapping: [⟨1 2 3 4], ⟨0 -5 -8 -14]]

Optimal tunings:

• 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]]

Optimal tunings:

• CTE: ~15/14 = 101.538

error map: ⟨0 -11.785, -2.041, 3.651, 13.296]

• CE: ~15/14 = 102.319 (preferred for dual-fifths 11-limit)

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 #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

Optimal ET sequence: 12

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

See also: Sensamagic clan

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