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 tunings:

• CTE: ~2 = 1200.000, ~27/25 = 100.752

error map: ⟨0.000 -5.717 +7.668]

• CWE: ~2 = 1200.000, ~27/25 = 100.798

error map: ⟨0.000 -5.946 +7.300]

• POTE: ~2 = 1200.000, ~27/25 = 100.838

error map: ⟨0.000 -6.145 +6.982]

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

Badness:

• Smith 0.139
• Dirichlet: 3.26

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.

If you are looking for the former canonical extension, see #Rip.

Subgroup: 2.3.5.7

Comma list: 126/125, 405/392

Mapping: [⟨1 2 3 4], ⟨0 -5 -8 -14]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~15/14 = 101.538

error map: ⟨0 -9.643 +1.385 +9.647]

• CWE: ~2 = 1200.000, ~15/14 = 101.777

error map: ⟨0 -10.841 -0.531 +6.294]

• CEE: ~2 = 1200.000, ~15/14 = 101.881

error map: ⟨0 -11.361 -1.364 +4.837]

Optimal ET sequence: 11cd, 12, 35, 47

Badness:

• Smith: 0.0601
• Dirichlet: 1.52

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: 45/44, 99/98, 126/125

Mapping: [⟨1 2 3 4 5], ⟨0 -5 -8 -14 -18]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~15/14 = 101.538

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

• CWE: ~2 = 1200.000, ~15/14 = 102.297

error map: ⟨0 -13.441 -4.691 -0.986 +7.333]

• CEE: ~2 = 1200.000, ~15/14 = 102.319

error map: ⟨0 -13.551 -4.868 -1.296 +6.935]

Optimal ET sequence: 11cdee, 12, 23de, 35

Badness:

• Smith: 0.0403
• Dirichlet: 1.33

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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~21/20 = 101.089

error map: ⟨0.000 -7.402 +4.970 +28.995]

• CWE: ~2 = 1200.000, ~21/20 = 100.109

error map: ⟨0.000 -2.501 +12.812 +30.956]

• POTE: ~2 = 1200.000, ~21/20 = 99.483

error map: ⟨0.000 +0.632 +17.825 +32.209]

Optimal ET sequence: 11c, 12

Badness (Smith): 0.0597

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 tunings:

• CTE: ~2 = 1200.000, ~21/20 = 101.923
• CWE: ~2 = 1200.000, ~21/20 = 100.320
• POTE: ~2 = 1200.000, ~21/20 = 99.385

Optimal ET sequence: 11c, 12

Badness (Smith): 0.0388

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 tunings:

• CTE: ~2 = 1200.000, ~21/20 = 102.376
• CWE: ~2 = 1200.000, ~21/20 = 99.762
• POTE: ~2 = 1200.000, ~21/20 = 98.572

Optimal ET sequence: 11c, 12f, 37ccddeeeeffff

Badness (Smith): 0.0316

Hemiripple

Subgroup: 2.3.5.7

Comma list: 49/48, 6561/6250

Mapping: [⟨1 2 3 3], ⟨0 -10 -16 -5]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~36/35 = 50.231

error map: ⟨0.000 -4.264 +9.991 -19.981]

• CWE: ~2 = 1200.000, ~36/35 = 50.593

error map: ⟨0.000 -7.883 +4.201 -21.790]

• POTE: ~2 = 1200.000, ~36/35 = 50.826

error map: ⟨0.000 -10.214 +0.472 -22.956]

Optimal ET sequence: 23d, 24, 47d

Badness (Smith): 0.175

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 tunings:

• CTE: ~2 = 1200.000, ~36/35 = 50.186
• CWE: ~2 = 1200.000, ~36/35 = 50.587
• POTE: ~2 = 1200.000, ~36/35 = 50.826

Optimal ET sequence: 23de, 24, 47de

Badness (Smith): 0.0668

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 tunings:

• CTE: ~2 = 1200.000, ~36/35 = 50.225
• CWE: ~2 = 1200.000, ~36/35 = 50.505
• POTE: ~2 = 1200.000, ~36/35 = 50.635

Optimal ET sequence: 23de, 24

Badness (Smith): 0.0466

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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~7/5 = 550.063

error map: ⟨0.000 -1.328 +14.690 -17.760]

• CWE: ~2 = 1200.000, ~7/5 = 549.998

error map: ⟨0.000 -1.976 +13.653 -18.861]

• POTE: ~2 = 1200.000, ~7/5 = 549.944

error map: ⟨0.000 -2.515 +12.791 -19.777]

Optimal ET sequence: 11cd, 13cd, 24

Badness (Smith): 0.190

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 tunings:

• CTE: ~2 = 1200.000, ~7/5 = 550.060
• CWE: ~2 = 1200.000, ~7/5 = 549.997
• POTE: ~2 = 1200.000, ~7/5 = 549.945

Optimal ET sequence: 11cdee, 13cdee, 24

Badness (Smith): 0.0827

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 tunings:

• CTE: ~2 = 1200.000, ~7/5 = 549.987
• CWE: ~2 = 1200.000, ~7/5 = 549.966
• POTE: ~2 = 1200.000, ~7/5 = 549.958

Optimal ET sequence: 11cdeef, 13cdeef, 24

Badness (Smith): 0.0499