Cathartic 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 canthartic family of rank-3 temperaments tempers out the cathartic comma (monzo: [4 -14 3 4⟩, ratio: 4802000/4782969), a 7-limit comma measuring about 6.9 cents.
Cathartic
Cathartic (formerly canou) features a period of an octave and generators of 3/2 and 81/70. The ~81/70 generator is about 255 cents wide, three of which make 14/9, and four make 9/5. It therefore splits the large septimal diesis, 49/48, into three equal parts, guaranteeing the existence of two interseptimal intervals related to the 35th harmonic.
A basic tuning option would be 99edo, although 80edo is even simpler and distinctive. More intricate tunings are provided by 311edo and 410edo, whereas the optimal patent val goes up to 1131edo, associating it with the amicable temperament.
9-, 14- and 19-note scales are highly characteristic for the temperament, with the abundance of 28/27 subminor seconds making it melodically active.
Subgroup: 2.3.5.7
Comma list: 4802000/4782969
Mapping: [⟨1 0 0 -1], ⟨0 1 2 2], ⟨0 0 -4 3]]
- mapping generators: ~2, ~3, ~81/70
Lattice basis:
- 3/2 length = 0.8110, 81/70 length = 0.5135
- Angle (3/2, 81/70) = 73.88 deg
- WE: ~2 = 1199.9597 ¢, ~3/2 = 702.3492 ¢, ~81/70 = 254.6168 ¢
- error map: ⟨-0.040 +0.354 -0.163 -0.317]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3455 ¢, ~81/70 = 254.6237 ¢
- error map: ⟨0.000 +0.390 -0.118 -0.264]
- 7-odd-limit: 3 +c/14, 5 and 7 just
- 9-odd-limit: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
Optimal ET sequence: 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b
Badness (Sintel): 4.95
Complexity spectrum: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
Undecimal cathartic
The fifth is in the range where a stack of four (i.e. a major third) can serve as ~19/15 and a stack of five (i.e. a major seventh) can serve as ~19/10, tempering out 1216/1215. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot 22/19, so it only makes sense to temper out their difference, 1540/1539. The implied 11-limit comma is the symbiotic comma.
Since the syntonic comma has been split in two, it is natural to map 19/17 to the mean of 9/8 and 10/9, tempering out 1445/1444, while the other 11-limit comma, 42875/42768 (S34⋅S352), suggests tempering out 595/594 (S34⋅S35), 1156/1155 (S34), and 1225/1224 (S35), which coincides with above. Finally, we can map 23/20 to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out 760/759. These extensions add little additional error, and 311edo remains an excellent tuning in all cases.
Subgroup: 2.3.5.7.11
Comma list: 19712/19683, 42875/42768
Mapping: [⟨1 0 0 -1 -7], ⟨0 1 2 2 7], ⟨0 0 -4 3 -3]]
- WE: ~2 = 1200.0568 ¢, ~3/2 = 702.2009 ¢, ~81/70 = 254.6291 ¢
- error map: ⟨+0.057 +0.303 -0.314 -0.480 +0.201]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1829 ¢, ~81/70 = 254.6186 ¢
- error map: ⟨0.0000 +0.228 -0.422 -0.604 +0.107]
Optimal ET sequence: 24, …, 75e, 94, 99e, 118, 193, 212, 311, 740, 1051d
Badness (Sintel): 2.45
Complexity spectrum: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5
2.3.5.7.11.17 subgroup
Subgroup: 2.3.5.7.11.17
Comma list: 595/594, 1156/1155, 19712/19683
Subgroup-val mapping: [⟨1 0 0 -1 -7 -5], ⟨0 1 2 2 7 6], ⟨0 0 -4 3 -3 -2]]
Optimal tunings:
- WE: ~2 = 1200.0485 ¢, ~3/2 = 702.2481 ¢, ~51/44 = 254.6343 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2325 ¢, ~51/44 = 254.6253 ¢
Optimal ET sequence: 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg
Badness (Sintel): 1.01
2.3.5.7.11.17.19 subgroup
Subgroup: 2.3.5.7.11.17.19
Comma list: 595/594, 969/968, 1156/1155, 1216/1215
Subgroup-val mapping: [⟨1 0 0 -1 -7 -5 -6], ⟨0 1 2 2 7 6 7], ⟨0 0 -4 3 -3 -2 -4]]
Optimal tunings:
- WE: ~2 = 1200.0444 ¢, ~3/2 = 702.2569 ¢, ~22/19 = 254.6305 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2421 ¢, ~22/19 = 254.6224 ¢
Optimal ET sequence: 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh
Badness (Sintel): 0.641
2.3.5.7.11.17.19.23 subgroup
Subgroup: 2.3.5.7.11.17.19.23
Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155
Mapping: [⟨1 0 0 -1 -7 -5 -6 4], ⟨0 1 2 2 7 6 7 1], ⟨0 0 -4 3 -3 -2 -4 -5]]
Optimal tunings:
- WE: ~2 = 1199.9778 ¢, ~3/2 = 702.2794 ¢, ~22/19 = 254.6572 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2883 ¢, ~22/19 = 254.6625 ¢
Optimal ET sequence: 24, 75e, 94, 99e, 118, 193, 217, 311
Badness (Sintel): 0.703
Canta
By adding 896/891, the pentacircle comma, 33/32 is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)⋅(364/363), named gentcanta in earlier materials.
Subgroup: 2.3.5.7.11
Comma list: 896/891, 472392/471625
Mapping: [⟨1 0 0 -1 6], ⟨0 1 2 2 -2], ⟨0 0 4 -3 -3]]
- WE: ~2 = 1199.0708 ¢, ~3/2 = 703.1969 ¢, ~64/55 = 254.4161 ¢
- error map: ⟨-0.929 +0.313 +0.557 -0.113 +1.820]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.5249 ¢, ~64/55 = 254.5492 ¢
- error map: ⟨0.000 +1.570 +2.539 +1.871 +5.280]
Optimal ET sequence: 75e, 80, 99e, 179e, 457bcddeeee
Badness (Sintel): 5.43
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 472392/471625
Mapping: [⟨1 0 0 -1 6 11], ⟨0 1 2 2 -2 -5], ⟨0 0 4 -3 -3 -3]]
Optimal tunings:
- WE: ~2 = 1199.0093 ¢, ~3/2 = 703.2884 ¢, ~64/55 = 254.4219 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8323 ¢, ~64/55 = 254.5887 ¢
Optimal ET sequence: 75e, 80, 99ef, 179ef, 462bccddeeeff
Badness (Sintel): 4.47
Semicathart
Semicathart (formerly semicanou) adds 9801/9800, the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that 99/70 = (81/70)⋅(11/9).
The other comma necessary to define it is 14641/14580, the semicathartisma, which is the difference between 121/120 and 243/242. By flattening the 11th harmonic by about one cent, it identifies 20/11 by three 11/9's stacked, so an octave can be divided into 11/9, 11/9, 11/9, and 11/10.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 14641/14580
Mapping: [⟨2 0 0 -2 1], ⟨0 1 2 2 2], ⟨0 0 -4 3 -1]]
- mapping generators: ~99/70, ~3, ~81/70
- WE: ~99/70 = 600.0142 ¢, ~3/2 = 702.4017 ¢, ~81/70 = 254.6228 ¢
- error map: ⟨+0.028 +0.475 +0.055 -0.126 -1.066]
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.4048 ¢, ~81/70 = 254.6179 ¢
- error map: ⟨0.0000 +0.450 +0.024 -0.163 -1.126]
Optimal ET sequence: 80, 94, 118, 198, 212, 292, 330e, 410
Badness (Sintel): 2.64