Cloudy clan
The cloudy clan tempers out the cloudy comma, 16807/16384, thus splitting the octave into 5 equal parts and mapping the harmonic 7 to 4\5.
Temperaments discussed elsewhere are:
- Blacksmith (+28/27 or 49/48) → Limmic temperaments
- Cloudtone (+81/80) → Meantone family
- Pental (+245/243) → Pental family
- Sengasec (+686/675) → Quintosec family
- Warlock (+3125/3072) → Magic family
Coblack
- See also: Trisedodge family #Coblack
In addition to the cloudy comma, the coblack temperament tempers out the starling comma (126/125). Coblack was named by Petr Pařízek in 2011, who considered it a counterpart of blacksmith[1].
Subgroup: 2.3.5.7
Comma list: 126/125, 16807/16384
Mapping: [⟨5 1 7 14], ⟨0 3 2 0]]
Wedgie: ⟨⟨15 10 0 -19 -42 -28]]
- CTE: ~8/7 = 1\5, ~48/35 = 553.8429 (~21/20 = 73.8429)
- POTE: ~8/7 = 1\5, ~48/35 = 553.044 (~21/20 = 73.044)
Optimal ET sequence: 15, 35, 50, 65, 115d
Badness: 0.107282
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 245/242, 385/384
Mapping: [⟨5 1 7 14 15], ⟨0 3 2 0 1]]
Optimal tunings:
- CTE: ~8/7 = 1\5, ~11/8 = 553.7951 (~21/20 = 73.7951)
- POTE: ~8/7 = 1\5, ~11/8 = 553.264 (~21/20 = 73.264)
Optimal ET sequence: 15, 35, 50, 65, 115d
Badness: 0.045070
Decic
The decic temperament (10 & 50) tempers out the cloudy comma, 16807/16384 and the marvel comma, 225/224 in the 7-limit, as well as the linus comma, [11 -10 -10 10⟩. This temperament is supported by 10-, 50-, and 60edo, and can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.
Subgroup: 2.3.5.7
Comma list: 225/224, 16807/16384
Mapping: [⟨10 0 39 28], ⟨0 1 -1 0]]
- mapping generators: ~15/14, ~3
Wedgie: ⟨⟨10 -10 0 -39 -28 28]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.696 (~49/48 = 21.304)
Optimal ET sequence: 10, 30b, 40, 50, 60, 110d, 170cdd
Badness: 0.089135
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 3087/3025
Mapping: [⟨10 0 39 28 3], ⟨0 1 -1 0 2]]
- mapping generators: ~15/14, ~3
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 696.791 (~56/55 = 23.209)
Optimal ET sequence: 10, 30be, 40, 50
Badness: 0.063900
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 196/195, 2200/2197
Mapping: [⟨10 0 39 28 3 37], ⟨0 1 -1 0 2 0]]
- mapping generators: ~14/13, ~3
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 696.993 (~78/77 = 23.007)
Optimal ET sequence: 10, 30be, 40, 50
Badness: 0.036880
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
Mapping: [⟨10 0 39 28 3 37 25], ⟨0 1 -1 0 2 0 1]]
- mapping generators: ~14/13, ~3
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 697.085 (~78/77 = 22.915)
Optimal ET sequence: 10, 30beg, 40, 50
Badness: 0.025064
Splendecic
Splendecic (10e & 50) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the fantastic rank-3 temperament, its name is a portmanteau of "splendid" (synonym of "fantastic") and "decic".
Subgroup: 2.3.5.7.11
Comma list: 225/224, 1617/1600, 2401/2376
Mapping: [⟨10 0 39 28 82], ⟨0 1 -1 0 -3]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.518 (~99/98 = 21.482)
Optimal ET sequence: 10e, 30bee, 40e, 50, 60e, 110de
Badness: 0.059802
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 1001/1000, 1188/1183
Mapping: [⟨10 0 39 28 82 37], ⟨0 1 -1 0 -3 0]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.365 (~91/90 = 21.635)
Optimal ET sequence: 10e, 30bee, 40e, 50, 60e, 110de
Badness: 0.037989
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
Mapping: [⟨10 0 39 28 82 37 25], ⟨0 1 -1 0 -3 0 1]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.375 (~91/90 = 21.625)
Optimal ET sequence: 10e, 30beeg, 40e, 50, 60e, 110deg
Badness: 0.026050
Prodecic
Prodecic (10 & 50e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the prodigy rank-3 temperament, its name is a portmanteau of "prodigy" and "decic".
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 5929/5832
Mapping: [⟨10 0 39 28 -13], ⟨0 1 -1 0 3]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.210 (~55/54 = 19.790)
Optimal ET sequence: 10, 40ee, 50e, 60e
Badness: 0.066600
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 275/273, 5929/5832
Mapping: [⟨10 0 39 28 -13 37], ⟨0 1 -1 0 3 0]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.503 (~91/90 = 19.497)
Optimal ET sequence: 10, 40ee, 50e, 60e
Badness: 0.041788
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
Mapping: [⟨10 0 39 28 -13 37 25], ⟨0 1 -1 0 3 0 1]]
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.474 (~91/90 = 19.526)
Optimal ET sequence: 10, 40ee, 50e, 60e
Badness: 0.027590
Pentadecal
- For the 5-limit version of this temperament, see High badness temperaments #Pentadecal.
The pentadecal temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), [-11 -15 15⟩. This temperament can be described as 15&60 temperament, tempering out the cloudy comma, 16807/16384 and the keema, 875/864 in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 875/864, 16807/16384
Mapping: [⟨15 0 11 42], ⟨0 1 1 0]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.223 (~126/125 = 19.777)
Optimal ET sequence: 15, 45, 60
Badness: 0.114833
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 385/384, 1372/1331
Mapping: [⟨15 0 11 42 52], ⟨0 1 1 0 0]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 702.733 (~56/55 = 17.267)
Optimal ET sequence: 15, 45, 60, 75de, 135de
Badness: 0.077420
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 144/143, 1372/1331
Mapping: [⟨15 0 11 42 52 8], ⟨0 1 1 0 0 2]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.715 (~91/90 = 18.285)
Optimal ET sequence: 15, 45f, 60, 135de, 195cddee
Badness: 0.051740
Quindecal
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440, 875/864
Mapping: [⟨15 0 11 42 28], ⟨0 1 1 0 1]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.318 (~126/125 = 19.682)
Optimal ET sequence: 15, 45e, 60e
Badness: 0.044405
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 196/195, 352/351, 875/864
Mapping: [⟨15 0 11 42 28 103], ⟨0 1 1 0 1 -2]]
- mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.647 (~65/64 = 18.353)
Optimal ET sequence: 15f, 45e, 60e
Badness: 0.055361
Quinkee
The quinkee temperament (15 & 40) has a period of 1/5 octave and tempers out the keega (1029/1000), and from this it derives its name.
Subgroup: 2.3.5.7
Comma list: 1029/1000, 6144/6125
Mapping: [⟨5 9 12 14], ⟨0 -3 -1 0]]
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 393.439 (~25/24 = 86.561)
Optimal ET sequence: 15, 40, 55, 125cd, 180ccdd
Badness: 0.184222
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175, 1029/1000
Mapping: [⟨5 9 12 14 18], ⟨0 -3 -1 0 -2]]
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 393.438 (~22/21 = 86.562)
Optimal ET sequence: 15, 40, 55, 125cd, 180ccdde
Badness: 0.065192
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 105/104, 121/120, 637/625
Mapping: [⟨5 9 12 14 18 20], ⟨0 -3 -1 0 -2 -4]]
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 392.798 (~22/21 = 87.202)
Optimal ET sequence: 15, 40, 55
Badness: 0.048440