Cloudy clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
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: mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


[[Optimal tuning]] ([[POTE]]): ~15/14 = 120.000{{c}}, ~3/2 = 698.696{{c}} (~49/48 = 21.304{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}


{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


[[Badness]] (Smith): 0.089135
[[Badness]] (Sintel): 2.26


=== 11-limit ===
=== 11-limit ===
Line 34: Line 38:
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 696.791{{c}} (~56/55 = 23.209{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})


{{Optimal ET sequence|legend=0| 10, 30be, 40, 50 }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness (Smith): 0.063900
Badness (Sintel): 2.11


==== 13-limit ====
==== 13-limit ====
Line 47: Line 53:
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 696.993{{c}} (~78/77 = 23.007{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


{{Optimal ET sequence|legend=0| 10, 30be, 40, 50 }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness (Smith): 0.036880
Badness (Sintel): 1.52


==== 17-limit ====
==== 17-limit ====
Line 60: Line 68:
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 697.085{{c}} (~78/77 = 22.915{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


{{Optimal ET sequence|legend=0| 10, 30beg, 40, 50 }}
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Badness (Smith): 0.025064
Badness (Sintel): 1.28


=== Splendecic ===
=== Splendecic ===
Line 75: Line 85:
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 698.518{{c}} (~99/98 = 21.482{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


{{Optimal ET sequence|legend=0| 10e, 30bee, 40e, 50, 60e, 110de }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Badness (Smith): 0.059802
Badness (Sintel): 1.98


==== 13-limit ====
==== 13-limit ====
Line 88: Line 100:
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 698.365{{c}} (~91/90 = 21.635{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


{{Optimal ET sequence|legend=0| 10e, 30bee, 40e, 50, 60e, 110de }}
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Badness (Smith): 0.037989
Badness (Sintel): 1.57


==== 17-limit ====
==== 17-limit ====
Line 101: Line 115:
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 698.375{{c}} (~91/90 = 21.625{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


{{Optimal ET sequence|legend=0| 10e, 30beeg, 40e, 50, 60e, 110deg }}
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Badness (Smith): 0.026050
Badness (Sintel): 1.33


=== Prodecic ===
=== Prodecic ===
Line 116: Line 132:
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 700.210{{c}} (~55/54 = 19.790{{c}})
Optimal tunings:
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


{{Optimal ET sequence|legend=0| 10, 40ee, 50e, 60e }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness (Smith): 0.066600
Badness (Sintel): 2.20


==== 13-limit ====
==== 13-limit ====
Line 129: Line 147:
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 700.503{{c}} (~91/90 = 19.497{{c}})
Optimal tunings:
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


{{Optimal ET sequence|legend=0| 10, 40ee, 50e, 60e }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness (Smith): 0.041788
Badness (Sintel): 1.73


==== 17-limit ====
==== 17-limit ====
Line 142: Line 162:
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
Mapping: {{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 tunings:
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


{{Optimal ET sequence|legend=0| 10, 40ee, 50e, 60e }}
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Badness (Smith): 0.027590
Badness (Sintel): 1.41


== Pentadecal ==
== Pentadecal ==

Revision as of 08:55, 25 June 2026

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 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:

Decic

Main article: Decic

Named by Xenllium in 2021, decic 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⟩. It maybe described as the 50 & 60 temperament, supported by 10-, 50-, and 60edo. It 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

Optimal tunings:

  • WE: ~15/14 = 120.1837 ¢, ~3/2 = 699.7654 ¢ (~49/48 = 21.3366 ¢)
error map: ⟨+1.837 -0.353 -0.753 -3.683]
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.8236 ¢ (~49/48 = 21.1764 ¢)
error map: ⟨0.000 -3.131 -5.137 -8.826]

Optimal ET sequence: 10, 30b, 40, 50, 60, 110d, 170cdd

Badness (Sintel): 2.26

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]]

Optimal tunings:

  • WE: ~15/14 = 120.1406 ¢, ~3/2 = 697.6075 ¢ (~56/55 = 23.2360 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.0142 ¢ (~56/55 = 22.9858 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 2.11

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]]

Optimal tunings:

  • WE: ~15/14 = 120.1166 ¢, ~3/2 = 697.6705 ¢ (~78/77 = 23.0289 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.1492 ¢ (~78/77 = 22.8508 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 1.52

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]]

Optimal tunings:

  • WE: ~15/14 = 120.1262 ¢, ~3/2 = 697.8185 ¢ (~78/77 = 22.9388 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 697.2757 ¢ (~78/77 = 22.7243 ¢)

Optimal ET sequence: 10, 40, 50

Badness (Sintel): 1.28

Splendecic

Splendecic (50 & 60e) 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 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 tunings:

  • WE: ~15/14 = 120.1874 ¢, ~3/2 = 699.6085 ¢ (~99/98 = 21.5156 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.3531 ¢ (~99/98 = 21.6469 ¢)

Optimal ET sequence: 10e, 40e, 50, 60e, 110de, 170cddee

Badness (Sintel): 1.98

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 tunings:

  • WE: ~15/14 = 120.1565 ¢, ~3/2 = 699.2756 ¢ (~91/90 = 21.6631 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.2480 ¢ (~91/90 = 21.7520 ¢)

Optimal ET sequence: 10e, 40e, 50, 60e, 110de

Badness (Sintel): 1.57

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 tunings:

  • WE: ~15/14 = 120.1571 ¢, ~3/2 = 699.2892 ¢ (~91/90 = 21.6536 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 698.3144 ¢ (~91/90 = 21.6856 ¢)

Optimal ET sequence: 10e, 50, 60e, 110deg

Badness (Sintel): 1.33

Prodecic

Prodecic (50e & 60e) 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 tunings:

  • WE: ~15/14 = 120.2024 ¢, ~3/2 = 701.3908 ¢ (~55/54 = 19.8237 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.5235 ¢ (~55/54 = 19.4765 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 2.20

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 tunings:

  • WE: ~15/14 = 120.1654 ¢, ~3/2 = 701.4683 ¢ (~91/90 = 19.5242 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.7175 ¢ (~91/90 = 19.2825 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 1.73

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 tunings:

  • WE: ~15/14 = 120.1577 ¢, ~3/2 = 701.3950 ¢ (~91/90 = 19.5514 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 700.6932 ¢ (~91/90 = 19.3068 ¢)

Optimal ET sequence: 10, 50e, 60e

Badness (Sintel): 1.41

Pentadecal

For the 5-limit version, see Miscellaneous 5-limit temperaments #Pentadecal.

Named by Xenllium in 2021, pentadecal tempers out the 15-5/3-comma ([-11 -15 15⟩) in the 5-limit. 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 = 80.000 ¢, ~3/2 = 700.223 ¢ (~126/125 = 19.777 ¢)

Optimal ET sequence: 15, 45, 60

Badness (Smith): 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]]

Optimal tuning (POTE): ~21/20 = 80.000 ¢, ~3/2 = 702.733 ¢ (~56/55 = 17.267 ¢)

Optimal ET sequence: 15, 45, 60, 75de, 135de

Badness (Smith): 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]]

Optimal tuning (POTE): ~21/20 = 80.000 ¢, ~3/2 = 701.715 ¢ (~91/90 = 18.285 ¢)

Optimal ET sequence: 15, 45f, 60, 135de, 195cddee

Badness (Smith): 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]]

Optimal tuning (POTE): ~21/20 = 80.000 ¢, ~3/2 = 700.318 ¢ (~126/125 = 19.682 ¢)

Optimal ET sequence: 15, 45e, 60e

Badness (Smith): 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]]

Optimal tuning (POTE): ~21/20 = 80.000 ¢, ~3/2 = 701.647 ¢ (~65/64 = 18.353 ¢)

Optimal ET sequence: 15f, 45e, 60e

Badness (Smith): 0.055361

Quinkee

Quinkee has a period of 1/5 octave and tempers out the keega (1029/1000), from which it derives its name. It may be described as the 15 & 40 temperament. It was named by Xenllium in 2021.

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 = 240.000 ¢, ~5/4 = 393.439 ¢ (~25/24 = 86.561 ¢)

Optimal ET sequence: 15, 40, 55, 125cd, 180ccdd

Badness (Smith): 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 = 240.000 ¢, ~5/4 = 393.438 ¢ (~22/21 = 86.562 ¢)

Optimal ET sequence: 15, 40, 55, 125cd, 180ccdde

Badness (Smith): 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 = 240.000 ¢, ~5/4 = 392.798 ¢ (~22/21 = 87.202 ¢)

Optimal ET sequence: 15, 40, 55

Badness (Smith): 0.048440

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