Keemic temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
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{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
: mapping generators: ~2, ~42/25
: mapping generators: ~2, ~42/25


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{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
: mapping generators: ~2, ~36/35


: Mapping generators: ~2, ~25/24
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1201.4060{{c}}, ~36/35 = 53.8791{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/24 = 53.816{{c}}
: [[error map]]: {{val| +1.406 -0.121 -6.348 +3.810 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 53.9055{{c}}
: error map: {{val| 0.000 -1.183 -8.975 +1.474 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 53.816{{c}} -->


{{Optimal ET sequence|legend=1| 22, 45, 67c }}
{{Optimal ET sequence|legend=1| 22, 45, 67c }}


[[Badness]] (Smith): 0.090769
[[Badness]] (Sintel): 2.30


== Barbad ==
== Barbad ==
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[[Comma list]]: 875/864, 16875/16807
[[Comma list]]: 875/864, 16875/16807


{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}
{{Mapping|legend=1| 1 -10 -5 -10 | 0 19 12 21 }}
: mapping generators: ~2, ~98/75


: Mapping generators: ~2, ~98/75
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1201.0462{{c}}, ~75/49 = 732.3071{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98/75 = 468.331{{c}}
: [[error map]]: {{val| +1.046 +1.418 -3.859 -0.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 731.7183{{c}}
: error map: {{val| 0.000 +0.692 -5.694 -2.742 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~75/49 = 731.669{{c}} -->


{{Optimal ET sequence|legend=0| 18, 23d, 41 }}
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}


[[Badness]] (Smith): 0.110448
[[Badness]] (Sintel): 2.80


=== 11-limit ===
=== 11-limit ===
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Comma list: 245/242, 540/539, 625/616
Comma list: 245/242, 540/539, 625/616


Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~98/75 = 468.367{{c}}
Optimal tunings:
* WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~75/49 = 731.633{{c}} -->


{{Optimal ET sequence|legend=0| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness (Smith): 0.050105
Badness (Sintel): 1.66


=== 13-limit ===
=== 13-limit ===
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Comma list: 144/143, 196/195, 245/242, 275/273
Comma list: 144/143, 196/195, 245/242, 275/273


Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}
Mapping: {{mapping| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 468.270{{c}}
Optimal tunings:
* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~20/13 = 731.730{{c}} -->


{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness (Smith): 0.039183
Badness (Sintel): 1.62


== Hyperkleismic ==
== Hyperkleismic ==
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{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
: mapping generators: ~2, ~6/5


: Mapping generators: ~2, ~6/5
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.0290{{c}}, ~6/5 = 323.7882{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 323.780{{c}}
: [[error map]]: {{val| +0.029 +2.358 -5.759 +2.597 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 323.7816{{c}}
: error map: {{val| 0.000 +2.332 -5.808 +2.519 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 323.780{{c}} -->


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


[[Badness]] (Smith): 0.157830
[[Badness]] (Sintel): 3.99


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 323.796{{c}}
Optimal tunings:
* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 323.796{{c}} -->


{{Optimal ET sequence|legend=0| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness (Smith): 0.065356
Badness (Sintel): 2.16


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 323.790{{c}}
Optimal tunings:
* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 323.790{{c}} -->


{{Optimal ET sequence|legend=0| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness (Smith): 0.035724
Badness (Sintel): 1.48


== Sevond ==
== Sevond ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''  
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''  


10/9 is tempered to be exactly 1\7 of an octave. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.
10/9 is tempered to be exactly 1\7. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
: mapping generators: ~10/9, ~3


: Mapping generators: ~10/9, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~10/9 = 171.4007{{c}}, ~3/2 = 705.4982{{c}}
[[Optimal tuning]] ([[POTE]]): ~10/9 = 171.429{{c}}, ~3/2 = 705.613{{c}}
: [[error map]]: {{val| -0.195 +3.348 -4.112 -0.499 }}
* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.6057{{c}}
: error map: {{val| 0.000 +3.651 -3.674 +0.071 }}
<!-- * [[POTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 705.613{{c}} -->


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=1| 7, …, 56, 63, 119 }}


[[Badness]] (Smith): 0.206592
[[Badness]] (Sintel): 5.23


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}


Optimal tuning (POTE): ~10/9 = 171.429{{c}}, ~3/2 = 705.518{{c}}
Optimal tunings:
* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}
<!-- * POTE: ~11/10 = 171.429{{c}}, ~3/2 = 705.518{{c}} -->


{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness (Smith): 0.070437
Badness (Sintel): 2.33


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


Optimal tuning (POTE): ~10/9 = 171.429{{c}}, ~3/2 = 705.344{{c}}
Optimal tunings:
* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}
<!-- * POTE: ~10/9 = 171.429{{c}}, ~3/2 = 705.344{{c}} -->


{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness (Smith): 0.041238
Badness (Sintel): 1.70


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Revision as of 10:58, 10 October 2025

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.

This is a collection of temperaments that temper out the keema (monzo: [-5 -3 3 1⟩, ratio: 875/864), with S-expression S5/S6. Its fundamental equivalence entails that 6/5 is sharpened so that it stacks three times to reach 7/4, and the interval between 6/5 and 5/4 is compressed so that 7/6–6/5–5/4–9/7 are set equidistant from each other. As the canonical extension of rank-3 keemic to the 11-limit tempers out the commas 100/99 and 385/384 (whereby (6/5)2 is identified with 16/11), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.

Full 7-limit keemic temperaments discussed elsewhere are:

Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.

Quasitemp

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

Quasitemp tempers out 2401/2400 in addition to 875/864 and may be described as the 37 & 41 temperament. It has a strong restriction to the 2.5/3.7/3 subgroup, called gariberttet, which is defined by tempering out 3125/3087. In gariberttet, three generators reach 5/3 and five reach 7/3, so that the generator itself has the interpretation of 25/21. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. In quasitemp, tempering out 875/864 entails that 8/7 is found after 9 generators, from which the mappings of 3 and 5 follow.

Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's ploidacot is iota-14-cot. This generator is equated to 22/13 for the 13-limit extension, tempering out 275/273.

Subgroup: 2.3.5.7

Comma list: 875/864, 2401/2400

Mapping: [⟨1 -9 -6 -4], ⟨0 14 11 9]]

mapping generators: ~2, ~42/25

Optimal tunings:

  • WE: ~2 = 1200.9237 ¢, ~42/25 = 907.9887 ¢
error map: ⟨+0.924 +1.573 -3.981 -0.623]
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.3471 ¢
error map: ⟨0.000 +0.905 -5.495 -2.702]

Optimal ET sequence: 4, …, 37, 41

Badness (Sintel): 1.53

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1375/1372

Mapping: [⟨1 -9 -6 -4 8], ⟨0 14 11 9 -6]]

Optimal tunings:

  • WE: ~2 = 1199.9585 ¢, ~42/25 = 907.4221 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.4521 ¢

Optimal ET sequence: 4, 37, 41, 119

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 275/273, 385/384

Mapping: [⟨1 -9 -6 -4 8 9], ⟨0 14 11 9 -6 -7]]

Optimal tunings:

  • WE: ~2 = 1199.4376 ¢, ~22/13 = 907.1175 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 907.5314 ¢

Optimal ET sequence: 4, 37, 41, 78, 119f

Badness (Sintel): 1.36

Quato

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 625/616

Mapping: [⟨1 -9 -6 -4 -23], ⟨0 14 11 9 35]]

Optimal tunings:

  • WE: ~2 = 1201.2729 ¢, ~42/25 = 908.1116 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.2109 ¢

Optimal ET sequence: 41, 127cd, 168cd

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 243/242, 275/273, 325/324

Mapping: [⟨1 -9 -6 -4 -23 -22], ⟨0 14 11 9 35 34]]

Optimal tunings:

  • WE: ~2 = 1201.4078 ¢, ~42/25 = 908.1362 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.1370 ¢

Optimal ET sequence: 41, 86ce

Badness (Sintel): 1.24

Chromo

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

Chromo represents the 13edf chain as a rank-2 temperament, with 6/5 and 5/4 mapped to 6 and 7 steps, respectively. Since the difference of those two intervals is abbreviated considerably from just, keemic provides the most meaningful 7-limit extension (setting 7/6, 6/5, 5/4, 9/7 equidistant) so that the temperament then approximates the 4:5:6:7 tetrad with 0:7:13:18 generator steps.

Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer escapade.

Subgroup: 2.3.5.7

Comma list: 875/864, 2430/2401

Mapping: [⟨1 1 2 2], ⟨0 13 7 18]]

mapping generators: ~2, ~36/35

Optimal tunings:

  • WE: ~2 = 1201.4060 ¢, ~36/35 = 53.8791 ¢
error map: ⟨+1.406 -0.121 -6.348 +3.810]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 53.9055 ¢
error map: ⟨0.000 -1.183 -8.975 +1.474]

Optimal ET sequence: 22, 45, 67c

Badness (Sintel): 2.30

Barbad

Subgroup: 2.3.5.7

Comma list: 875/864, 16875/16807

Mapping: [⟨1 -10 -5 -10], ⟨0 19 12 21]]

mapping generators: ~2, ~98/75

Optimal tunings:

  • WE: ~2 = 1201.0462 ¢, ~75/49 = 732.3071 ¢
error map: ⟨+1.046 +1.418 -3.859 -0.838]
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 731.7183 ¢
error map: ⟨0.000 +0.692 -5.694 -2.742]

Optimal ET sequence: 18, 23d, 41

Badness (Sintel): 2.80

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 540/539, 625/616

Mapping: [⟨1 -10 -5 -10 -13], ⟨0 19 12 21 27]]

Optimal tunings:

  • WE: ~2 = 1200.8513 ¢, ~75/49 = 732.1519 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 731.6740 ¢

Optimal ET sequence: 18e, 23de, 41

Badness (Sintel): 1.66

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 245/242, 275/273

Mapping: [⟨1 -10 -5 -10 -13 -3], ⟨0 19 12 21 27 11]]

Optimal tunings:

  • WE: ~2 = 1199.7960 ¢, ~20/13 = 731.6053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 731.7208 ¢

Optimal ET sequence: 18e, 23de, 41

Badness (Sintel): 1.62

Hyperkleismic

Subgroup: 2.3.5.7

Comma list: 875/864, 51200/50421

Mapping: [⟨1 -3 -2 2], ⟨0 17 16 3]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.0290 ¢, ~6/5 = 323.7882 ¢
error map: ⟨+0.029 +2.358 -5.759 +2.597]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 323.7816 ¢
error map: ⟨0.000 +2.332 -5.808 +2.519]

Optimal ET sequence: 26, 37, 63

Badness (Sintel): 3.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 2420/2401

Mapping: [⟨1 -3 -2 2 4], ⟨0 17 16 3 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9010 ¢, ~6/5 = 323.7691 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 323.7931 ¢

Optimal ET sequence: 26, 37, 63

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 275/273, 385/384

Mapping: [⟨1 -3 -2 2 4 1], ⟨0 17 16 3 -2 10]]

Optimal tunings:

  • WE: ~2 = 1200.0524 ¢, ~6/5 = 323.8039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 323.7912 ¢

Optimal ET sequence: 26, 37, 63

Badness (Sintel): 1.48

Sevond

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sevond (5-limit).

10/9 is tempered to be exactly 1\7. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.

Subgroup: 2.3.5.7

Comma list: 875/864, 327680/321489

Mapping: [⟨7 0 -6 53], ⟨0 1 2 -3]]

mapping generators: ~10/9, ~3

Optimal tunings:

  • WE: ~10/9 = 171.4007 ¢, ~3/2 = 705.4982 ¢
error map: ⟨-0.195 +3.348 -4.112 -0.499]
  • CWE: ~10/9 = 171.4286 ¢, ~3/2 = 705.6057 ¢
error map: ⟨0.000 +3.651 -3.674 +0.071]

Optimal ET sequence: 7, …, 56, 63, 119

Badness (Sintel): 5.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 6655/6561

Mapping: [⟨7 0 -6 53 2], ⟨0 1 2 -3 2]]

Optimal tunings:

  • WE: ~11/10 = 171.3859 ¢, ~3/2 = 705.3421 ¢
  • CWE: ~11/10 = 171.4286 ¢, ~3/2 = 705.4973 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Sintel): 2.33

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 352/351, 385/384

Mapping: [⟨7 0 -6 53 2 37], ⟨0 1 2 -3 2 -1]]

Optimal tunings:

  • WE: ~11/10 = 171.4163 ¢, ~3/2 = 705.2930 ¢
  • CWE: ~11/10 = 171.4286 ¢, ~3/2 = 705.3402 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Sintel): 1.70

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