Keemic temperaments: Difference between revisions

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Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 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.  
Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 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.  


The generator is equated to [[13/11]] for the 13-limit extension, tempering out [[275/273]].  
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
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 875/864, 2401/2400
[[Comma list]]: 875/864, 2401/2400


{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}
{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}


: Mapping generators: ~2, ~25/21
: mapping generators: ~2, ~42/25


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 292.710{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9237{{c}}, ~42/25 = 907.9887{{c}}
: [[error map]]: {{val| +0.924 +1.573 -3.981 -0.623 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}
: error map: {{val| 0.000 +0.905 -5.495 -2.702 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~42/25 = 907.290{{c}} -->


{{Optimal ET sequence|legend=1| 4, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}


[[Badness]] (Smith): 0.060269
[[Badness]] (Sintel): 1.53


=== 11-limit ===
=== 11-limit ===
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Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}
Mapping: {{mapping| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 292.547{{c}}
Optimal tunings:
* WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 907.453{{c}} -->


{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}


Badness (Smith): 0.043209
Badness (Sintel): 1.43


==== 13-limit ====
==== 13-limit ====
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Comma list: 100/99, 196/195, 275/273, 385/384
Comma list: 100/99, 196/195, 275/273, 385/384


Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
Mapping: {{mapping| 1 -9 -6 -4 8 9 | 0 14 11 9 -6 -7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 292.457{{c}}
Optimal tunings:
* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 907.543{{c}} -->


{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}


Badness (Smith): 0.032913
Badness (Sintel): 1.36


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


Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}
Mapping: {{mapping| 1 -9 -6 -4 -23 | 0 14 11 9 35 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 292.851{{c}}
Optimal tunings:
* WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 907.149{{c}} -->


{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}


Badness (Smith): 0.041170
Badness (Sintel): 1.36


==== 13-limit ====
==== 13-limit ====
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Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}
Mapping: {{mapping| 1 -9 -6 -4 -23 -22 | 0 14 11 9 35 34 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 292.928{{c}}
Optimal tunings:
* WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 907.072{{c}} -->


{{Optimal ET sequence|legend=0| 41, 86ce, 127cd }}
{{Optimal ET sequence|legend=0| 41, 86ce }}


Badness (Smith): 0.030081
Badness (Sintel): 1.24


== Chromo ==
== Chromo ==

Revision as of 10:37, 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, ~25/24

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/24 = 53.816 ¢

Optimal ET sequence: 22, 45, 67c

Badness (Smith): 0.090769

Barbad

Subgroup: 2.3.5.7

Comma list: 875/864, 16875/16807

Mapping: [⟨1 9 7 11], ⟨0 -19 -12 -21]]

Mapping generators: ~2, ~98/75

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98/75 = 468.331 ¢

Optimal ET sequence: 18, 23d, 41

Badness (Smith): 0.110448

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 9 7 11 14], ⟨0 -19 -12 -21 -27]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98/75 = 468.367 ¢

Optimal ET sequence: 18e, 23de, 41, 228ccdd

Badness (Smith): 0.050105

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 9 7 11 14 8], ⟨0 -19 -12 -21 -27 -11]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/10 = 468.270 ¢

Optimal ET sequence: 18e, 23de, 41

Badness (Smith): 0.039183

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 tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 323.780 ¢

Optimal ET sequence: 26, 37, 63

Badness (Smith): 0.157830

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 tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 323.796 ¢

Optimal ET sequence: 26, 37, 63

Badness (Smith): 0.065356

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 tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 323.790 ¢

Optimal ET sequence: 26, 37, 63

Badness (Smith): 0.035724

Sevond

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.

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 tuning (POTE): ~10/9 = 171.429 ¢, ~3/2 = 705.613 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Smith): 0.206592

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 tuning (POTE): ~10/9 = 171.429 ¢, ~3/2 = 705.518 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Smith): 0.070437

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 tuning (POTE): ~10/9 = 171.429 ¢, ~3/2 = 705.344 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Smith): 0.041238

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