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Keemic temperaments: Difference between revisions

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These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]]. Keemic temperaments include [[Jubilismic clan #Doublewide|doublewide]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Porcupine|porcupine]], [[Gamelismic clan #Superkleismic|superkleismic]], [[Magic family #Magic|magic]], [[Kleismic family #Keemun|keemun]], and [[Sycamore family #Sycamore|sycamore]]. Discussed below are quasitemp and barbad.
{{Technical data page}}
This is a collection of [[rank-2 temperament|linear]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -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]])<sup>2</sup> 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:
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Dicot|Dicot family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Monkey]] (+5120/5103) → [[Tetracot family #Monkey|Tetracot family]]
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Undeka]]'' (+3200/3087) → [[11th-octave temperaments #Undeka|11th-octave temperaments]]
 
Discussed below are quasitemp, chromo, barbad, pentadecal, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].


== Quasitemp ==
== Quasitemp ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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 {{nowrap| 37 & 41 }} temperament. It is characterized by equating the interval between the pental and septimal thirds ([[36/35]]) with the classical chromatic semitone ([[25/24]]), and by tempering together the septimal dieses of [[49/48]] and [[50/49]]. In that sense, it is opposed to [[orwellismic temperaments]], in particular [[myna]], where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.
 
Quasitemp can also be thought of as a [[strong extension]] of the 2.5/3.7/3-subgroup temperament 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. 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 2401/2400
[[Comma list]]: 875/864, 2401/2400


[[Mapping]]: [{{val| 1 5 5 5 }}, {{val| 0 -14 -11 -9 }}]
{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
 
: mapping generators: ~2, ~42/25
{{Multival|legend=1| 14 11 9 -15 -25 -10 }}


[[POTE generator]]: ~25/21 = 292.710
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 23: Line 47:
Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


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


POTE generator: ~25/21 = 292.547
Optimal tunings:  
* WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}


Vals: {{Val list| 4, 37, 41, 119 }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}


Badness: 0.043209
Badness (Sintel): 1.43


==== 13-limit ====
==== 13-limit ====
Line 36: Line 62:
Comma list: 100/99, 196/195, 275/273, 385/384
Comma list: 100/99, 196/195, 275/273, 385/384


POTE generator: ~13/11 = 292.457
Mapping: {{mapping| 1 -9 -6 -4 8 9 | 0 14 11 9 -6 -7 }}


Mapping: [{{val| 1 5 5 5 2 2 }}, {{val| 0 -14 -11 -9 6 7 }}]
Optimal tunings:  
* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}


POTE generator: ~13/11 = 292.457
{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}


Vals: {{Val list| 4, 37, 41, 78, 119f }}
Badness (Sintel): 1.36
 
Badness: 0.032913


=== Quato ===
=== Quato ===
Line 51: Line 77:
Comma list: 243/242, 441/440, 625/616
Comma list: 243/242, 441/440, 625/616


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


POTE generator: ~25/21 = 292.851
Optimal tunings:  
* WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}


Vals: {{Val list| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}


Badness: 0.041170
Badness (Sintel): 1.36


==== 13-limit ====
==== 13-limit ====
Line 64: Line 92:
Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


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


POTE generator: ~13/11 = 292.928
Optimal tunings:  
* WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}


Vals: {{Val list| 41, 86ce, 127cd }}
{{Optimal ET sequence|legend=0| 41, 86ce }}


Badness: 0.030081
Badness (Sintel): 1.24


== Chromo ==
== Chromo ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Chromo]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Chromo]].''


Subgroup: 2.3.5.7
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
[[Comma list]]: 875/864, 2430/2401


[[Mapping]]: [{{Val|1 1 2 2}}, {{Val|0 13 7 18}}]
{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
: mapping generators: ~2, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4060{{c}}, ~36/35 = 53.8791{{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 }}
 
{{Optimal ET sequence|legend=1| 22, 45, 67c }}
 
[[Badness]] (Sintel): 2.30
 
== Barbad ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 16875/16807
 
{{Mapping|legend=1| 1 -10 -5 -10 | 0 19 12 21 }}
: mapping generators: ~2, ~98/75
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0462{{c}}, ~75/49 = 732.3071{{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 }}
 
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}
 
[[Badness]] (Sintel): 2.80
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 540/539, 625/616
 
Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}
 
Optimal tunings:
* WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}
 
Optimal tunings:
* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}
 
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
 
Badness (Sintel): 1.62
 
== 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 ({{monzo| -11 -15 15 }}) in the 5-limit. This temperament can be described as {{nowrap| 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|legend=1| 15 0 11 42 | 0 1 1 0 }}
: mapping generators: ~21/20, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 80.1141{{c}}, ~3/2 = 700.2213{{c}} (~126/125 = 19.8053{{c}})
: [[error map]]: {{val| +1.711 +0.977 -2.127 -4.035 }}
* [[CWE]]: ~21/20 = 80.0000{{c}}, ~3/2 = 701.2357{{c}} (~126/125 = 19.7643{{c}})
: error map: {{val| 0.000 -0.719 -5.078 -8.826 }}
 
{{Optimal ET sequence|legend=1| 15, 45, 60 }}
 
[[Badness]] (Sintel): 2.91
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 1372/1331
 
Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}
 
Optimal tunings:
* WE: ~21/20 = 80.0213{{c}}, ~3/2 = 702.9194{{c}} (~56/55 = 17.2721{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 702.8751{{c}} (~56/55 = 17.1249{{c}})
 
{{Optimal ET sequence|legend=0| 15, 45, 60, 75de, 135de }}
 
Badness (Sintel): 2.56


[[POTE generator]]: ~25/24 = 53.816
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 22, 45, 67c }}
Comma list: 100/99, 105/104, 144/143, 1372/1331


[[Badness]]: 0.090769
Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}


== Undeka ==
Optimal tunings:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Undeka]].''
* WE: ~21/20 = 80.0207{{c}}, ~3/2 = 701.8966{{c}} (~91/90 = 18.2900{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8545{{c}} (~91/90 = 18.1455{{c}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 15, 45f, 60, 135de, 195cddee }}


[[Comma list]]: 875/864, 3200/3087
Badness (Sintel): 2.14


[[Mapping]]: [{{Val|11 0 8 31}}, {{Val|0 1 1 0}}]
=== Quindecal ===
Subgroup: 2.3.5.7.11


{{Multival|legend=1|11 11 0 -8 -31 -31}}
Comma list: 121/120, 441/440, 875/864


[[POTE generator]]: ~3/2 = 708.792
Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}


{{Val list|legend=1| 11c, 22 }}
Optimal tunings:
* WE: ~21/20 = 80.1322{{c}}, ~3/2 = 701.4751{{c}} (~126/125 = 19.7148{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.5453{{c}} (~126/125 = 18.4547{{c}})


[[Badness]]: 0.141782
{{Optimal ET sequence|legend=0| 15, 45e, 60e }}
 
Badness (Sintel): 1.47
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 196/195, 352/351, 875/864
 
Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}
 
Optimal tunings:
* WE: ~21/20 = 80.1015{{c}}, ~3/2 = 702.5374{{c}} (~126/125 = 18.3763{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8504{{c}} (~126/125 = 18.1496{{c}})
 
{{Optimal ET sequence|legend=0| 15f, 60e, 135deef }}
 
Badness (Sintel): 2.29
 
== Hyperkleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 51200/50421
 
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0290{{c}}, ~6/5 = 323.7882{{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 }}
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
[[Badness]] (Sintel): 3.99


=== 11-limit ===
=== 11-limit ===
Sugbroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 352/343, 385/384
Comma list: 100/99, 385/384, 2420/2401


Mapping: [{{Val|11 0 8 31 38}}, {{Val|0 1 1 0 0}}]
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}


POTE generator: ~3/2 = 706.768
Optimal tunings:
* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}


Vals: {{Val list| 11c, 22 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness: 0.068672
Badness (Sintel): 2.16


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 65/63, 100/99, 169/165, 352/343
Comma list: 100/99, 169/168, 275/273, 385/384


Mapping: [{{Val|11 0 8 31 38 23}}, {{Val|0 1 1 0 0 1}}]
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}


POTE generator: ~3/2 = 707.764
Optimal tunings:
* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}


Vals: {{Val list| 11cf, 22 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness: 0.056528
Badness (Sintel): 1.48


== Barbad ==
== Sevond ==
Subgroup: 2.3.5.7
: ''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.


[[Comma list]]: 875/864, 16875/16807
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 9 7 11 }}, {{val| 0 -19 -12 -21 }}]
[[Comma list]]: 875/864, 327680/321489


{{Multival|legend=1| 19 12 21 -25 -20 15 }}
{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
: mapping generators: ~10/9, ~3


[[POTE generator]]: ~98/75 = 468.331
[[Optimal tuning]]s:  
* [[WE]]: ~10/9 = 171.4007{{c}}, ~3/2 = 705.4982{{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 }}


{{Val list|legend=1| 18, 23d, 41 }}
{{Optimal ET sequence|legend=1| 7, …, 56, 63, 119 }}


[[Badness]]: 0.110448
[[Badness]] (Sintel): 5.23


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 245/242, 540/539, 625/616
Comma list: 100/99, 385/384, 6655/6561


Mapping: [{{val| 1 9 7 11 14 }}, {{val| 0 -19 -12 -21 -27 }}]
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}


POTE generator: ~98/75 = 468.367
Optimal tunings:  
* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}


Vals: {{Val list| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness: 0.050105
Badness (Sintel): 2.33


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 196/195, 245/242, 275/273
Comma list: 100/99, 169/168, 352/351, 385/384


Mapping: [{{val| 1 9 7 11 14 8 }}, {{val| 0 -19 -12 -21 -27 -11 }}]
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


POTE generator: ~13/10 = 468.270
Optimal tunings:
* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}


Vals: {{Val list| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness: 0.039183
Badness (Sintel): 1.70


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Keemic_temperaments"