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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, 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
Line 8: Line 29:
[[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, ~42/25
: mapping generators: ~2, ~25/21
 
{{Multival|legend=1| 14 11 9 -15 -25 -10 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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 25: Line 47:
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 = 1\1, ~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}}


{{Optimal ET sequence|legend=1| 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 38: Line 62:
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 = 1\1, ~13/11 = 292.457
Optimal tunings:
* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}


{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}


Badness: 0.032913
Badness (Sintel): 1.36


=== Quato ===
=== Quato ===
Line 51: Line 77:
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 = 1\1, ~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}}


{{Optimal ET sequence|legend=1| 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: {{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 = 1\1, ~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}}


{{Optimal ET sequence|legend=1| 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]].''
 
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
[[Subgroup]]: 2.3.5.7
Line 80: Line 114:


{{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 = 1\1, ~25/24 = 53.816
: [[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 }}
{{Optimal ET sequence|legend=1| 22, 45, 67c }}


[[Badness]]: 0.090769
[[Badness]] (Sintel): 2.30
 
== Undeka ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Undeka]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 3200/3087
 
{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
 
: mapping generators: ~21/20, ~3
 
{{Multival|legend=1| 11 11 0 -8 -31 -31 }}
 
[[Optimal tuning]] ([[POTE]]): ~21/20 = 1\11, ~3/2 = 708.792
 
{{Optimal ET sequence|legend=1| 11c, 22 }}
 
[[Badness]]: 0.141782
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 352/343, 385/384
 
Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}
 
Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 706.768
 
{{Optimal ET sequence|legend=1| 11c, 22 }}
 
Badness: 0.068672
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 65/63, 100/99, 169/165, 352/343
 
Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}
 
Optimal tuning (POTE): ~13/12 = 1\11, ~3/2 = 707.764
 
{{Optimal ET sequence|legend=1| 11cf, 22 }}
 
Badness: 0.056528


== Barbad ==
== Barbad ==
Line 139: Line 131:
[[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


{{Multival|legend=1| 19 12 21 -25 -20 15 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1201.0462{{c}}, ~75/49 = 732.3071{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331
: [[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=1| 18, 23d, 41 }}
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}


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


=== 11-limit ===
=== 11-limit ===
Line 156: Line 149:
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 = 1\1, ~98/75 = 468.367
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=1| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness: 0.050105
Badness (Sintel): 1.66


=== 13-limit ===
=== 13-limit ===
Line 169: Line 164:
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 = 1\1, ~13/10 = 468.270
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=1| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness: 0.039183
Badness (Sintel): 1.62


== Hyperkleismic ==
== Hyperkleismic ==
Line 183: Line 180:


{{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


{{Multival|legend=1| 17 16 3 -14 -43 -38 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.0290{{c}}, ~6/5 = 323.7882{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780
: [[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 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


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


=== 11-limit ===
=== 11-limit ===
Line 201: Line 199:
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 = 1\1, ~6/5 = 323.796
Optimal tunings:
* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness: 0.065356
Badness (Sintel): 2.16


=== 13-limit ===
=== 13-limit ===
Line 214: Line 214:
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 = 1\1, ~6/5 = 323.790
Optimal tunings:
* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness: 0.035724
Badness (Sintel): 1.48


== Sevond ==
== 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
[[Subgroup]]: 2.3.5.7


Line 226: Line 232:


{{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]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 242: Line 251:
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 = 1\7, ~3/2 = 705.518
Optimal tunings:
* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness: 0.070437
Badness (Sintel): 2.33


=== 13-limit ===
=== 13-limit ===
Line 255: Line 266:
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 = 1\7, ~3/2 = 705.344
Optimal tunings:
* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness: 0.041238
Badness (Sintel): 1.70


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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