Keemic temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
m fixed double redirect
Move pentadecal here
 
(32 intermediate revisions by 6 users not shown)
Line 1: Line 1:
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]], [[Shibboleth family #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
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
[[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 }}


{{Multival|legend=1| 14 11 9 -15 -25 -10 }}
{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710
[[Badness]] (Sintel): 1.53
 
{{Optimal ET sequence|legend=1| 4, 37, 41 }}
 
[[Badness]]: 0.060269


=== 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]].''


== Barbad ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 3200/3087
[[Comma list]]: 875/864, 16875/16807


{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
{{Mapping|legend=1| 1 -10 -5 -10 | 0 19 12 21 }}
 
: mapping generators: ~2, ~98/75
: 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 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=1| 11c, 22 }}
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}


[[Badness]]: 0.141782
[[Badness]] (Sintel): 2.80


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


Comma list: 100/99, 352/343, 385/384
Comma list: 245/242, 540/539, 625/616


Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}
Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}


Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 706.768
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| 11c, 22 }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness: 0.068672
Badness (Sintel): 1.66


=== 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: 144/143, 196/195, 245/242, 275/273
 
Mapping: {{mapping| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}


Mapping: {{mapping| 11 0 8 31 38 23 | 0 1 1 0 0 1 }}
Optimal tunings:  
* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}


Optimal tuning (POTE): ~13/12 = 1\11, ~3/2 = 707.764
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


{{Optimal ET sequence|legend=1| 11cf, 22 }}
Badness (Sintel): 1.62


Badness: 0.056528
== 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.


== Barbad ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


: mapping generators: ~2, ~98/75
Comma list: 100/99, 105/104, 144/143, 1372/1331


{{Multival|legend=1| 19 12 21 -25 -20 15 }}
Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331
Optimal tunings:
* 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}})


{{Optimal ET sequence|legend=1| 18, 23d, 41 }}
{{Optimal ET sequence|legend=0| 15, 45f, 60, 135de, 195cddee }}


[[Badness]]: 0.110448
Badness (Sintel): 2.14


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


Comma list: 245/242, 540/539, 625/616
Comma list: 121/120, 441/440, 875/864


Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367
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}})


{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=0| 15, 45e, 60e }}


Badness: 0.050105
Badness (Sintel): 1.47


=== 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: 121/120, 196/195, 352/351, 875/864


Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}
Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270
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=1| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=0| 15f, 60e, 135deef }}


Badness: 0.039183
Badness (Sintel): 2.29


== Hyperkleismic ==
== Hyperkleismic ==
Line 183: Line 262:


{{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 281:
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 296:
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 ==
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.
: ''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 228: Line 314:


{{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 244: Line 333:
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 257: Line 348:
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]]

Latest revision as of 10:21, 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.

This is a collection of linear 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, pentadecal, hyperkleismic, and sevond, in the order of increasing TE logflat badness.

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 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

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

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 tunings:

  • WE: ~21/20 = 80.1141 ¢, ~3/2 = 700.2213 ¢ (~126/125 = 19.8053 ¢)
error map: ⟨+1.711 +0.977 -2.127 -4.035]
  • CWE: ~21/20 = 80.0000 ¢, ~3/2 = 701.2357 ¢ (~126/125 = 19.7643 ¢)
error map: ⟨0.000 -0.719 -5.078 -8.826]

Optimal ET sequence: 15, 45, 60

Badness (Sintel): 2.91

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 tunings:

  • WE: ~21/20 = 80.0213 ¢, ~3/2 = 702.9194 ¢ (~56/55 = 17.2721 ¢)
  • CWE: ~21/20 = 80.0000 ¢, ~3/2 = 702.8751 ¢ (~56/55 = 17.1249 ¢)

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

Badness (Sintel): 2.56

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 tunings:

  • WE: ~21/20 = 80.0207 ¢, ~3/2 = 701.8966 ¢ (~91/90 = 18.2900 ¢)
  • CWE: ~21/20 = 80.0000 ¢, ~3/2 = 701.8545 ¢ (~91/90 = 18.1455 ¢)

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

Badness (Sintel): 2.14

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 tunings:

  • WE: ~21/20 = 80.1322 ¢, ~3/2 = 701.4751 ¢ (~126/125 = 19.7148 ¢)
  • CWE: ~21/20 = 80.0000 ¢, ~3/2 = 701.5453 ¢ (~126/125 = 18.4547 ¢)

Optimal ET sequence: 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: [⟨15 0 11 42 28 103], ⟨0 1 1 0 1 -2]]

Optimal tunings:

  • WE: ~21/20 = 80.1015 ¢, ~3/2 = 702.5374 ¢ (~126/125 = 18.3763 ¢)
  • CWE: ~21/20 = 80.0000 ¢, ~3/2 = 701.8504 ¢ (~126/125 = 18.1496 ¢)

Optimal ET sequence: 15f, 60e, 135deef

Badness (Sintel): 2.29

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

Retrieved from "https://en.xen.wiki/index.php?title=Keemic_temperaments&oldid=232745"