Xenharmonic Wiki

Keemic temperaments: Difference between revisions

← Older edit
VisualWikitext
Move pentadecal here
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "
 
(6 intermediate revisions by the same user not shown)
Line 9: Line 9:
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Wesley]]'' (+405/392) → [[Wesley family #Septimal wesley|Wesley family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Fifives]]'' (+83349/81920) → [[Fifive family #Fifives|Fifive family]]
* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[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]].
Discussed below are quasitemp, chromo, barbad, pentadecal, undeka, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].


== Quasitemp ==
== Quasitemp ==
Line 47: Line 48:
Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


Mapping: {{mapping| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}
{{Mapping|legend=0| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 62: Line 63:
Comma list: 100/99, 196/195, 275/273, 385/384
Comma list: 100/99, 196/195, 275/273, 385/384


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


Optimal tunings:  
Optimal tunings:  
Line 77: Line 78:
Comma list: 243/242, 441/440, 625/616
Comma list: 243/242, 441/440, 625/616


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


Optimal tunings:  
Optimal tunings:  
Line 92: Line 93:
Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


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


Optimal tunings:  
Optimal tunings:  
Line 149: Line 150:
Comma list: 245/242, 540/539, 625/616
Comma list: 245/242, 540/539, 625/616


Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}
{{Mapping|legend=0| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}


Optimal tunings:  
Optimal tunings:  
Line 164: Line 165:
Comma list: 144/143, 196/195, 245/242, 275/273
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|legend=0| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}


Optimal tunings:  
Optimal tunings:  
Line 179: Line 180:
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.
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.


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 196: Line 198:
[[Badness]] (Sintel): 2.91
[[Badness]] (Sintel): 2.91


=== 11-limit ===
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13
 
Comma list: 105/104, 325/324, 15625/15379
 
{{Mapping|legend=0| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}
 
Optimal tunings:
* WE: ~21/20 = 80.1133{{c}}, ~3/2 = 700.1871{{c}} (~91/90 = 20.8325{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 700.2086{{c}} (~91/90 = 19.7914{{c}})
 
{{Optimal ET sequence|legend=0| 15, 30bcff, 45f, 60 }}
 
Badness (Sintel): 1.80
 
=== Quindecal ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 385/384, 1372/1331
Comma list: 121/120, 441/440, 875/864


Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}
{{Mapping|legend=0| 15 0 11 42 28 | 0 1 1 0 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~21/20 = 80.0213{{c}}, ~3/2 = 702.9194{{c}} (~56/55 = 17.2721{{c}})
* WE: ~21/20 = 80.1322{{c}}, ~3/2 = 701.4751{{c}} (~126/125 = 19.7148{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 702.8751{{c}} (~56/55 = 17.1249{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.5453{{c}} (~126/125 = 18.4547{{c}})


{{Optimal ET sequence|legend=0| 15, 45, 60, 75de, 135de }}
{{Optimal ET sequence|legend=0| 15, 45e, 60e }}


Badness (Sintel): 2.56
Badness (Sintel): 1.47


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


Comma list: 100/99, 105/104, 144/143, 1372/1331
Comma list: 105/104, 121/120, 275/273, 325/324


Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}
{{Mapping|legend=0| 15 0 11 42 28 8 | 0 1 1 0 1 2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~21/20 = 80.0207{{c}}, ~3/2 = 701.8966{{c}} (~91/90 = 18.2900{{c}})
* WE: ~21/20 = 80.1359{{c}}, ~3/2 = 700.2567{{c}} (~91/90 = 20.9661{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8545{{c}} (~91/90 = 18.1455{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 700.2955{{c}} (~91/90 = 19.7045{{c}})


{{Optimal ET sequence|legend=0| 15, 45f, 60, 135de, 195cddee }}
{{Optimal ET sequence|legend=0| 15, 30bceff, 45ef, 60e }}


Badness (Sintel): 2.14
Badness (Sintel): 1.34


=== Quindecal ===
== Undeka ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit 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
 
[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 108.9318{{c}}, ~3/2 = 707.7579{{c}}
: [[error map]]: {{val| -1.750 +4.053 -8.852 +8.059 }}
* [[CWE]]: ~21/20 = 109.0909{{c}}, ~3/2 = 707.7526{{c}}
: error map: {{val| 0.000 +5.798 -5.834 +12.992 }}
 
{{Optimal ET sequence|legend=1| 11b, 22 }}
 
[[Badness]] (Sintel): 3.59
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 441/440, 875/864
Comma list: 100/99, 352/343, 385/384


Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}
{{Mapping|legend=0| 11 0 8 31 38 | 0 1 1 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~21/20 = 80.1322{{c}}, ~3/2 = 701.4751{{c}} (~126/125 = 19.7148{{c}})
* WE: ~21/20 = 109.0321{{c}}, ~3/2 = 706.3870{{c}}
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.5453{{c}} (~126/125 = 18.4547{{c}})
* CWE: ~21/20 = 109.0909{{c}}, ~3/2 = 706.4785{{c}}


{{Optimal ET sequence|legend=0| 15, 45e, 60e }}
{{Optimal ET sequence|legend=0| 11c, 22 }}


Badness (Sintel): 1.47
Badness (Sintel): 2.27


==== 13-limit ====
=== 2.3.5.7.11.17 subgroup ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.17


Comma list: 121/120, 196/195, 352/351, 875/864
Comma list: 85/84, 100/99, 121/119, 385/384


Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}
{{Mapping|legend=0| 11 0 8 31 38 45 | 0 1 1 0 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~21/20 = 80.1015{{c}}, ~3/2 = 702.5374{{c}} (~126/125 = 18.3763{{c}})
* WE: ~17/16 = 109.0232{{c}}, ~3/2 = 706.5074{{c}}
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8504{{c}} (~126/125 = 18.1496{{c}})
* CWE: ~17/16 = 109.0909{{c}}, ~3/2 = 706.6786{{c}}


{{Optimal ET sequence|legend=0| 15f, 60e, 135deef }}
{{Optimal ET sequence|legend=0| 11c, 22 }}


Badness (Sintel): 2.29
Badness (Sintel): 1.82


== Hyperkleismic ==
== Hyperkleismic ==
Line 279: Line 316:
Comma list: 100/99, 385/384, 2420/2401
Comma list: 100/99, 385/384, 2420/2401


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


Optimal tunings:  
Optimal tunings:  
Line 294: Line 331:
Comma list: 100/99, 169/168, 275/273, 385/384
Comma list: 100/99, 169/168, 275/273, 385/384


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


Optimal tunings:  
Optimal tunings:  
Line 331: Line 368:
Comma list: 100/99, 385/384, 6655/6561
Comma list: 100/99, 385/384, 6655/6561


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


Optimal tunings:  
Optimal tunings:  
Line 346: Line 383:
Comma list: 100/99, 169/168, 352/351, 385/384
Comma list: 100/99, 169/168, 352/351, 385/384


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


Optimal tunings:  
Optimal tunings:  
Line 356: Line 393:
Badness (Sintel): 1.70
Badness (Sintel): 1.70


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