Xenharmonic Wiki

Amity family: Difference between revisions

← Older edit
VisualWikitext
Re-sort the amicable extensions as amorous is more notable
Move chromat and witcher here
 
Line 29: Line 29:


=== Overview to extensions ===
=== Overview to extensions ===
The second comma to extend the 5-limit amity include [[4375/4374]] for septimal amity, [[225/224]] for houborizic, 65625/65536 for paramity, [[126/125]] for accord, [[245/243]] for bamity, [[2430/2401]] for hamity, [[1029/1024]] for gamity, [[10976/10935]] for chromat, [[703125/702464]] for trinity, [[2401/2400]] for amicable, [[2100875/2097152]] for calamity, [[420175/419904]] for witcher, and [[16875/16807]] for familia.  
The second comma to extend 5-limit amity include [[4375/4374]] for septimal amity, [[65625/65536]] for paramity, [[126/125]] for accord, and [[225/224]] for houborizic. Those all use the same generator as the 5-limit temperament.  


Temperaments discussed elsewhere include:
Bamity adds [[245/243]] with a 1/2-octave period. Hamity adds [[2430/2401]] and splits the generator in two. Gamity adds [[1029/1024]]. Chromat adds [[10976/10935]]. Trinity adds [[703125/702464]]. Those split the generator in three. Amicable adds [[2401/2400]], splitting the generator in four. Witcher adds [[420175/419904]]. Calamity adds [[2100875/2097152]]. Familia adds [[16875/16807]]. Those split the generator in five.  
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)
 
The rest are considered below.  


== Septimal amity ==
== Septimal amity ==
Line 739: Line 735:


Badness (Sintel): 1.01
Badness (Sintel): 1.01
== Chromat ==
Chromat has a period of 1/3 octave and tempers out the [[hemimage comma]] (10976/10935) and the [[triwellisma]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 10976/10935, 235298/234375
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
[[Badness]] (Sintel): 1.46
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 10976/10935
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
Optimal tunings:
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}
Badness (Sintel): 1.67
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 10976/10935
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
Optimal tunings:
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}
Badness (Sintel): 1.90
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
Optimal tunings:
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}
Badness (Sintel): 1.61
==== Catachrome ====
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
Optimal tunings:
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
Badness (Sintel): 1.81
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
Optimal tunings:
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
Badness (Sintel): 1.54
==== Chromic ====
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 729/728, 1875/1859
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
Optimal tunings:
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
Badness (Sintel): 2.06
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
Optimal tunings:
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
Badness (Sintel): 1.58
=== Hemichromat ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 10976/10935, 102487/102400
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
Optimal tunings:
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}
Badness (Sintel): 2.22
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
Optimal tunings:
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
Badness (Sintel): 1.38


== Amicable ==
== Amicable ==
Line 945: Line 1,096:


Badness (Sintel): 1.17
Badness (Sintel): 1.17
== Witcher ==
Witcher tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 99 & 166 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 420175/419904, 1600000/1594323
{{Mapping|legend=1| 1 -7 -20 -2 | 0 25 65 14 }}
: mapping generators: ~2, ~80/63
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8698{{c}}, ~80/63 = 412.0531{{c}}
: [[error map]]: {{val| -0.130 +0.283 -0.260 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/63 = 412.0956{{c}}
: error map: {{val| 0.000 +0.435 -0.100 +0.512 }}
{{Optimal ET sequence|legend=1| 32c, 67c, 99, 265, 364, 463, 562, 1025 }}
[[Badness]] (Sintel): 2.04
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4356, 1240029/1239040
Mapping: {{mapping| 1 -7 -20 -2 -35 | 0 25 65 14 112 }}
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~80/63 = 412.0758{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/63 = 412.0788{{c}}
{{Optimal ET sequence|legend=0| 99e, 166, 265 }}
Badness (Sintel): 2.73
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4375/4356, 10648/10647
Mapping: {{mapping| 1 -7 -20 -2 -35 -43 | 0 25 65 14 112 136 }}
Optimal tunings:
* WE: ~2 = 1200.0660{{c}}, ~33/26 = 412.0928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/26 = 412.0709{{c}}
{{Optimal ET sequence|legend=0| 99ef, 166, 265 }}
Badness (Sintel): 2.40
=== Semiwitcher ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 41503/41472, 585640/583443
Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }}
: mapping generators: ~99/70, ~80/63
Optimal tunings:
* WE: ~99/70 = 599.9717{{c}}, ~80/63 = 412.0766{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~80/63 = 412.0951{{c}}
{{Optimal ET sequence|legend=0| 32c, 166, 198, 364, 562, 926 }}
Badness (Sintel): 2.92
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280
Mapping: {{mapping| 2 11 25 10 16 24 | 0 25 65 14 29 53 }}
Optimal tunings:
* WE: ~99/70 = 599.9666{{c}}, ~33/26 = 412.0726{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/26 = 412.0944{{c}}
{{Optimal ET sequence|legend=0| 32cf, 166, 198, 364, 562, 926 }}
Badness (Sintel): 1.57
=== Hitch ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 117649/117612, 1600000/1594323
Mapping: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }}
: mapping generators: ~2, ~891/560
Optimal tunings:
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}}
{{Optimal ET sequence|legend=0| 67c, 198, 463, 661, 1124bde }}
Badness (Sintel): 3.09
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323
Mapping: {{mapping| 1 -32 -85 -16 31 -134 | 0 50 130 28 -41 205 }}
Optimal tunings:
* WE: ~2 = 1199.8415{{c}}, ~891/560 = 805.9451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0508{{c}}
{{Optimal ET sequence|legend=0| 198, 463, 661 }}
Badness (Sintel): 2.51
==== Hitcher ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
Mapping: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }}
Optimal tunings:
* WE: ~2 = 1199.7690{{c}}, ~78/49 = 805.8958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/49 = 806.0500{{c}}
{{Optimal ET sequence|legend=0| 67c, 198, 463f, 661ff }}
Badness (Sintel): 2.46


== Calamity ==
== Calamity ==
Retrieved from "https://en.xen.wiki/w/Amity_family"