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Vishnu family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''vishnu family''' [[tempering out|tempers out]] the [[vishnu comma]] ({{monzo|legend=1| 23 6 -14 }}, [[ratio]]: 6 115 295 232 / 6 103 515 625), a [[5-limit]] comma of about 3 [[cent]]s. It is the amount by which seven just chromatic semitones of [[25/24]] fall short of a just perfect fourth of [[4/3]], or in other words (4/3)/(25/24)<sup>7</sup>.
The '''vishnu family''' [[tempering out|tempers out]] the [[vishnu comma]] ({{monzo|legend=1| 23 6 -14 }}, [[ratio]]: 6 115 295 232 / 6 103 515 625), an [[unnoticeable comma]] of about 3.34 [[cent]]s in size.  


== Vishnu ==
== Vishnu ==
The generator for the vishnu temperament is a [[25/24|classical chromatic semitone]], seven of which make a [[4/3|perfect fourth]], and [[5/4]] is found as a half-octave period minus three generators. Its [[ploidacot]] is diploid epsilon-heptacot. It is a member of the [[diaschismic–gothmic equivalence continuum]] with {{nowrap| ''n'' {{=}} 7/2 }}; thus it is most naturally used as a 34-form.
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 11: Line 13:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~78125/55296 = 600.0000{{c}}, ~25/24 = 71.1556{{c}}
* [[WE]]: ~78125/55296 = 599.9771{{c}}, ~25/24 = 71.1405{{c}}
* [[POTE]]: ~78125/55296 = 600.000{{c}}, ~25/24 = 71.143{{c}}
: [[error map]]: {{val| -0.046 -0.030 +0.150 }}
* [[CWE]]: ~78125/55296 = 600.0000{{c}}, ~25/24 = 71.1482{{c}}
: error map: {{val| 0.000 +0.007 +0.242 }}


{{Optimal ET sequence|legend=1| 16, 34, 84, 118, 270, 388, 506, 894, 1400, 3306c, 4706cc }}
{{Optimal ET sequence|legend=1| 16, 34, 84, 118, 270, 388, 506, 894, 1400, 3306c, 4706cc }}


[[Badness]] (Smith): 0.031181
[[Badness]] (Sintel): 0.731
 
=== Overview to extensions ===
The second comma in the comma list defines which 7-limit family member we are looking at. Septimal vishnu (118 & 152) adds [[4375/4374]]. Narayana (84 & 118) adds [[65625/65536]]. Vishnean (16 & 50) adds [[126/125]]. There is also an unnamed 50 & 84 temperament, which tempers out [[225/224]]. These all use the same generators as classical vishnu.
 
Trivish (84 & 186) adds [[250047/250000]], splitting the semi-octave period in three. Quarvish (68 & 270) adds [[2401/2400]], splitting the generator in four. Decavish (50 & 220) adds [[420175/419904]], splitting the semi-octave period in five.


== Septimal vishnu ==
== Septimal vishnu ==
Septimal vishnu may be described as the {{nowrap| 118 & 152 }} temperament, tempering out 4375/4374, the [[ragisma]], and 29360128/29296875, the [[quasiorwellisma]], aside from the vishnu comma. [[270edo]] is an excellent tuning for this temperament, with generator 16\270 (71.{{overline|1}} cents), but it can be tuned to [[422edo]], [[692edo]], or [[962edo]]. It extends naturally to the [[11-limit]] by identifying a stack of four [[5/4]]'s as [[11/9]], tempering out [[5632/5625]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 26: Line 37:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~10125/7168 = 600.0000{{c}}, ~25/24 = 71.1175{{c}}
* [[WE]]: ~10125/7168 = 599.9486{{c}}, ~25/24 = 71.1014{{c}}
* [[POTE]]: ~10125/7168 = 600.000{{c}}, ~25/24 = 71.107{{c}}
: [[error map]]: {{val| -0.103 +0.129 +0.125 -0.092 }}
* [[CWE]]: ~10125/7168 = 600.0000{{c}}, ~25/24 = 71.1106{{c}}
: error map: {{val| 0.000 +0.271 +0.354 +0.081 }}


{{Optimal ET sequence|legend=1| 34d, 84d, 118, 152, 270, 1232, 1502, 1772c, 2042bc, 2312bc, 2582bc, 4894bbccd }}
{{Optimal ET sequence|legend=1| 34d, 84d, 118, 152, 270, 1232, 1502, 1772c, 2042bc, 2312bc, 2582bc, 4894bbccd }}


[[Badness]] (Smith): 0.041912
[[Badness]] (Sintel): 1.06


=== 11-limit ===
=== 11-limit ===
Line 41: Line 54:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1142{{c}}
* WE: ~99/70 = 599.9549{{c}}, ~25/24 = 71.0994{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~25/24 = 71.105{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1077{{c}}


{{Optimal ET sequence|legend=0| 34d, 84de, 118, 152, 270, 962, 1232, 1502, 2734bcd, 4236bbccdd }}
{{Optimal ET sequence|legend=0| 34d, 84de, 118, 152, 270, 962, 1232, 1502 }}


Badness (Smith): 0.014180
Badness (Sintel): 0.469
 
==== Acyuta ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 5632/5625
 
Mapping: {{mapping| 2 4 5 10 10 17 | 0 -7 -3 -37 -26 -81 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9475{{c}}, ~25/24 = 71.0947{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1028{{c}}
 
{{Optimal ET sequence|legend=0| 118f, 152f, 270, 692, 962, 1654cd, 2616bcdf }}
 
Badness (Sintel): 0.653


==== Ananta ====
==== Ananta ====
Line 56: Line 84:


Optimal tuning:  
Optimal tuning:  
* CTE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1173{{c}}
* WE: ~99/70 = 599.9940{{c}}, ~25/24 = 71.1163{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~25/24 = 71.117{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1170{{c}}


{{Optimal ET sequence|legend=0| 118, 152, 270, 928, 1198, 1468 }}
{{Optimal ET sequence|legend=0| 118, 152, 270, 928, 1198, 1468 }}


Badness (Smith): 0.023678
Badness (Sintel): 0.978
 
== Narayana ==
Named by [[Xenllium]] in 2021, narayana tempers out 65625/65536, the [[horwell comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 84 & 118 }} temperament.


==== Acyuta ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 65625/65536, 321489/320000
 
{{Mapping|legend=1| 2 4 5 3 | 0 -7 -3 22 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.1141{{c}}, ~25/24 = 71.2783{{c}}
: [[error map]]: {{val| +0.228 -0.447 +0.422 -0.362 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~25/24 = 71.2659{{c}}
: error map: {{val| 0.000 -0.816 -0.111 -0.976 }}
 
{{Optimal ET sequence|legend=1| 34, 84, 118, 202, 320, 522 }}
 
[[Badness]] (Sintel): 2.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 1716/1715, 2080/2079, 3025/3024, 5632/5625
Comma list: 441/440, 8019/8000, 65625/65536


Mapping: {{mapping| 2 4 5 10 10 17 | 0 -7 -3 -37 -26 -81 }}
Mapping: {{mapping| 2 4 5 3 3 | 0 -7 -3 22 33 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1079{{c}}
* WE: ~99/70 = 600.0992{{c}}, ~25/24 = 71.2593{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~25/24 = 71.101{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.2501{{c}}


{{Optimal ET sequence|legend=0| 118f, 152f, 270, 692, 962, 1654cd, 2616bcdf }}
{{Optimal ET sequence|legend=0| 34e, 84, 118, 202, 320 }}


Badness (Smith): 0.015805
Badness (Sintel): 1.52


== Vishnean ==
== Vishnean ==
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, vishnean is a low-complexity low-accuracy extension of vishnu that tempers out 126/125, the [[starling comma]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 86: Line 135:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~735/512 = 600.0000{{c}}, ~25/24 = 71.4956{{c}}
* [[WE]]: ~735/512 = 600.8754{{c}}, ~25/24 = 71.8834{{c}}
* [[POTE]]: ~735/512 = 600.000{{c}}, ~25/24 = 71.779{{c}}
: [[error map]]: {{val| +1.751 -1.637 +2.413 -5.031 }}
* [[CWE]]: ~735/512 = 600.0000{{c}}, ~25/24 = 71.7141{{c}}
: error map: {{val| 0.000 -3.954 -1.456 -10.255 }}


{{Optimal ET sequence|legend=1| 16, 34, 50, 134dd }}
{{Optimal ET sequence|legend=1| 16, 34, 50, 134dd }}


[[Badness]] (Smith): 0.161398
[[Badness]] (Sintel): 4.08


=== 11-limit ===
=== 11-limit ===
Line 101: Line 152:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/44 = 600.0000{{c}}, ~25/24 = 71.6218{{c}}
* WE: ~63/44 = 600.7144{{c}}, ~25/24 = 72.0091{{c}}
* POTE: ~63/44 = 600.000{{c}}, ~25/24 = 71.924{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~25/24 = 71.8588{{c}}


{{Optimal ET sequence|legend=0| 16, 34, 50 }}
{{Optimal ET sequence|legend=0| 16, 34, 50 }}


Badness (Smith): 0.071990
Badness (Sintel): 2.38


=== 13-limit ===
=== 13-limit ===
Line 116: Line 167:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.0000{{c}}, ~25/24 = 71.6369{{c}}
* WE: ~55/39 = 600.5983{{c}}, ~25/24 = 71.9975{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~25/24 = 71.926{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/24 = 71.8699{{c}}


{{Optimal ET sequence|legend=0| 16, 34, 50 }}
{{Optimal ET sequence|legend=0| 16, 34, 50 }}


Badness (Smith): 0.040739
Badness (Sintel): 1.68
 
== Trivish ==
Trivish tempers out 250047/250000, the [[landscape comma]], with a 1/6-octave period. It may be described as the {{nowrap| 84 & 186 }} temperament. It is more significant in the [[11-limit]], where it tempers out the [[kalisma]] and the [[olympia]]. In the [[13-limit]], the generator can be taken as a [[~]][[14/13]]. [[270edo]] is an obvious tuning for it, but it can be tuned to [[354edo]], [[624edo]] or [[894edo]]. It was named by [[Flora Canou]] in 2026 for the fact that it splits the period of vishnu in three.  


== Narayana ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 321489/320000
[[Comma list]]: 250047/250000, 134217728/133984375


{{Mapping|legend=1| 2 4 5 3 | 0 -7 -3 22 }}
{{Mapping|legend=1| 6 5 12 22 | 0 7 3 -8 }}
: mapping generators: ~18375/16384, ~2205/2048


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~567/400 = 600.0000{{c}}, ~25/24 = 71.2706{{c}}
* [[WE]]: ~18375/16384 = 199.9885{{c}}, ~2205/2048 = 128.8583{{c}}
* [[POTE]]: ~567/400 = 600.000{{c}}, ~25/24 = 71.265{{c}}
: [[error map]]: {{val| -0.069 -0.005 +0.123 +0.054 }}
* [[CWE]]: ~18375/16384 = 200.0000{{c}}, ~2205/2048 = 128.8643{{c}}
: error map: {{val| 0.000 +0.095 +0.279 +0.256 }}


{{Optimal ET sequence|legend=1| 34, 84, 118, 202, 320, 522 }}
{{Optimal ET sequence|legend=1| 84, 186, 270, 624, 894, 1164 }}


[[Badness]] (Smith): 0.094295
[[Badness]] (Sintel): 1.51


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


Comma list: 441/440, 8019/8000, 65625/65536
Comma list: 9801/9800, 131072/130977, 151263/151250
 
Mapping: {{mapping| 6 5 12 22 33 | 0 7 3 -8 -19 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9885{{c}}, ~320/297 = 128.8582{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~320/297 = 128.8661{{c}}
 
{{Optimal ET sequence|legend=0| 84, 186, 270, 624, 894, 1164 }}
 
Badness (Sintel): 0.807
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 4096/4095, 34398/34375
 
Mapping: {{mapping| 6 5 12 22 33 28 | 0 7 3 -8 -19 -9 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9922{{c}}, ~14/13 = 128.8565{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~14/13 = 128.8621{{c}}
 
{{Optimal ET sequence|legend=0| 84, 186, 270, 624, 894, 2058f }}
 
Badness (Sintel): 0.791
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1521/1520, 1716/1715, 2080/2079, 2376/2375, 3250/3249


Mapping: {{mapping| 2 4 5 3 3 | 0 -7 -3 22 33 }}
Mapping: {{mapping| 6 5 12 22 33 28 30 | 0 7 3 -8 -19 -9 -7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.2609{{c}}
* WE: ~55/49 = 199.9909{{c}}, ~14/13 = 128.8564{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~25/24 = 71.248{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~14/13 = 128.8630{{c}}


{{Optimal ET sequence|legend=0| 34e, 84, 118, 202, 320 }}
{{Optimal ET sequence|legend=0| 84, 186, 270, 624, 894 }}


Badness (Smith): 0.045908
Badness (Sintel): 0.591


== Quarvish ==
== Quarvish ==
Quarvish tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 68 & 270 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 159: Line 247:


{{Mapping|legend=1| 2 4 5 6 | 0 -28 -12 -13 }}
{{Mapping|legend=1| 2 4 5 6 | 0 -28 -12 -13 }}
: mapping generators: ~78125/55296, ~110592/109375


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~78125/55296 = 600.0000{{c}}, ~7/5 = 582.2115{{c}} (~110592/109375 = 17.7885{{c}})
* [[WE]]: ~78125/55296 = 599.9874{{c}}, ~110592/109375 = 17.7862{{c}}
* [[POTE]]: ~78125/55296 = 600.000{{c}}, ~7/5 = 582.213{{c}} (~110592/109375 = 17.787{{c}})
: [[error map]]: {{val| -0.025 -0.020 +0.189 -0.123 }}
* [[CWE]]: ~78125/55296 = 600.0000{{c}}, ~110592/109375 = 17.7873{{c}}
: error map: {{val| 0.000 +0.002 +0.239 -0.060 }}


{{Optimal ET sequence|legend=1| 68, 134, 202, 270, 742, 1012 }}
{{Optimal ET sequence|legend=1| 68, 134, 202, 270, 742, 1012 }}


[[Badness]] (Smith): 0.045029
[[Badness]] (Sintel): 1.14


=== 11-limit ===
=== 11-limit ===
Line 176: Line 267:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.0000{{c}}, ~7/5 = 582.2122{{c}} (~99/98 = 17.7878{{c}})
* WE: ~99/70 = 599.9884{{c}}, ~99/98 = 17.7865{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~7/5 = 582.213{{c}} (~99/98 = 17.787{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~99/98 = 17.7870{{c}}


{{Optimal ET sequence|legend=0| 68, 134, 202, 270, 742, 1012 }}
{{Optimal ET sequence|legend=0| 68, 134, 202, 270, 742, 1012 }}


Badness (Smith): 0.019868
Badness (Sintel): 0.657


== Decavish ==
== Decavish ==
The decavish temperament (50 & 270) tempers out the wizma, {{monzo| -6 -8 2 5 }} = 420175/419904 and the [[linus comma]], {{monzo| 11 -10 -10 10 }} = 578509309952/576650390625.
Decavish tempers out the [[wizma]] as well as the [[linus comma]], and may be described as the {{nowrap| 50 & 220 }} temperament. It was named by [[Xenllium]] in 2021 for the fact that its period is 1/10 octave.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 194: Line 285:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 120.0000{{c}}, ~54/49 = 168.8587{{c}} (~25/24 = 71.1413{{c}}, or ~36/35 = 48.8587{{c}})
* [[WE]]: ~15/14 = 119.9957{{c}}, ~54/49 = 168.8645{{c}} (~36/35 = 48.8689{{c}})
* [[POTE]]: ~15/14 = 120.000{{c}}, ~54/49 = 168.871{{c}} (~25/24 = 71.129{{c}}, or ~36/35 = 48.871{{c}})
: [[error map]]: {{val| -0.043 +0.071 +0.197 -0.242 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~54/49 = 168.8658{{c}} (~36/35 = 48.8658{{c}})
: error map: {{val| 0.000 +0.106 +0.284 -0.168 }}


{{Optimal ET sequence|legend=1| 50, …, 220, 270, 1400, 1670, 1940, 2210c, 2480c }}
{{Optimal ET sequence|legend=1| 50, …, 220, 270, 1400, 1670, 1940, 2210c, 2480c }}


[[Badness]] (Smith): 0.114531
[[Badness]] (Sintel): 2.90


=== 11-limit ===
=== 11-limit ===
Line 209: Line 302:


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 120.0000{{c}}, ~54/49 = 168.8583{{c}} (~25/24 = 71.1417{{c}}, or ~36/35 = 48.8583{{c}})
* WE: ~15/14 = 119.9997{{c}}, ~54/49 = 168.8587{{c}} (~36/35 = 48.8590{{c}})
* POTE: ~15/14 = 120.000{{c}}, ~54/49 = 168.859{{c}} (~25/24 = 71.141{{c}}, or ~36/35 = 48.859{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~54/49 = 168.8589{{c}} (~36/35 = 48.8589{{c}})


{{Optimal ET sequence|legend=0| 50, …, 220, 270, 860, 1130, 1400 }}
{{Optimal ET sequence|legend=0| 50, …, 220, 270, 860, 1130, 1400 }}


Badness (Smith): 0.036938
Badness (Sintel): 1.22


=== 13-limit ===
=== 13-limit ===
Line 224: Line 317:


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 120.0000{{c}}, ~54/49 = 168.8583{{c}} (~25/24 = 71.1417{{c}}, or ~36/35 = 48.8583{{c}})
* WE: ~15/14 = 120.0037{{c}}, ~54/49 = 168.8537{{c}} (~36/35 = 48.8500{{c}})
* POTE: ~15/14 = 120.000{{c}}, ~54/49 = 168.859{{c}} (~25/24 = 71.151{{c}}, or ~36/35 = 48.859{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~54/49 = 168.8506{{c}} (~36/35 = 48.8506{{c}})
 
{{Optimal ET sequence|legend=0| 50, …, 220, 270, 590, 860, 1130 }}


{{Optimal ET sequence|legend=0| 50, 220, 270, 590, 860, 1130 }}
Badness (Sintel): 0.905


Badness (Smith): 0.021906
== References ==
<references/>


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