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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 19: Line 21:


[[Badness]] (Sintel): 0.731
[[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]]. Vishy (16 & 18) adds [[50/49]]. 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 42: Line 51:
Comma list: 3025/3024, 4375/4374, 5632/5625
Comma list: 3025/3024, 4375/4374, 5632/5625


Mapping: {{mapping| 2 4 5 10 10 | 0 -7 -3 -37 -26 }}
{{Mapping|legend=0| 2 4 5 10 10 | 0 -7 -3 -37 -26 }}


Optimal tunings:  
Optimal tunings:  
Line 51: Line 60:


Badness (Sintel): 0.469
Badness (Sintel): 0.469
==== Acyuta ====
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 5632/5625
{{Mapping|legend=0| 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 57: Line 81:
Comma list: 1001/1000, 3025/3024, 4096/4095, 4375/4374
Comma list: 1001/1000, 3025/3024, 4096/4095, 4375/4374


Mapping: {{mapping| 2 4 5 10 10 1 | 0 -7 -3 -37 -26 54 }}
{{Mapping|legend=0| 2 4 5 10 10 1 | 0 -7 -3 -37 -26 54 }}


Optimal tuning:  
Optimal tuning:  
Line 67: Line 91:
Badness (Sintel): 0.978
Badness (Sintel): 0.978


==== Acyuta ====
== Narayana ==
Subgroup: 2.3.5.7.11.13
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.
 
[[Subgroup]]: 2.3.5.7
 
[[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|legend=0| 2 4 5 3 3 | 0 -7 -3 22 33 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 599.9475{{c}}, ~25/24 = 71.0947{{c}}
* WE: ~99/70 = 600.0992{{c}}, ~25/24 = 71.2593{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.1028{{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 (Sintel): 0.653
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]]. It is also supported unideally by the [[84edo|84d]] ({{nowrap| {{=}} 34 + 50 }}) val, as the 7 is flat but the 11 is sharp such that they have opposing tendencies.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 104: Line 149:
Comma list: 126/125, 385/384, 1350/1331
Comma list: 126/125, 385/384, 1350/1331


Mapping: {{mapping| 2 4 5 5 8 | 0 -7 -3 5 -9 }}
{{Mapping|legend=0| 2 4 5 5 8 | 0 -7 -3 5 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 119: Line 164:
Comma list: 105/104, 126/125, 144/143, 847/845
Comma list: 105/104, 126/125, 144/143, 847/845


Mapping: {{mapping| 2 4 5 5 8 8 | 0 -7 -3 5 -9 -5 }}
{{Mapping|legend=0| 2 4 5 5 8 8 | 0 -7 -3 5 -9 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 129: Line 174:
Badness (Sintel): 1.68
Badness (Sintel): 1.68


== Narayana ==
== Vishy ==
Named by [[Godtone]] in 2026 after its inaccurate but simple interpretation of its half-octave period as [[7/5]][[~]][[10/7]], vishy tempers out [[50/49]] and represents the absolute simplest way to reach prime 7 by interpreting three ~25/24 generators as a significantly flat [[~]][[8/7]]. Its mapping might be deemed favorable because of all primes being reached in a negative number of generators, with the most complex, 11, being reached in -9 (as in vishnean). In fact, this temperament's mapping is identical to that of [[vishnean]] up to a simpler, higher-damage mapping of 7 in the same direction as all the other primes.
 
The extreme tuning 50edo tunes 11 approximately pure and 7 a little better via the 50d val (= 16 + 34d), but damages 3, 7/3 and 11/7 significantly more than CWE, and has a dubiously flat-tending 5-limit. However, in light of this, one might consider [[84edo]] via the 84dd val (= 34d + 50d), where 7 is ~1.7{{c}} sharper but everything else is significantly more accurate, with the exception of a sharper 11; especially, the 5-limit is approximately in-tune, so that with its more accurate 11 (and marginally more accurate 7) it is a good alternative to 34edo.
 
Therefore, it should be noted that as only the mapping of 7 differs from vishnean, using both mappings in any of [[34edo]], [[50edo]] and [[84edo]] (with [[2.3.5.11 subgroup]] fixed) could be seen as the primary and opportunistic utility of this temperament, by using whichever mapping of 7 happens to be more convenient or convincing in a given harmonic situation (if indeed the 7 is felt to be accurate enough).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 110592/109375
 
{{Mapping|legend=1| 2 4 5 6 | 0 -7 -3 -3 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8702{{c}}, ~25/24 = 70.6439{{c}}
: [[error map]]: {{val| -2.260 -0.982 -3.895 +12.463 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~25/24 = 71.9109{{c}}
: error map: {{val| 0.000 +0.969 +0.654 +18.141 }}
 
{{Optimal ET sequence|legend=0| 16, 18, 34d }}
 
Badness (Sintel): 3.96
 
=== 11-limit ===
Other than [[176/175]], 11-limit vishy also tempers out [[625/616]] and [[1331/1323]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 176/175, 864/847
 
{{Mapping|legend=0| 2 4 5 6 8 | 0 -7 -3 -3 -9 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8234{{c}}, ~25/24 = 70.7100{{c}}
: error map: {{val| -2.353 -1.631 -4.327 +11.984 +2.879 }}
* CWE: ~7/5 = 600.0000{{c}}, ~25/24 = 71.1044{{c}}
: error map: {{val| 0.000 -6.099 -2.375 +15.113 +0.498 }}
 
{{Optimal ET sequence|legend=0| 16, 18e, 34d }}
 
Badness (Sintel): 2.03
 
== 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.
 
[[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:  
* [[WE]]: ~567/400 = 600.1141{{c}}, ~25/24 = 71.2783{{c}}
* [[WE]]: ~18375/16384 = 199.9885{{c}}, ~2205/2048 = 128.8583{{c}}
: [[error map]]: {{val| +0.228 -0.447 +0.422 -0.362 }}
: [[error map]]: {{val| -0.069 -0.005 +0.123 +0.054 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~25/24 = 71.2659{{c}}
* [[CWE]]: ~18375/16384 = 200.0000{{c}}, ~2205/2048 = 128.8643{{c}}
: error map: {{val| 0.000 -0.816 -0.111 -0.976 }}
: 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]] (Sintel): 2.39
[[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|legend=0| 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|legend=0| 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|legend=0| 6 5 12 22 33 28 30 | 0 7 3 -8 -19 -9 -7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.0992{{c}}, ~25/24 = 71.2593{{c}}
* WE: ~55/49 = 199.9909{{c}}, ~14/13 = 128.8564{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/24 = 71.2501{{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 (Sintel): 1.52
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 184: Line 306:
Comma list: 2401/2400, 9801/9800, 172032/171875
Comma list: 2401/2400, 9801/9800, 172032/171875


Mapping: {{mapping| 2 4 5 6 6 | 0 -28 -12 -13 31 }}
{{Mapping|legend=0| 2 4 5 6 6 | 0 -28 -12 -13 31 }}


Optimal tunings:  
Optimal tunings:  
Line 195: Line 317:


== Decavish ==
== Decavish ==
Decavish tempers out the [[wizma]] as well as the [[linus comma]], and may be described as the {{nowrap| 50 & 270 }} temperament.  
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 219: Line 341:
Comma list: 9801/9800, 41503/41472, 172032/171875
Comma list: 9801/9800, 41503/41472, 172032/171875


Mapping: {{mapping| 10 6 19 14 36 | 0 7 3 10 -1 }}
{{Mapping|legend=0| 10 6 19 14 36 | 0 7 3 10 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 234: Line 356:
Comma list: 1001/1000, 4225/4224, 4459/4455, 6656/6655
Comma list: 1001/1000, 4225/4224, 4459/4455, 6656/6655


Mapping: {{mapping| 10 6 19 14 36 37 | 0 7 3 10 -1 0 }}
{{Mapping|legend=0| 10 6 19 14 36 37 | 0 7 3 10 -1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 244: Line 366:
Badness (Sintel): 0.905
Badness (Sintel): 0.905


== 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"