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{{Technical data page}}
'''The Nexus''' is a collection of temperaments of different ranks, including subgroup temperaments, that all temper out the [[nexus comma]] – the unnoticeable comma with the ratio 1771561/1769472, and a monzo of {{monzo|-16 -3 0 0 6}}. Among the known branches of this collection are nexus, schisnex, nexiquart, altinex, and tribilo.
'''The Nexus''' is a collection of temperaments of different ranks, including subgroup temperaments, that all temper out the [[nexus comma]] – the unnoticeable comma with the ratio 1771561/1769472, and a monzo of {{monzo|-16 -3 0 0 6}}. Among the known branches of this collection are nexus, schisnex, nexiquart, altinex, and tribilo.


Because the nexus comma is the difference between (121/96)<sup>3</sup> and 2/1, all nexus temperaments divide 2/1 into three equal parts of (approximately) 400 cents.
Because the nexus comma is the difference between (121/96)<sup>3</sup> and 2/1, all nexus temperaments divide 2/1 into three equal parts of (approximately) 400 cents.


= Rank-4 temperaments =
== Rank-4 temperaments ==
See [[Catalog of rank-4 temperaments #Nexus (1771561/1769472)]].
 
The parent temperament of this whole collection is a rank-4 temperament. However, there are other rank-4 extensions which have yet to be explored.
The parent temperament of this whole collection is a rank-4 temperament. However, there are other rank-4 extensions which have yet to be explored.


== Nexus ==
== Rank-3 temperaments ==
{{main| Nexus family }}
From the rank-4 parent temperament, there are several rank-3 temperaments that can be derived.


The parent temperament for the Nexus is Nexus temperament itself. In this temperament, the 5-limit and 7-limit parts are represented by independent generators.
Temperaments not discussed here include:
* ''[[Tyr]]'' (+3025/3024) → [[Landscape family #Tyr]]


Subgroup: 2.3.5.7.11
=== Schisnex ===
Named by [[Aura]] in 2021, schisnex adds the [[nexus comma]] to the list of commas tempered out. Thus the generator for [[7/1|7]] is indepedent.  


[[Comma list]]: [[1771561/1769472]]
[[Subgroup]]: 2.3.5.7.11


[[Mapping]]: [{{val| 3 0 0 0 8}}, {{val| 0 2 0 0 1}}, {{val| 0 0 1 0 0}}, {{val| 0 0 0 1 0}}]
[[Comma list]]: 32805/32768, 1771561/1769472


Mapping generators: ~121/96, ~1331/768, ~5/1, ~7/1
{{Mapping|legend=1| 3 0 45 0 8 | 0 2 -16 0 1 | 0 0 0 1 0 }}
: mapping generators: ~121/96, ~1331/768, ~7


{{Multival|legend=1|rank=4| 6 0 0 3 -16 }}
[[Optimal tuning]]s:
* [[WE]]: ~121/96 = 400.0305{{c}}, ~1331/768 = 950.9426{{c}}, ~7/4 = 968.6428{{c}}
: [[error map]]: {{val| +0.092 -0.070 -0.022 -0.000 -0.131 }}
* [[CWE]]: ~121/96 = 400.0000{{c}}, ~1331/768 = 950.8678{{c}}, ~7/4 = 968.6270{{c}}
: error map: {{val| 0.000 -0.219 -0.199 -0.199 -0.450 }}


[[POTE generator]]s: ~1331/768 = 950.9518 or ~12/11 = 150.9518, ~5/4 = 386.2068, ~7/4 = 968.6966
{{Optimal ET sequence|legend=1| 24, 48c, 111cd, 135, 159, 183, 342, 1209, 1551e, 1893e, 2235ce, 5703bbcddeeee }}


{{Optimal ET sequence|legend=1| 9, 15, 24, 33cd, 39d, 48, 57, 63, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}
[[Badness]] (Sintel): 6.19


[[Badness]]: 0.588 × 10<sup>-6</sup>
=== Nexiquart ===
This temperament can be seen as adding the [[quartisma]] to the list of tempered out commas on account of the quartisma being an unnoticeable comma for adding the 7-limit to the mix.


= Rank-3 temperaments =
[[Subgroup]]: 2.3.5.7.11
From the rank-4 parent temperament, there are several rank-3 temperaments that can be derived.


Temperaments not discussed here include:
[[Comma list]]: 19712/19683, 1771561/1769472
* ''[[Tyr]]'', {3025/3024, 102487/102400} → [[Lehmerismic temperaments #Tyr]]


== Schisnex ==
{{Mapping|legend=1| 3 0 0 -32 8 | 0 2 0 17 1 | 0 0 1 0 0 }}
This temperament adds the [[schisma]] to the list of tempered out commas on account of the schisma being an unnoticeable comma for adding the 5-limit to the mix.
: mapping generators: ~121/96, ~1331/768, ~5


Subgroup: 2.3.5.7.11
[[Optimal tuning]]s:  
* [[WE]]: ~121/96 = 399.9912{{c}}, ~1331/768 = 951.0868{{c}}, ~5/4 = 386.3663{{c}}
: [[error map]]: {{val| -0.026 +0.219 -0.000 -0.069 -0.301 }}
* [[CWE]]: ~121/96 = 400.0000{{c}}, ~1331/768 = 951.1062{{c}}, ~5/4 = 386.3573{{c}}
: error map: {{val| 0.000 +0.257 +0.044 -0.021 -0.212 }}


[[Comma list]]: 32805/32768, 1771561/1769472
{{Optimal ET sequence|legend=1| 24, 48d, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b }}


[[Mapping]]: [{{val| 3 0 45 0 8 }}, {{val| 0 2 -16 0 1 }}, {{val| 0 0 0 1 0 }}]
[[Badness]] (Sintel): 7.54


{{Multival|legend=1|rank=3| 0 6 0 -48 0 -3 -90 0 16 -173 }}
== Rank-2 temperaments ==
Rank-2 temperaments in the Nexus include [[septiennealimmal clan #Hemiennealimmal|hemiennealimmal]], [[misty family #Hemimist|hemimist]], [[schismatic family #Hemiterm|hemiterm]], [[schismatic family #Altinex|altinex]], [[canopic clan #Mirkat|mirkat]], [[kleismic family #Tritikleismic|tritikleismic]], [[vulture family #Terture|terture]], and [[landscape microtemperaments #Avicenna|avicenna]].


[[POTE generator]]s: ~1331/768 = 950.8701, ~7/4 = 968.5689
Considered below is the 2.3.11-subgroup temperament tribilo.  


{{Optimal ET sequence|legend=1| 24, 48c, 111cd, 135, 159, 183, 342, 1209, 1551e, 1893e, 2235ce }}
=== Tribilo ===
Tribilo is the pure rank-2 2.3.11-subgroup temperament that tempers out the nexus comma – its name comes from the nexus comma being referred to as the "tribilo comma" in [[color notation]].


[[Badness]]: 5.155 × 10<sup>-3</sup>
[[Subgroup]]: 2.3.11


== Nexiquart ==
[[Comma list]]: 1771561/1769472
This temperament can be seen as adding the [[quartisma]] to the list of tempered out commas on account of the quartisma being an unnoticeable comma for adding the 7-limit to the mix.


Subgroup: 2.3.5.7.11
{{Mapping|legend=2| 3 0 8 | 0 2 1 }}
: mapping generators: ~121/96, ~1331/768


[[Comma list]]: 19712/19683, 1771561/1769472
[[Optimal tuning]]s:  
* [[WE]]: ~121/96 = 400.0154{{c}}, ~1331/768 = 950.9883{{c}}
* [[CWE]]: ~121/96 = 400.0000{{c}}, ~1331/768 = 950.9685{{c}}


[[Mapping]]: [{{val| 3 0 0 -32 8 }}, {{val| 0 2 0 17 1 }}, {{val| 0 0 1 0 0 }}]
{{Optimal ET sequence|legend=1| 9, 15, 24, 87, 111, 135, 159, 1407, 1566, 1725e, 1884e, 2043e, 2202e, 2361e, 2520e, 2679e }}


{{Multival|legend=1|rank=3| 6 0 0 -51 -3 0 -64 16 0 -168 }}
[[Badness]] (Sintel): 0.127


[[POTE generator]]s: ~1331/768 = 951.1077, ~5/4 = 386.3748
=== Triblet ===
Subgroup: 2.3.11.13


{{Optimal ET sequence|legend=1| 24, 48d, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b }}
Comma list: 352/351, 1771561/1769472


[[Badness]]: 6.275 × 10<sup>-3</sup>
Subgroup-val mapping: {{mapping| 3 0 8 23 | 0 2 1 -5 }}


= Rank-2 temperaments =
Optimal tunings:
Rank-2 temperaments in the Nexus include [[Ragismic microtemperaments #Hemiennealimmal|hemiennealimmal]], [[Misty family #Hemimist|hemimist]], [[Schismatic family #Hemiterm|hemiterm]], [[Schismatic family #Altinex|altinex]], [[Mirkwai clan #Mirkat|mirkat]], [[Kleismic family #Tritikleismic|tritikleismic]], [[Landscape microtemperaments #Terture|terture]], and [[Landscape microtemperaments #Avicenna|avicenna]].  
* WE: ~121/96 = 399.8965{{c}}, ~1331/768 = 951.2352{{c}}
* CWE: ~121/96 = 400.0000{{c}}, ~1331/768 = 951.4779{{c}}


Considered below is the 2.3.11 subgroup temperament tribilo.
{{Optimal ET sequence|legend=0| 9f, 15f, 24, 63, 87, 111, 198, 309f, 507f }}


== Tribilo ==
Badness (Sintel): 0.613
{{Main| Tribilo family }}


Tribilo is the pure Rank-2 2.3.11 temperament that tempers out the nexus comma – its name comes from the nexus comma being referred to as the "tribilo comma" in [[color notation]].
=== Trilobite ===
Subgroup: 2.3.11.13


Subgroup: 2.3.11
Comma list: 85293/85184, 1771561/1769472


[[Comma list]]: 1771561/1769472
Subgroup-val mapping: {{mapping| 3 0 8 42 | 0 2 1 -13 }}
 
[[Sval]] [[mapping]]: [{{val| 3 0 8 }}, {{val| 0 2 1 }}]
 
Sval mapping generators: ~121/96, ~1331/768


[[POTE generator]]: ~1331/768 = 950.9518
Optimal tunings:  
* WE: ~121/96 = 400.0489{{c}}, ~1331/768 = 950.8972{{c}}
* CWE: ~121/96 = 400.0000{{c}}, ~1331/768 = 951.7766{{c}}


{{Optimal ET sequence|legend=1| 9, 15, 24, 87, 111, 135, 159, 1407, 1566, 1725e, 1884e, 2043e, 2202e, 2361e, 2520e, 2679e }}
{{Optimal ET sequence|legend=0| 24, 111f, 135f, 159, 183, 390, 573, 963f, 1536bef }}


Badness:
Badness (Sintel): 0.724


[[Category:Commatic realms]]
[[Category:Commatic realms]]
[[Category:The Nexus| ]] <!-- main article -->
[[Category:The Nexus| ]] <!-- main article -->
[[Category:Nexus]]
[[Category:Nexus]]
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