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== Rank-4 temperaments ==
== Rank-4 temperaments ==
{{main| Nexus family }}
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 ===
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.
Subgroup: 2.3.5.7.11
[[Comma list]]: [[1771561/1769472]]
[[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}}]
Mapping generators: ~121/96, ~1331/768, ~5/1, ~7/1
{{Multival|legend=1|rank=4| 6 0 0 3 -16 }}
[[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| 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]]: 0.588 × 10<sup>-6</sup>


== Rank-3 temperaments ==
== Rank-3 temperaments ==
Line 33: Line 13:


Temperaments not discussed here include:  
Temperaments not discussed here include:  
* ''[[Tyr]]'', {3025/3024, 102487/102400} → [[Lehmerismic temperaments #Tyr]]
* ''[[Tyr]]'' (+3025/3024) → [[Landscape family #Tyr]]


=== Schisnex ===
=== Schisnex ===
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.
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.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 32805/32768, 1771561/1769472
[[Comma list]]: 32805/32768, 1771561/1769472


[[Mapping]]: [{{val| 3 0 45 0 8 }}, {{val| 0 2 -16 0 1 }}, {{val| 0 0 0 1 0 }}]
{{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=3| 0 6 0 -48 0 -3 -90 0 16 -173 }}


[[POTE generator]]s: ~1331/768 = 950.8701, ~7/4 = 968.5689
[[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 }}


{{Optimal ET sequence|legend=1| 24, 48c, 111cd, 135, 159, 183, 342, 1209, 1551e, 1893e, 2235ce }}
{{Optimal ET sequence|legend=1| 24, 48c, 111cd, 135, 159, 183, 342, 1209, 1551e, 1893e, 2235ce, 5703bbcddeeee }}


[[Badness]]: 5.155 × 10<sup>-3</sup>
[[Badness]] (Sintel): 6.19


=== Nexiquart ===
=== 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.
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
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 19712/19683, 1771561/1769472
[[Comma list]]: 19712/19683, 1771561/1769472


[[Mapping]]: [{{val| 3 0 0 -32 8 }}, {{val| 0 2 0 17 1 }}, {{val| 0 0 1 0 0 }}]
{{Mapping|legend=1| 3 0 0 -32 8 | 0 2 0 17 1 | 0 0 1 0 0 }}
: mapping generators: ~121/96, ~1331/768, ~5


{{Multival|legend=1|rank=3| 6 0 0 -51 -3 0 -64 16 0 -168 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~121/96 = 399.9912{{c}}, ~1331/768 = 951.0868{{c}}, ~5/4 = 386.3663{{c}}
[[POTE generator]]s: ~1331/768 = 951.1077, ~5/4 = 386.3748
: [[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 }}


{{Optimal ET sequence|legend=1| 24, 48d, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b }}
{{Optimal ET sequence|legend=1| 24, 48d, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b }}


[[Badness]]: 6.275 × 10<sup>-3</sup>
[[Badness]] (Sintel): 7.54


== Rank-2 temperaments ==
== Rank-2 temperaments ==
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]].  
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]].  


Considered below is the 2.3.11 subgroup temperament tribilo.  
Considered below is the 2.3.11-subgroup temperament tribilo.  


=== Tribilo ===
=== Tribilo ===
{{Main| Nexus clan }}
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]].
 
[[Subgroup]]: 2.3.11
 
[[Comma list]]: 1771561/1769472
 
{{Mapping|legend=2| 3 0 8 | 0 2 1 }}
: mapping generators: ~121/96, ~1331/768


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]].
[[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}}


Subgroup: 2.3.11
{{Optimal ET sequence|legend=1| 9, 15, 24, 87, 111, 135, 159, 1407, 1566, 1725e, 1884e, 2043e, 2202e, 2361e, 2520e, 2679e }}


[[Comma list]]: 1771561/1769472
[[Badness]] (Sintel): 0.127
 
=== Triblet ===
Subgroup: 2.3.11.13
 
Comma list: 352/351, 1771561/1769472
 
Subgroup-val mapping: {{mapping| 3 0 8 23 | 0 2 1 -5 }}
 
Optimal tunings:
* WE: ~121/96 = 399.8965{{c}}, ~1331/768 = 951.2352{{c}}
* CWE: ~121/96 = 400.0000{{c}}, ~1331/768 = 951.4779{{c}}
 
{{Optimal ET sequence|legend=0| 9f, 15f, 24, 63, 87, 111, 198, 309f, 507f }}
 
Badness (Sintel): 0.613
 
=== Trilobite ===
Subgroup: 2.3.11.13


[[Sval]] [[mapping]]: [{{val| 3 0 8 }}, {{val| 0 2 1 }}]
Comma list: 85293/85184, 1771561/1769472


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


[[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]]

Latest revision as of 08:35, 17 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 [-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)3 and 2/1, all nexus temperaments divide 2/1 into three equal parts of (approximately) 400 cents.

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.

Rank-3 temperaments

From the rank-4 parent temperament, there are several rank-3 temperaments that can be derived.

Temperaments not discussed here include:

Schisnex

Named by Aura in 2021, schisnex adds the nexus comma to the list of commas tempered out. Thus the generator for 7 is indepedent.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 1771561/1769472

Mapping: [⟨3 0 45 0 8], ⟨0 2 -16 0 1], ⟨0 0 0 1 0]]

mapping generators: ~121/96, ~1331/768, ~7

Optimal tunings:

  • WE: ~121/96 = 400.0305 ¢, ~1331/768 = 950.9426 ¢, ~7/4 = 968.6428 ¢
error map: ⟨+0.092 -0.070 -0.022 -0.000 -0.131]
  • CWE: ~121/96 = 400.0000 ¢, ~1331/768 = 950.8678 ¢, ~7/4 = 968.6270 ¢
error map: ⟨0.000 -0.219 -0.199 -0.199 -0.450]

Optimal ET sequence: 24, 48c, 111cd, 135, 159, 183, 342, 1209, 1551e, 1893e, 2235ce, 5703bbcddeeee

Badness (Sintel): 6.19

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.

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 1771561/1769472

Mapping: [⟨3 0 0 -32 8], ⟨0 2 0 17 1], ⟨0 0 1 0 0]]

mapping generators: ~121/96, ~1331/768, ~5

Optimal tunings:

  • WE: ~121/96 = 399.9912 ¢, ~1331/768 = 951.0868 ¢, ~5/4 = 386.3663 ¢
error map: ⟨-0.026 +0.219 -0.000 -0.069 -0.301]
  • CWE: ~121/96 = 400.0000 ¢, ~1331/768 = 951.1062 ¢, ~5/4 = 386.3573 ¢
error map: ⟨0.000 +0.257 +0.044 -0.021 -0.212]

Optimal ET sequence: 24, 48d, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b

Badness (Sintel): 7.54

Rank-2 temperaments

Rank-2 temperaments in the Nexus include hemiennealimmal, hemimist, hemiterm, altinex, mirkat, tritikleismic, terture, and avicenna.

Considered below is the 2.3.11-subgroup temperament tribilo.

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.

Subgroup: 2.3.11

Comma list: 1771561/1769472

Subgroup-val mapping: [⟨3 0 8], ⟨0 2 1]]

mapping generators: ~121/96, ~1331/768

Optimal tunings:

  • WE: ~121/96 = 400.0154 ¢, ~1331/768 = 950.9883 ¢
  • CWE: ~121/96 = 400.0000 ¢, ~1331/768 = 950.9685 ¢

Optimal ET sequence: 9, 15, 24, 87, 111, 135, 159, 1407, 1566, 1725e, 1884e, 2043e, 2202e, 2361e, 2520e, 2679e

Badness (Sintel): 0.127

Triblet

Subgroup: 2.3.11.13

Comma list: 352/351, 1771561/1769472

Subgroup-val mapping: [⟨3 0 8 23], ⟨0 2 1 -5]]

Optimal tunings:

  • WE: ~121/96 = 399.8965 ¢, ~1331/768 = 951.2352 ¢
  • CWE: ~121/96 = 400.0000 ¢, ~1331/768 = 951.4779 ¢

Optimal ET sequence: 9f, 15f, 24, 63, 87, 111, 198, 309f, 507f

Badness (Sintel): 0.613

Trilobite

Subgroup: 2.3.11.13

Comma list: 85293/85184, 1771561/1769472

Subgroup-val mapping: [⟨3 0 8 42], ⟨0 2 1 -13]]

Optimal tunings:

  • WE: ~121/96 = 400.0489 ¢, ~1331/768 = 950.8972 ¢
  • CWE: ~121/96 = 400.0000 ¢, ~1331/768 = 951.7766 ¢

Optimal ET sequence: 24, 111f, 135f, 159, 183, 390, 573, 963f, 1536bef

Badness (Sintel): 0.724

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