The Nexus

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:

• Tyr (+3025/3024) → Landscape family #Tyr

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