The Nexus

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

Main article: Nexus family

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: [⟨3 0 0 0 8], ⟨0 2 0 0 1], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

Mapping generators: ~121/96, ~1331/768, ~5/1, ~7/1

Wedgie: ⟨⟨⟨⟨6 0 0 3 -16]]]]

POTE generators: ~1331/768 = 950.9518 or ~12/11 = 150.9518, ~5/4 = 386.2068, ~7/4 = 968.6966

Optimal ET sequence: 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^-6

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, 102487/102400} → Lehmerismic temperaments #Tyr

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.

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]]

Wedgie: ⟨⟨⟨0 6 0 -48 0 -3 -90 0 16 -173]]]

POTE generators: ~1331/768 = 950.8701, ~7/4 = 968.5689

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

Badness: 5.155 × 10^-3

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]]

Wedgie: ⟨⟨⟨6 0 0 -51 -3 0 -64 16 0 -168]]]

POTE generators: ~1331/768 = 951.1077, ~5/4 = 386.3748

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

Badness: 6.275 × 10^-3

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

Main article: Nexus clan

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.

Subgroup: 2.3.11

Comma list: 1771561/1769472

Sval mapping: [⟨3 0 8], ⟨0 2 1]]

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

POTE generator: ~1331/768 = 950.9518

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

Badness: