The Nexus

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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

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

Main article: Nexus family

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

Vals9, 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:

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

Vals24, 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

Vals24, 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, mirkat, tritikleismic, terture, and avicenna.

Considered below are altinex and the 2.3.11 subgroup temperament tribilo.

Altinex

See also: Schismatic family

Subgroup: 2.3.5.7.11

Comma list: 385/384, 14700/14641, 19712/19683

Mapping: [3 0 45 -32 8], 0 2 -16 17 1]]

Mapping generators: ~121/96, ~121/70

POTE generators: ~121/70 = 950.9629

Vals24, 135, 159

Badness: 0.1012

Tribilo

Main article: 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.

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

Vals9, 15, 24, 87, 111, 135, 159, 1407, 1566, 1725e, 1884e, 2043e, 2202e, 2361e, 2520e, 2679e

Badness: