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
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
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
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: