The Nexus: Difference between revisions
→Rank-3 temperaments: +data |
+data |
||
| Line 1: | Line 1: | ||
'''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 {{monzo|-16 -3 0 0 6}}. Among the known branches of this collection are nexus, schisnex, nexiquart, altinex, and tribilo. | '''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 {{monzo|-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) | Because the nexus comma is the difference between (121/96)<sup>3</sup> and 2/1, all nexus temperaments divide 2/1 into three equal parts of (approximately) 400 cents. | ||
Temperaments not discussed here include: | Temperaments not discussed here include: | ||
* ''[[ | * ''[[tyr]]'', {3025/3024, 102487/102400} → [[Lehmerismic temperaments #Tyr]] | ||
= Rank-4 temperaments = | = Rank-4 temperaments = | ||
| Line 12: | Line 12: | ||
{{main| Nexus family }} | {{main| Nexus family }} | ||
The parent temperament for the Nexus is Nexus temperament itself. | 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 | Subgroup: 2.3.5.7.11 | ||
| Line 24: | Line 24: | ||
{{Multival|legend=1|rank=4| 6 0 0 3 -16 }} | {{Multival|legend=1|rank=4| 6 0 0 3 -16 }} | ||
[[POTE generator]]s: ~1331/768 = 950.9518, ~5/4 = 386.2068, ~7/4 = 968.6966 | [[POTE generator]]s: ~1331/768 = 950.9518 or ~12/11 = 150.9518, ~5/4 = 386.2068, ~7/4 = 968.6966 | ||
{{Val list|legend=1| 9, 15, 24, 39d, 48, 57, 63, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566 }} | {{Val list|legend=1| 9, 15, 24, 39d, 48, 57, 63, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566 }} | ||
| Line 32: | Line 32: | ||
= Rank-3 temperaments = | = Rank-3 temperaments = | ||
From the rank-4 parent temperament, there are several rank-3 temperaments that can be derived. | From the rank-4 parent temperament, there are several rank-3 temperaments that can be derived. | ||
== Schisnex == | == Schisnex == | ||
| Line 46: | Line 44: | ||
{{Multival|legend=1|rank=3| 0 6 0 -48 0 -3 -90 0 16 -173 }} | {{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 | [[POTE generator]]s: ~1331/768 = 950.8701 or ~12/11 = 150.8701, ~7/4 = 968.5689 | ||
{{Val list|legend=1| 24, 48, 111cd, 135, 159, 183, 342 }} | {{Val list|legend=1| 24, 48, 111cd, 135, 159, 183, 342 }} | ||
| Line 63: | Line 61: | ||
{{Multival|legend=1|rank=3| 6 0 0 -51 -3 0 -64 16 0 -168 }} | {{Multival|legend=1|rank=3| 6 0 0 -51 -3 0 -64 16 0 -168 }} | ||
[[POTE generator]]s: ~1331/768 = 951.1077, ~5/4 = 386.3748 | [[POTE generator]]s: ~1331/768 = 951.1077 or ~12/11 = 151.1077, ~5/4 = 386.3748 | ||
{{Val list|legend=1| 24, 48, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b }} | {{Val list|legend=1| 24, 48, 87d, 111, 159, 246, 270, 1215, 1485, 1755, 2025, 2295b }} | ||
| Line 70: | Line 68: | ||
= Rank-2 temperaments = | = Rank-2 temperaments = | ||
So far, among the only known rank-2 temperaments in the Nexus are | So far, among the only known rank-2 temperaments in the Nexus are altinex and tribilo. | ||
== Altinex == | == Altinex == | ||
Subgroup: 2.3.5.7.11 | |||
[[Comma list]]: 385/384, 14700/14641, 19712/19683 | |||
[[Mapping]]: [{{val| 3 0 45 -32 8 }}, {{val| 0 2 -16 17 1 }}] | |||
Mapping generators: ~121/96, ~121/70 | |||
[[POTE generator]]s: ~121/70 = 950.9629 or ~12/11 = 150.9629 | |||
{{Val list|legend=1| 24, 135, 159 }} | |||
Badness: | [[Badness]]: 0.1012 | ||
== Tribilo == | == Tribilo == | ||
| Line 93: | Line 93: | ||
Subgroup: 2.3.11 | Subgroup: 2.3.11 | ||
Comma list: 1771561/1769472 | [[Comma list]]: 1771561/1769472 | ||
[[Sval]] [[mapping]]: [{{val| 3 0 8}}, {{val| 0 2 1}}] | |||
Sval mapping generators: ~121/96, ~1331/768 | |||
[[POTE generator]]: ~1331/768 = 950.9518 or ~12/11 = 150.9518 | |||
{{Val list|legend=1| 9, 15, 24, 87, 111, 135, 159, 1407, 1566, 1725e, 1884e, 2043e, 2202e, 2361e, 2520e, 2679e }} | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Commatic realm]] | [[Category:Commatic realm]] | ||
[[Category:Nexus]] | [[Category:Nexus]] | ||
Revision as of 05:25, 9 June 2021
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.
Temperaments not discussed here include:
- tyr, {3025/3024, 102487/102400} → Lehmerismic temperaments #Tyr
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
Badness: 0.588 × 10-6
Rank-3 temperaments
From the rank-4 parent temperament, there are several rank-3 temperaments that can be derived.
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 or ~12/11 = 150.8701, ~7/4 = 968.5689
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 or ~12/11 = 151.1077, ~5/4 = 386.3748
Badness: 6.275 × 10-3
Rank-2 temperaments
So far, among the only known rank-2 temperaments in the Nexus are altinex and tribilo.
Altinex
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 or ~12/11 = 150.9629
Badness: 0.1012
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 or ~12/11 = 150.9518