2601/2600: Difference between revisions

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== Temperaments ==
== Temperaments ==
Tempering out this comma in the 17-limit results in the rank-6 '''sextantonismic temperament''', or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic temperament'''. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and enables the [[sextantonismic chords]]. If [[140625/140608]] is also added to the comma list, the sixth-tone above becomes literally a sixth of [[9/8]] and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out [[9801/9800]] and/or [[12376/12375]] since 2601/2600 = (9801/9800)(12376/12375)<sup>2</sup>(140625/140608).  
Tempering out this comma in the 17-limit results in the rank-6 '''sextantonismic temperament''', or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic temperament'''. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and enables the [[sextantonismic chords]]. If [[140625/140608]] is also added to the comma list, the sixth-tone above becomes literally a sixth of [[9/8]] and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out [[9801/9800]] and/or [[12376/12375]] since 2601/2600 = (9801/9800)(12376/12375)<sup>2</sup>(140625/140608).  
=== Sextantonic ===
[[Subgroup]]: 2.3.5.13.17
{{Mapping|legend=2| 1 0 0 2 1 | 0 1 0 0 -1 | 0 0 0 0 1 | 0 0 0 2 1 }}
: sval mapping generators: ~2, ~3, ~5, ~51/20
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~51/40 = 420.3045
{{Optimal ET sequence|legend=1| 12, 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg }}


=== Sextantonismic ===
=== Sextantonismic ===
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<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]]
<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]]
=== Sextantonic ===
[[Subgroup]]: 2.3.5.13.17
{{Mapping|legend=2| 1 0 0 2 1 | 0 1 0 0 -1 | 0 0 0 0 1 | 0 0 0 2 1 }}
: sval mapping generators: ~2, ~3, ~5, ~51/20
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~51/40 = 420.3045
{{Optimal ET sequence|legend=1| 12, 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg }}


== Etymology ==
== Etymology ==

Revision as of 10:24, 28 March 2024

Interval information
Ratio 2601/2600
Factorization 2-3 × 32 × 5-2 × 13-1 × 172
Monzo [-3 2 -2 0 0 -1 2⟩
Size in cents 0.6657312¢
Name sextantonisma
Color name 17oo3ugg2, sosothugugu 2nd,
Sosothugugu comma
FJS name [math]\displaystyle{ \text{d2}^{17,17}_{5,5,13} }[/math]
Special properties square superparticular,
reduced
Tenney height (log2 nd) 22.6891
Weil height (log2 max(n, d)) 22.6897
Wilson height (sopfr(nd)) 69
Comma size unnoticeable
S-expression S51
Open this interval in xen-calc

2601/2600, the sextantonisma, is a 17-limit (also 2.3.5.13.17 subgroup) superparticular comma measuring about 0.67 cents. It may be properly described as the septendecimal sixth-tones comma, since it is the difference between 51/50 and 52/51, the two 17-limit sixth-tones. Another prominent identity is the little gap between 18/13 and a stack of two 20/17's.

Commatic relations

In terms of commas, it is the difference between the following pairs:

Temperaments

Tempering out this comma in the 17-limit results in the rank-6 sextantonismic temperament, or in the 2.3.5.13.17 subgroup, the rank-4 sextantonic temperament. In either case 26/25 is split into two equal parts, each representing 51/50~52/51, and enables the sextantonismic chords. If 140625/140608 is also added to the comma list, the sixth-tone above becomes literally a sixth of 9/8 and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out 9801/9800 and/or 12376/12375 since 2601/2600 = (9801/9800)(12376/12375)2(140625/140608).

Sextantonic

Subgroup: 2.3.5.13.17

Sval mapping: [⟨1 0 0 2 1], ⟨0 1 0 0 -1], ⟨0 0 0 0 1], ⟨0 0 0 2 1]]

sval mapping generators: ~2, ~3, ~5, ~51/20

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~51/40 = 420.3045

Optimal ET sequence: 12, 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg

Sextantonismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 0 1 2 ],
⟨ 0 1 0 0 0 0 -1 ],
⟨ 0 0 1 0 0 0 1 ],
⟨ 0 0 0 1 0 0 0 ],
⟨ 0 0 0 0 1 0 0 ],
⟨ 0 0 0 0 0 2 1 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11, ~51/20

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9252, ~5/4 = 386.3777, ~7/4, ~11/8, ~51/40 = 420.3045

Optimal ET sequence: 17cg, 22f, 26, 29g, 31, 38df, 43, 46, 60e, 65d, 68, 72, 103, 111, 140, 171, 183, 217, 243e, 282, 311, 354, 400, 422, 460, 494, 742, 814, 954, 1236, 1696, 2190g, 4069g*

* optimal patent val: 2932

Etymology

The sextantonisma was named by Flora Canou in 2023. It is a contraction of septendecimal sixth-tones comma into a single word consisting of Latin sextans ("sixth") and tonus ("tone").

See also

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