2601/2600: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = septendecimal sixth-tones comma
| Name = sextantonisma
| Color name = 17oo3ugg2, sosothugugu 2nd,<br>Sosothugugu comma
| Color name = 17oo3ugg2, sosothugugu 2nd,<br>Sosothugugu comma
| Comma = yes
| Comma = yes
}}
}}
'''2601/2600''' is a [[17-limit]] (also 2.3.5.13.17 subgroup) [[superparticular]] [[comma]] measuring about 0.67 [[cent]]s. 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.  
'''2601/2600''', the '''sextantonisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.5.13.17-[[subgroup]]) [[superparticular]] [[comma]] measuring about 0.666 [[cent]]s. 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. It also represents 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:  
In terms of commas, it is the difference between the following pairs:  
* [[289/288]] and [[325/324]]
* [[289/288]] and [[325/324]] *
* [[561/560]] and [[715/714]]
* [[561/560]] and [[715/714]]
* [[833/832]] and [[1225/1224]]
* [[833/832]] and [[1225/1224]]
Line 14: Line 15:
* [[1701/1700]] and [[4914/4913]]
* [[1701/1700]] and [[4914/4913]]
* [[2401/2400]] and [[31213/31212]]
* [[2401/2400]] and [[31213/31212]]
* [[2431/2430]] and [[37180/37179]]
<nowiki/>* relation within the 2.3.5.13.17 subgroup


== Temperaments ==
== Temperaments ==
Tempering out this comma results in [[26/25]] being split into two equal parts, each representing 51/50~52/51, and enables the related essentially tempered chords.  
[[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 [[sextantonismic chords]] are enabled.
 
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 }}
: mapping generators: ~2, ~3, ~5, ~51/20
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0165{{c}}, ~3/2 = 701.9109{{c}}, ~5/4 = 386.3400{{c}}, ~51/40 = 420.2767{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9128{{c}}, ~5/4 = 386.3554{{c}}, ~51/40 = 420.2871{{c}}
 
{{Optimal ET sequence|legend=1| 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg }}
 
[[Badness]] (Sintel): 0.0654
 
=== Sextantonismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 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]]s:
* [[WE]]: ~2 = 1200.0165{{c}}, ~3/2 = 701.9109{{c}}, ~5/4 = 386.3400{{c}}, ~7/4 = 968.7969{{c}}, ~11/8 = 551.2685{{c}}, ~51/40 = 420.2767{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9128{{c}}, ~5/4 = 386.3554{{c}}, ~7/4 = 968.8019{{c}}, ~11/8 = 551.2883{{c}}, ~51/40 = 420.2871{{c}}
 
{{Optimal ET sequence|legend=1| 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 }}<nowiki>*</nowiki>
 
<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]]
 
[[Badness]] (Sintel): 0.689
 
== Etymology ==
The sextantonisma was named by [[Flora Canou]] in 2023. It is a contraction of ''sixth-tones comma'' into a single word consisting of Latin ''sextans'' ("sixth") and ''tonus'' ("tone"). This comma was chosen as the sixth-tones comma because the sixth-tones it separates lie in the middle of the harmonic series segment of sixth-tones, 48::54.  


== See also ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]
[[Category:Sextantonismic]]
[[Category:Commas named for the intervals they stack]]

Latest revision as of 07:40, 29 March 2026

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 norm (log2 nd) 22.6891
Weil norm (log2 max(n, d)) 22.6897
Wilson norm (sopfr(nd)) 69
Comma size unnoticeable
S-expression S51
Open this interval in xen-calc

2601/2600, the sextantonisma, is an unnoticeable 17-limit (also 2.3.5.13.17-subgroup) superparticular comma measuring about 0.666 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. It also represents 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:

* relation within the 2.3.5.13.17 subgroup

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 sextantonismic chords are enabled.

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

Subgroup-val mapping: [⟨1 0 0 2 1], ⟨0 1 0 0 -1], ⟨0 0 0 0 1], ⟨0 0 0 2 1]]

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

Optimal tunings:

  • WE: ~2 = 1200.0165 ¢, ~3/2 = 701.9109 ¢, ~5/4 = 386.3400 ¢, ~51/40 = 420.2767 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9128 ¢, ~5/4 = 386.3554 ¢, ~51/40 = 420.2871 ¢

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

Badness (Sintel): 0.0654

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 tunings:

  • WE: ~2 = 1200.0165 ¢, ~3/2 = 701.9109 ¢, ~5/4 = 386.3400 ¢, ~7/4 = 968.7969 ¢, ~11/8 = 551.2685 ¢, ~51/40 = 420.2767 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9128 ¢, ~5/4 = 386.3554 ¢, ~7/4 = 968.8019 ¢, ~11/8 = 551.2883 ¢, ~51/40 = 420.2871 ¢

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

Badness (Sintel): 0.689

Etymology

The sextantonisma was named by Flora Canou in 2023. It is a contraction of sixth-tones comma into a single word consisting of Latin sextans ("sixth") and tonus ("tone"). This comma was chosen as the sixth-tones comma because the sixth-tones it separates lie in the middle of the harmonic series segment of sixth-tones, 48::54.

See also

Retrieved from "https://en.xen.wiki/index.php?title=2601/2600&oldid=227066"