1156/1155: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 1156/1155
| Name = quadrantonisma
| Monzo = 2 -1 -1 -1 -1 0 2
| Color name =17oo1urg2, sosolurugu 2nd,<br>Sosolurugu comma
| Cents = 1.49826
| Comma = yes
| Name = septendecimal quartertones comma
| Color name =
| Sound =  
}}
}}
'''1156/1155''' is a [[17-limit]] no-13 [[superparticular]] [[comma]] measuring about 1.41 [[cent]]s. It may be properly described as the ''septendecimal quartertones comma'', since it is the difference between [[34/33]] and [[35/34]], the two 17-limit quartertones.  
'''1156/1155''', the '''quadrantonisma''', is a [[17-limit]] no-13 [[superparticular]] [[comma]] measuring about 1.41 [[cent]]s. It may be properly described as the ''septendecimal quartertones comma'', since it is the difference between [[34/33]] and [[35/34]], the two 17-limit quartertones.  


In terms of commas, it is the difference between [[289/288]] and [[385/384]], and between [[936/935]] and [[4914/4913]]. It factors into ([[2080/2079]])([[2601/2600]]), or ([[1275/1274]])([[12376/12375]]).
== Commatic relations ==
In terms of commas, it is the difference between the following pairs:
* [[289/288]] and [[385/384]]
* [[442/441]] and [[715/714]]
* [[561/560]] and [[1089/1088]]
* [[595/594]] and [[1225/1224]]
* [[936/935]] and [[4914/4913]]
 
It factors into the following pairs:
* [[2080/2079]] and [[2601/2600]]
* [[1275/1274]] and [[12376/12375]]


== Temperaments ==
== Temperaments ==
Tempering out this comma results in [[35/33]] being split into two equal parts, each representing 34/33~35/34, and enables the related essentially tempered chords. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is located right between [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone.  
Tempering out this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone.
 
=== Quadrantonic ===
[[Subgroup]]: 2.3.5.7.11.17
 
[[Sval]] [[mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 1 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || -2 || -1 || ]]
|}
 
: sval mapping generators: ~2, ~3, ~5, ~7, ~34/11
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186
 
{{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 27eg, 39dg, 43, 46, 65d, 68, 72, 118, 171, 183, 239, 282, 301, 311, 400, 422, 472, 494, 894, 1012g, 1205, 1388 }}
 
=== Quadrantonismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 1 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || -2 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}
 
: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186, ~13/8
 
{{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 26, 27eg, 29g, 39dfg, 43, 46, 65d, 68, 72, 111, 121, 140, 171, 183, 217, 282, 301, 311, 354, 400, 422, 494, 894, 1012g, 1133, 1205, 1506g, 1627e }}
 
== Etymology ==
The quadrantonisma was named by [[Flora Canou]] in 2023. It is a contraction of ''septendecimal quartertones comma'' into a single word consisting of Latin ''quadrans'' ("fourth") and ''tonus'' ("tone").  


== See also ==
== See also ==
Line 18: Line 76:
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:17-limit]]
[[Category:Quadrantonismic]]
[[Category:Unnoticeable comma]]
[[Category:Commas named for the intervals they stack]]
[[Category:Superparticular]]

Latest revision as of 01:55, 16 November 2024

Interval information
Ratio 1156/1155
Factorization 22 × 3-1 × 5-1 × 7-1 × 11-1 × 172
Monzo [2 -1 -1 -1 -1 0 2⟩
Size in cents 1.498255¢
Name quadrantonisma
Color name 17oo1urg2, sosolurugu 2nd,
Sosolurugu comma
FJS name [math]\displaystyle{ \text{d2}^{17,17}_{5,7,11} }[/math]
Special properties square superparticular,
reduced
Tenney height (log2 nd) 20.3486
Weil height (log2 max(n, d)) 20.3499
Wilson height (sopfr(nd)) 64
Comma size unnoticeable
S-expression S34
Open this interval in xen-calc

1156/1155, the quadrantonisma, is a 17-limit no-13 superparticular comma measuring about 1.41 cents. It may be properly described as the septendecimal quartertones comma, since it is the difference between 34/33 and 35/34, the two 17-limit quartertones.

Commatic relations

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

It factors into the following pairs:

Temperaments

Tempering out this comma in the 17-limit results in the rank-6 quadrantonismic temperament, or in the 2.3.5.7.11.17 subgroup, the rank-5 quadrantonic temperament. In either case 35/33 is split into two equal parts, each representing 34/33~35/34, and enables quadrantonismic chords. If 9801/9800 is also added to the comma list, the quartertone above becomes literally a quarter of 9/8 and is tuned exactly middle of 33/32, the undecimal quartertone, and 36/35, the septimal quartertone.

Quadrantonic

Subgroup: 2.3.5.7.11.17

Sval mapping:

[⟨ 1 0 0 0 0 -1 ],
⟨ 0 1 0 0 1 1 ],
⟨ 0 0 1 0 1 1 ],
⟨ 0 0 0 1 1 1 ],
⟨ 0 0 0 0 -2 -1 ]]
sval mapping generators: ~2, ~3, ~5, ~7, ~34/11

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186

Optimal ET sequence: 17cg, 19eg, 22, 27eg, 39dg, 43, 46, 65d, 68, 72, 118, 171, 183, 239, 282, 301, 311, 400, 422, 472, 494, 894, 1012g, 1205, 1388

Quadrantonismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 0 0 -1 ],
⟨ 0 1 0 0 1 0 1 ],
⟨ 0 0 1 0 1 0 1 ],
⟨ 0 0 0 1 1 0 1 ],
⟨ 0 0 0 0 -2 0 -1 ],
⟨ 0 0 0 0 0 1 0 ]]
mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186, ~13/8

Optimal ET sequence: 17cg, 19eg, 22, 26, 27eg, 29g, 39dfg, 43, 46, 65d, 68, 72, 111, 121, 140, 171, 183, 217, 282, 301, 311, 354, 400, 422, 494, 894, 1012g, 1133, 1205, 1506g, 1627e

Etymology

The quadrantonisma was named by Flora Canou in 2023. It is a contraction of septendecimal quartertones comma into a single word consisting of Latin quadrans ("fourth") and tonus ("tone").

See also

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