1156/1155: Difference between revisions

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== Temperaments ==
== 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.
[[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 ===
=== Quadrantonic ===
[[Subgroup]]: 2.3.5.7.11.17
[[Subgroup]]: 2.3.5.7.11.17


[[Sval]] [[mapping]]: <br>
[[Subgroup val|Subgroup-val]] [[mapping]]: <br>
{| class="right-all"
{| class="right-all"
|-
|-
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|}
|}


: sval mapping generators: ~2, ~3, ~5, ~7, ~34/11
: 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 tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3991{{c}}, ~7/4 = 968.9508{{c}}, ~17/11 = 752.9186{{c}}


{{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 }}
{{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 }}
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: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13  
: 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 tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9948{{c}}, ~5/4 = 386.3991{{c}}, ~7/4 = 968.9508{{c}}, ~17/11 = 752.9186{{c}}, ~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 }}
{{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 }}

Revision as of 17:02, 22 January 2026

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 norm (log2 nd) 20.3486
Weil norm (log2 max(n, d)) 20.3499
Wilson norm (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

Subgroup-val 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 ]]
mapping generators: ~2, ~3, ~5, ~7, ~34/11

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~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 = 1200.0000 ¢, ~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 quartertones comma into a single word consisting of Latin quadrans ("fourth") and tonus ("tone"). This comma was chosen as the quartertones comma because the quartertones it separates lie in the middle of the harmonic series segment of quartertones, 32::36.

See also

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