1156/1155: Difference between revisions

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'''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.  
'''1156/1155''', the '''quadrantonisma''', is an [[unnoticeable comma|unnoticeable]] [[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.  


== Commatic relations ==
== Commatic relations ==
Line 19: Line 19:


== 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 [[quadrantonismic chords]] are enabled.  


=== Quadrantonismic ===
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. This tempers the [[harmonic series segment]] of quartertones, 32::36, to reduce it to three equidistant elements: 33/32, 34/33~35/34, 36/35.
[[Subgroup]]: 2.3.5.7.11.13.17
 
Alternatively, 1089/1088 ({{S|33}}) can be added to the comma list, which reduces the segment to two distinct elements: 33/32~34/33~35/34, 36/35; 1225/1224 ({{S|35}}) works similarly, resulting in 33/32, 34/33~35/34~36/35. Tempering out both 1089/1088 and 1225/1224 while observing 1156/1155 is another major option, resulting in 33/32~34/33, 35/34~36/35, and merging all these temperaments will lead to [[uniwiz]], a rank-3 temperament with a single quartertone representing all the differently sized quartertones in the 2.3.5.7.11.17-subgroup.
 
=== Quadrantonic ===
[[Subgroup]]: 2.3.5.7.11.17


[[Mapping]]: <br>
[[Subgroup val|Subgroup-val]] [[mapping]]: <br>
{| class="right-all"
{| class="right-all"
|-
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -1 || ],
| [⟨ || 1 || 0 || 0 || 0 || -4 || -3 || ],
|-
|-
| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ],
| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ],
|-
|-
| ⟨ || 0 || 0 || 1 || 0 || 1 || 0 || 1 || ],
| ⟨ || 0 || 0 || 1 || 0 || 1 || 1 || ],
|-
|-
| ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ],
| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || -2 || 0 || -1 || ],
|-
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
| ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]]
|}
|}
: mapping generators: ~2, ~3, ~5, ~7, ~22/17


: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9696{{c}}, ~3/2 = 702.0236{{c}}, ~5/4 = 386.4564{{c}}, ~7/4 = 969.0065{{c}}, ~22/17 = 447.0219{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0185{{c}}, ~5/4 = 386.4333{{c}}, ~7/4 = 968.9917{{c}}, ~22/17 = 447.0568{{c}}


[[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, 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, 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 }}
[[Badness]] (Sintel): 0.337


=== Quadrantonic ===
=== Quadrantonismic ===
[[Subgroup]]: 2.3.5.7.11.17
[[Subgroup]]: 2.3.5.7.11.13.17


[[Sval]] [[mapping]]: <br>
[[Mapping]]: <br>
{| class="right-all"
{| class="right-all"
|-
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ],
| [⟨ || 1 || 0 || 0 || 0 || -4 || 0 || -3 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ],
|-
|-
| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ],
| ⟨ || 0 || 0 || 1 || 0 || 1 || 0 || 1 || ],
|-
|-
| ⟨ || 0 || 0 || 1 || 0 || 1 || 1 || ],
| ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ],
|-
|-
| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ],
| ⟨ || 0 || 0 || 0 || 0 || 2 || 0 || 1 || ],
|-
|-
| ⟨ || 0 || 0 || 0 || 0 || -2 || -1 || ]]
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}
|}
: mapping generators: ~2, ~3, ~5, ~7, ~22/17, ~13


: sval mapping generators: ~2, ~3, ~5, ~7, ~34/11
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9696{{c}}, ~3/2 = 702.0236{{c}}, ~5/4 = 386.4564{{c}}, ~7/4 = 969.0065{{c}}, ~22/17 = 447.0219{{c}}, ~13/8 = 840.6188{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0185{{c}}, ~5/4 = 386.4333{{c}}, ~7/4 = 968.9917{{c}}, ~22/17 = 447.0568{{c}}, ~13/8 = 840.5850{{c}}


[[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, 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, 27eg, 39dg, 43, 46, 65d, 68, 72, 118, 171, 183, 239, 282, 301, 311, 400, 422, 472, 494, 894, 1012g, 1205, 1388 }}
[[Badness]] (Sintel): 0.836


== Etymology ==
== 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").  
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 ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:Quadrantonismic]]
[[Category:Quadrantonismic]]
[[Category:Commas named for the intervals they stack]]

Latest revision as of 07:33, 29 March 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 an unnoticeable 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 quadrantonismic chords are enabled.

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. This tempers the harmonic series segment of quartertones, 32::36, to reduce it to three equidistant elements: 33/32, 34/33~35/34, 36/35.

Alternatively, 1089/1088 (S33) can be added to the comma list, which reduces the segment to two distinct elements: 33/32~34/33~35/34, 36/35; 1225/1224 (S35) works similarly, resulting in 33/32, 34/33~35/34~36/35. Tempering out both 1089/1088 and 1225/1224 while observing 1156/1155 is another major option, resulting in 33/32~34/33, 35/34~36/35, and merging all these temperaments will lead to uniwiz, a rank-3 temperament with a single quartertone representing all the differently sized quartertones in the 2.3.5.7.11.17-subgroup.

Quadrantonic

Subgroup: 2.3.5.7.11.17

Subgroup-val mapping:

[⟨ 1 0 0 0 -4 -3 ],
⟨ 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, ~22/17

Optimal tunings:

  • WE: ~2 = 1199.9696 ¢, ~3/2 = 702.0236 ¢, ~5/4 = 386.4564 ¢, ~7/4 = 969.0065 ¢, ~22/17 = 447.0219 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0185 ¢, ~5/4 = 386.4333 ¢, ~7/4 = 968.9917 ¢, ~22/17 = 447.0568 ¢

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

Badness (Sintel): 0.337

Quadrantonismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 -4 0 -3 ],
⟨ 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, ~22/17, ~13

Optimal tunings:

  • WE: ~2 = 1199.9696 ¢, ~3/2 = 702.0236 ¢, ~5/4 = 386.4564 ¢, ~7/4 = 969.0065 ¢, ~22/17 = 447.0219 ¢, ~13/8 = 840.6188 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0185 ¢, ~5/4 = 386.4333 ¢, ~7/4 = 968.9917 ¢, ~22/17 = 447.0568 ¢, ~13/8 = 840.5850 ¢

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

Badness (Sintel): 0.836

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

Retrieved from "https://en.xen.wiki/index.php?title=1156/1155&oldid=227064"