1156/1155: Difference between revisions

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| Comma = yes

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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 ==

Line 19:

== 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"

|-

| [⟨ || 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 || 2 || 1 || ]]

|-

~~| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0~~ || ]]

|}

: 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]]: 2.3.5.7.11.13.17

Line 52:

Line 57:

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ],

| [⟨ || 1 || 0 || 0 || 0 || -4 || 0 || -3 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ],

| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 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

: ~~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 ==

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 ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

[[Category:Quadrantonismic]]

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

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
-'''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 ==
@@ Line 19: / Line 19: @@
 == 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"
 |-
-| [⟨ || 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 || 2 || 1 || ]]
-|-
-| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
 |}
+: 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]]: 2.3.5.7.11.13.17
@@ Line 52: / Line 57: @@
 {| class="right-all"
 |-
-| [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ],
+| [⟨ || 1 || 0 || 0 || 0 || -4 || 0 || -3 || ],
 |-
-| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ],
+| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 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
-: 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 ==
-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 ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Quadrantonismic]]
+[[Category:Commas named for the intervals they stack]]