1156/1155 - Revision history

FloraC: /* Temperaments */ grammar

2026-03-29T07:33:59Z

Temperaments: grammar

← Older revision		Revision as of 07:33, 29 March 2026
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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]].		[[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.		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.

FloraC: /* Temperaments */ expand

2026-03-25T10:04:34Z

Temperaments: expand

← Older revision		Revision as of 10:04, 25 March 2026
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 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. 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 ({{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 ===		=== Quadrantonic ===

FloraC: Update data

2026-03-25T09:20:21Z

Update data

← Older revision		Revision as of 09:20, 25 March 2026
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	{\| class="right-all"		{\| class="right-all"
	\|-		\|-
	\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -1 \|\| ],		\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| -4 \|\| -3 \|\| ],
	\|-		\|-
	\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| ],		\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| ],
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| 1 \|\| ],		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| 1 \|\| ],
	\|-		\|-
	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| -1 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 2 \|\| 1 \|\| ]]
	\|}		\|}
	: mapping generators: ~2, ~3, ~5, ~7, ~34/11		: mapping generators: ~2, ~3, ~5, ~7, ~22/17

	[[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 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 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 }}

			[[Badness]] (Sintel): 0.337

	=== Quadrantonismic ===		=== Quadrantonismic ===
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	{\| class="right-all"		{\| class="right-all"
	\|-		\|-
	\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -1 \|\| ],		\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| -4 \|\| 0 \|\| -3 \|\| ],
	\|-		\|-
	\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 1 \|\| ],		\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 1 \|\| ],
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| 0 \|\| 1 \|\| ],		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| 0 \|\| 1 \|\| ],
	\|-		\|-
	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| 0 \|\| -1 \|\| ],		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 2 \|\| 0 \|\| 1 \|\| ],
	\|-		\|-
	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| ]]
	\|}		\|}
	: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13		: mapping generators: ~2, ~3, ~5, ~7, ~22/17, ~13

	[[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 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 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 }}

			[[Badness]] (Sintel): 0.836

	== Etymology ==		== Etymology ==

FloraC: Misc. cleanup

2026-03-25T09:12:30Z

Misc. cleanup

← Older revision		Revision as of 09:12, 25 March 2026
Line 4:		Line 4:
	\| Comma = yes		\| 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 ==		== Commatic relations ==
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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 ===
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| -1 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| -1 \|\| ]]
	\|}		\|}

	: mapping generators: ~2, ~3, ~5, ~7, ~34/11		: mapping generators: ~2, ~3, ~5, ~7, ~34/11

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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| ]]
	\|}		\|}

	: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13		: mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13

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

FloraC: Cleanup

2026-01-22T17:02:14Z

Cleanup

← Older revision		Revision as of 17:02, 22 January 2026
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 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"
	\|-		\|-
Line 38:		Line 38:
	\|}		\|}

	: ~~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 }}

FloraC: + reason why this one is the quadrantonisma

2026-01-22T17:00:09Z

+ reason why this one is the quadrantonisma

← Older revision		Revision as of 17:00, 22 January 2026
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	== 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 ==

BudjarnLambeth at 01:55, 16 November 2024

2024-11-16T01:55:47Z

← Older revision		Revision as of 01:55, 16 November 2024
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	[[Category:Quadrantonismic]]		[[Category:Quadrantonismic]]
			[[Category:Commas named for the intervals they stack]]

FloraC: /* Temperaments */ swap them to impress the reader of the "simpler" temp first

2024-03-28T10:27:47Z

Temperaments: swap them to impress the reader of the "simpler" temp first

@@ Line 20: / Line 20: @@
 == 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.
 === Quadrantonismic ===
@@ Line 45: / Line 68: @@
 {{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 ==

FloraC: /* Temperaments */ correct the subgroup

2023-08-15T10:45:04Z

Temperaments: correct the subgroup

← Older revision		Revision as of 10:45, 15 August 2023
Line 47:		Line 47:

	=== Quadrantonic ===		=== Quadrantonic ===
	[[Subgroup]]: 2.3.5.7.11~~.13~~.17		[[Subgroup]]: 2.3.5.7.11.17

	[[Sval]] [[mapping]]: <br>		[[Sval]] [[mapping]]: <br>

FloraC: /* Temperaments */ correct keys

2023-08-13T13:16:05Z

Temperaments: correct keys

← Older revision		Revision as of 13:16, 13 August 2023
Line 49:		Line 49:
	[[Subgroup]]: 2.3.5.7.11.13.17		[[Subgroup]]: 2.3.5.7.11.13.17

	[[~~Mapping~~]]: <br>		[[Sval]] [[mapping]]: <br>
	{\| class="right-all"		{\| class="right-all"
	\|-		\|-
Line 63:		Line 63:
	\|}		\|}

	: mapping generators: ~2, ~3, ~5, ~7, ~34/11		: 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 tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186