No-threes subgroup temperaments - Revision history

FloraC: /* No-threes no-fives subgroup temperaments */ move arsenic here

2026-07-29T09:20:56Z

No-threes no-fives subgroup temperaments: move arsenic here

← Older revision		Revision as of 09:20, 29 July 2026
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	[[Badness]] (Sintel): 0.386		[[Badness]] (Sintel): 0.386

			=== Arsenic ===
			Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.

			[[Subgroup]]: 2.11.13.17.19

			[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432

			{{Mapping\|legend=2\| 33 0 8 249 26 \| 0 1 1 -1 1 }}
			: mapping generators: ~247/242, ~11

			[[Optimal tuning]]s:
			* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
			* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}

			{{Optimal ET sequence\|legend=1\| 33, 165, 198, 231, 495h }}

			[[Badness]] (Sintel): 2.30

	=== Berylic ===		=== Berylic ===

FloraC: Delete tricesimoprimal miracloid. + todo for no-3 naiad

2026-07-03T16:20:05Z

Delete tricesimoprimal miracloid. + todo for no-3 naiad

@@ Line 342: / Line 342: @@
 [[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
 === No-threes naiad (rank-3) ===
 {{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}

FloraC: /* No-threes no-fives subgroup temperaments */

2026-07-02T12:40:30Z

No-threes no-fives subgroup temperaments

← Older revision		Revision as of 12:40, 2 July 2026
Line 579:		Line 579:
	Temperaments discussed elsewhere include		Temperaments discussed elsewhere include
	* Orgone → [[Orgonia #Orgone\|Orgonia]]		* Orgone → [[Orgonia #Orgone\|Orgonia]]
	* Berylic → [[4th-octave temperaments #Berylic\|4th-octave temperaments]]
	* 21-23-commatic → [[21st-octave temperaments #21-23-commatic\|21st-octave temperaments]]		* 21-23-commatic → [[21st-octave temperaments #21-23-commatic\|21st-octave temperaments]]
	* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic\|31st-octave temperaments]]		* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic\|31st-octave temperaments]]

FloraC: Move berylic here

2026-07-02T12:37:30Z

Move berylic here

← Older revision		Revision as of 12:37, 2 July 2026
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	[[Badness]] (Sintel): 0.386		[[Badness]] (Sintel): 0.386

			=== Berylic ===
			Berylic tempers out the [[berylisma]] in the 2.11.37 subgroup, representing the fact that [[44/37]] is a {{w\|continued fraction}} convergent to 2<sup>1/4</sup> the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low [[badness]], being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by edo systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.

			If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407\|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.

			[[Subgroup]]: 2.11.37

			[[Comma list]]: 1874161/1874048

			{{Mapping\|legend=2\| 4 0 7 \| 0 1 1 }}
			: mapping generators: ~44/37, ~11

			[[Optimal tuning]]s:
			* [[WE]]: ~44/37 = 300.0003{{c}}, ~11/8 = 551.3211{{c}}
			: [[error map]]: {{val\| +0.001 +0.007 -0.017 }}
			* [[CWE]]: ~44/37 = 300.0000{{c}}, ~11/8 = 551.3237{{c}}
			: error map: {{val\| 0.000 +0.006 -0.020 }}

			{{Optimal ET sequence\|legend=1\| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}

			[[Badness]] (Sintel): 0.00188

	=== Mavericks ===		=== Mavericks ===

FloraC: /* Temperaments with a 2.5.7 gene */ move pakkanen to the comma page as this page is mainly for rank-2

2026-07-02T12:06:14Z

Temperaments with a 2.5.7 gene: move pakkanen to the comma page as this page is mainly for rank-2

@@ Line 364: / Line 364: @@
 Badness (Sintel): 4.51
-=== Pakkanen ===
+=== No-threes naiad (rank-3) ===
 {{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}

FloraC: /* No-threes no-fives subgroup temperaments */ move argument here

2026-07-02T11:35:58Z

No-threes no-fives subgroup temperaments: move argument here

← Older revision		Revision as of 11:35, 2 July 2026
Line 624:		Line 624:

	Badness (Sintel): 0.0309		Badness (Sintel): 0.0309

			=== Argument ===
			Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment\|augment]].

			[[Subgroup]]: 2.7.11

			[[Comma list]]: 1372/1331

			{{Mapping\|legend=2\| 3 0 2 \| 0 1 1 }}
			: mapping generators: ~14/11, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~14/11 = 399.8041{{c}}, ~7/4 = 963.1666{{c}}
			: [[error map]]: {{val\| -0.588 -6.835 +10.281 }}
			* [[CWE]]: ~14/11 = 400.0000{{c}}, ~4/4 = 962.7466{{c}}
			: error map: {{val\| 0.000 -6.079 +11.429 }}

			{{Optimal ET sequence\|legend=1\| 6, 9, 15, 36, 51e, 66e }}

			[[Badness]] (Sintel): 0.475

	=== Score ===		=== Score ===

FloraC: /* No-threes no-fives subgroup temperaments */ sort by badness

2026-07-02T11:30:30Z

No-threes no-fives subgroup temperaments: sort by badness

@@ Line 787: / Line 787: @@
 [[Badness]] (Sintel): 0.175
 === Blackbirds ===
@@ Line 811: / Line 834: @@
 [[Badness]] (Sintel): 0.668
 === Yamablu ===

FloraC: /* No-threes no-fives subgroup temperaments */ + missing entries

2026-07-02T11:29:19Z

No-threes no-fives subgroup temperaments: + missing entries

← Older revision		Revision as of 11:29, 2 July 2026
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	=== Score ===		=== Score ===
	{{See also\| Chromatic pairs #Score }}		{{See also\| Chromatic pairs #Score }}

			Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.

	[[Subgroup]]: 2.7.11.13		[[Subgroup]]: 2.7.11.13
Line 650:		Line 652:
	Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].		Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].

	[[Subgroup]]: 2.7.11~~.13~~		[[Subgroup]]: 2.7.11

	[[Comma list]]: ~~1573~~/~~1568, 15488/15379~~		[[Comma list]]: 214358881/210827008

	{{Mapping\|legend=2\| 1 0 1 3 \| 0 8 7 2 }}		{{Mapping\|legend=2\| 1 0 1 \| 0 8 7 }}

	{{Mapping\|legend=3\| 1 0 0 0 1 3 \| 0 0 0 8 7 2 }}		{{Mapping\|legend=3\| 1 0 0 0 1 \| 0 0 0 8 7 }}
	: mapping generators: ~2, ~14/11		: mapping generators: ~2, ~14/11

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = ~~1199~~.~~8668~~{{c}}, ~14/11 = 421.~~2623~~{{c}}		* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}		: [[error map]]: {{val\| +0.189 +1.303 -2.266 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2365{{c}}
			: error map: {{val\| 0.000 +1.066 -2.662 }}

			{{Optimal ET sequence\|legend=1\| 17, 20, 37, 57, 94, 151 }}

			[[Badness]] (Sintel): 1.73

			==== 2.7.11.13 subgroup ====
			Subgroup: 2.7.11.13

			Comma list: 1573/1568, 15488/15379

			Subgroup-val mapping: {{mapping\| 1 0 1 3 \| 0 8 7 2 }}

			Gencom mapping: {{mapping\|\| 1 0 0 0 1 3 \| 0 0 0 8 7 2 }}

			Optimal tunings:
			* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}

	{{Optimal ET sequence\|legend=1\| 17, 20, 37, 57, 94, 225 }}		{{Optimal ET sequence\|legend=0\| 17, 20, 37, 57, 94, 225 }}

	[[Badness]] (Sintel): 0.307		Badness (Sintel): 0.307

	=== Voltage ===		=== Voltage ===
Line 815:		Line 836:

	=== Yamablu ===		=== Yamablu ===
	Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Genchain mode numbering\|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].		Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The [[Kite's Genchain mode numbering\|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].

	[[Subgroup]]: 2.11.13.17.19		[[Subgroup]]: 2.11.13.17.19

FloraC: Switch to Sintel's badness, WE & CWE tunings (4/4)

2026-07-02T11:13:48Z

Switch to Sintel's badness, WE & CWE tunings (4/4)

Show changes

FloraC: /* Temperaments with a higher 2.5.p gene */ sort by badness

2026-07-02T08:56:03Z

Temperaments with a higher 2.5.p gene: sort by badness

@@ Line 437: / Line 437: @@
 Temperaments discussed elsewhere include:
 * Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
 === Wizz ===
@@ Line 499: / Line 479: @@
 [[Badness]] (Sintel): 0.564
 === Sephiroth ===

← Older revision		Revision as of 12:40, 2 July 2026
Line 579:		Line 579:
	Temperaments discussed elsewhere include		Temperaments discussed elsewhere include
	* Orgone → [[Orgonia #Orgone\|Orgonia]]		* Orgone → [[Orgonia #Orgone\|Orgonia]]
	* Berylic → [[4th-octave temperaments #Berylic\|4th-octave temperaments]]
	* 21-23-commatic → [[21st-octave temperaments #21-23-commatic\|21st-octave temperaments]]		* 21-23-commatic → [[21st-octave temperaments #21-23-commatic\|21st-octave temperaments]]
	* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic\|31st-octave temperaments]]		* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic\|31st-octave temperaments]]