No-threes subgroup temperaments - Revision history

Eliora: /* Overview by mapping of 5 */

2026-08-16T16:41:49Z

Overview by mapping of 5

← Older revision		Revision as of 16:41, 16 August 2026
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	Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].		Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
	* For no-fives, see [[#No-threes no-fives subgroup temperaments]].		* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
	* ~~French decimal and trader have~~ a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.		* Trader has a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
	* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.		* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
	* Insect has a ~55/32 generator, three of which give the ~5/1.		* Insect has a ~55/32 generator, three of which give the ~5/1.

Eliora: per discussion, pointless temperament even by 1789edo lore standards (and it is not even jacobin)

2026-08-16T16:40:30Z

per discussion, pointless temperament even by 1789edo lore standards (and it is not even jacobin)

← Older revision		Revision as of 16:40, 16 August 2026
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	Badness (Sintel): 1.29		Badness (Sintel): 1.29

	~~=== French decimal ===~~
	French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.

	~~[[Subgroup]]: 2.5.7~~

	~~[[Comma list]]: {{monzo\| 372 -159 -1 }}~~

	~~{{Mapping\|legend=2\| 1 0 372 \| 0 1 -159 }}~~
	~~: mapping generators: ~2, ~5~~

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
	~~: [[error map]]: {{val\| -0.010 +0.023 +0.000 }}~~
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
	~~: error map: {{val\| 0.000 +0.046 +0.019 }}~~

	~~{{Optimal ET sequence\|legend=1\| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}~~

	~~[[Badness]] (Sintel): 148~~

	~~==== 2.5.7.11 subgroup ====~~
	~~Subgroup: 2.5.7.11~~

	~~Comma list: {{monzo\| -49 8 17 -5 }}, {{monzo\| 45 -27 10 -3 }}~~

	~~Subgroup-val mapping: {{mapping\| 1 0 372 1255 \| 0 1 -159 -539 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}

	~~{{Optimal ET sequence\|legend=0\| 264, 997e, 1261e, 1525, 1789 }}~~

	~~Badness (Sintel): 52.2~~

	~~==== 2.5.7.11.13 subgroup ====~~
	~~Subgroup: 2.5.7.11.13~~

	~~Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625~~

	~~Subgroup-val mapping: {{mapping\| 1 0 372 1255 -398 \| 0 1 -159 -539 173 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}

	~~{{Optimal ET sequence\|legend=0\| 261, 1261e, 1525, 1789 }}~~

	~~Badness (Sintel): 10.5~~

	=== Bastille ===		=== Bastille ===

FloraC: /* Sephiroth */ + link to submerged

2026-08-08T13:19:04Z

Sephiroth: + link to submerged

← Older revision		Revision as of 13:19, 8 August 2026
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	{{See also\| Chromatic pairs #Sephiroth }}		{{See also\| Chromatic pairs #Sephiroth }}

	Sephiroth is the ~~no-7~~ [[restriction]] of [[muggles]].		Sephiroth is the common [[restriction]] of [[muggles]] and [[submerged]].

	[[Subgroup]]: 2.5.11		[[Subgroup]]: 2.5.11

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
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	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
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	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 ===

← Older revision		Revision as of 12:40, 2 July 2026
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	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]]