No-twos subgroup temperaments - Revision history

CompactStar: /* Threneb */

2026-10-20T22:40:18Z

Threneb

← Older revision		Revision as of 22:40, 20 October 2026
Line 804:		Line 804:

	=== Threneb ===		=== Threneb ===
	Named by [[User:CompactStar\|CompactStar]] in 2026, Threneb is another weak extension that tempers out the threedie ([[2197/2187]]), equating [[13/9]] to 1 step of [[3edt]], and making that the new period.		Named by [[User:CompactStar\|CompactStar]] in 2026, Threneb is another weak extension that tempers out the threedie ([[2197/2187]]), equating [[13/9]] to 1 step of [[3edt]], and making that the new period. Unfortunately, the smaller MOS scales generated by this are very [[hard]].

	[[Subgroup]]: 3.5.11.13		[[Subgroup]]: 3.5.11.13

CompactStar: /* Deneb */

2026-10-20T22:39:01Z

Deneb

← Older revision		Revision as of 22:39, 20 October 2026
Line 801:		Line 801:

	Badness (Sintel): 0.838		Badness (Sintel): 0.838


			=== Threneb ===
			Named by [[User:CompactStar\|CompactStar]] in 2026, Threneb is another weak extension that tempers out the threedie ([[2197/2187]]), equating [[13/9]] to 1 step of [[3edt]], and making that the new period.

			[[Subgroup]]: 3.5.11.13

			[[Comma list]]: 6655/6561, 2197/2187

			{{Mapping\|legend=3\| 3 0 8 7 \| 0 -3 1 0 }}
			: mapping generators: ~13/9, ~243/143

			[[Optimal tuning]]s:
			* [[WE]]: ~3 = 634.520{{c}}, ~243/143 = 928.583{{c}}
			: [[error map]]: {{val\| +1.606 -0.565 -3.739 +1.114}}
			* [[CWE]]: ~3 = 634.520{{c}}, ~243/143 = 927.930{{c}}
			: error map: {{val\| 0.000 -2.559 -7.368 -2.633 }}

			[[Optimal ET sequence]]: TBA

			[[Badness]] (Sintel): 0.646

	== Electra ==		== Electra ==

FloraC: /* Vega */

2026-10-19T11:45:32Z

Vega

← Older revision		Revision as of 11:45, 19 October 2026
Line 199:		Line 199:

	{{Mapping\|legend=3\| 1 0 4 \| 0 2 -3 }}		{{Mapping\|legend=3\| 1 0 4 \| 0 2 -3 }}
	: mapping generators: ~3, ~25/21		: mapping generators: ~3, ~81/35

	[[Optimal tuning]]s:		[[Optimal tuning]]s:

FloraC: Document vega

2026-10-19T07:14:54Z

Document vega

← Older revision		Revision as of 07:14, 19 October 2026
Line 192:		Line 192:

	Badness (Sintel): 0.177		Badness (Sintel): 0.177

			== Vega ==
			[[Subgroup]]: 3.5.7

			[[Comma list]]: 6561/6125

			{{Mapping\|legend=3\| 1 0 4 \| 0 2 -3 }}
			: mapping generators: ~3, ~25/21

			[[Optimal tuning]]s:
			* [[WE]]: ~3 = 1891.9431{{c}}, ~81/35 = 1397.0975{{c}}
			: [[error map]]: {{val\| -10.012 +7.881 +7.654 }}
			* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/35 = 1404.6664{{c}}
			: error map: {{val\| 0.000 +23.019 +24.995 }}

			[[Optimal ET sequence]]: [[4edt\|b4]], [[11edt\|b11d]], [[15edt\|b15]], [[19edt\|b19]], [[23edt\|b23]], [[42edt\|b42d]], [[65edt\|b65cd]]

			[[Badness]] (Sintel): 1.34

	== Sirius ==		== Sirius ==

CompactStar: /* Overview by mapping of 5 */

2026-10-18T04:12:29Z

Overview by mapping of 5

← Older revision		Revision as of 04:12, 18 October 2026
Line 7:		Line 7:
	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]].

	* Arcturus, antares, aldebaran, delta centauri and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.		* Arcturus, antares, aldebaran, delta centauri and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals.
	* BPS has a ~9/7 generator, two of which give the ~5/3.		* BPS has a ~9/7 generator, two of which give the ~5/3.
	* Sirius has a ~25/21 generator, three of which give the ~5/3.		* Sirius has a ~25/21 generator, three of which give the ~5/3.

CompactStar: /* Overview by mapping of 5 */

2026-10-18T04:12:21Z

Overview by mapping of 5

← Older revision		Revision as of 04:12, 18 October 2026
Line 7:		Line 7:
	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]].

	* Arcturus, antares, aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.		* Arcturus, antares, aldebaran, delta centauri and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
	* BPS has a ~9/7 generator, two of which give the ~5/3.		* BPS has a ~9/7 generator, two of which give the ~5/3.
	* Sirius has a ~25/21 generator, three of which give the ~5/3.		* Sirius has a ~25/21 generator, three of which give the ~5/3.

FloraC: Graphs/diagrams have already been merged

2026-10-17T09:41:18Z

Graphs/diagrams have already been merged

← Older revision		Revision as of 09:41, 17 October 2026
Line 1,200:		Line 1,200:
	* [[Catfish7]]		* [[Catfish7]]
	* [[Catfish9]]		* [[Catfish9]]

	~~= Graphs =~~
	~~See: [[Catalog of 3.5.7 subgroup rank two temperaments #Graphs]]~~

	~~== Projective tuning space diagrams ==~~
	~~See: [[Catalog of 3.5.7 subgroup rank two temperaments #Projective tuning space diagrams]]~~

	[[Category:Catalogs of subgroup temperaments]]		[[Category:Catalogs of subgroup temperaments]]
	[[Category:Catalogs of rank-2 temperaments]]		[[Category:Catalogs of rank-2 temperaments]]

FloraC: Merge from polaris

2026-10-17T09:25:46Z

Merge from polaris

← Older revision		Revision as of 09:25, 17 October 2026
Line 44:		Line 44:

	=== Polturus ===		=== Polturus ===
	This extension of arcturus adds [[polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.		This extension of arcturus adds [[#Polaris\|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down.

	Subgroup: 3.5.7.11		Subgroup: 3.5.7.11
Line 459:		Line 459:

	== Miaplacidus ==		== Miaplacidus ==
	Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig\|link=~~true~~\|3L 4s<3/1>}}, {{mos scalesig\|link=~~true~~\|3L 7s<3/1>}}, {{mos scalesig\|link=~~true~~\|3L 10s<3/1>}}, and {{mos scalesig\|link=~~true~~\|13L 3s<3/1>}}. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]].		Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig\|link=1\|3L 4s<3/1>}}, {{mos scalesig\|link=1\|3L 7s<3/1>}}, {{mos scalesig\|link=1\|3L 10s<3/1>}}, and {{mos scalesig\|link=1\|13L 3s<3/1>}}. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]].

	[[Subgroup]]: 3.5.7		[[Subgroup]]: 3.5.7
Line 634:		Line 634:
	= 3.5.11-subgroup temperaments =		= 3.5.11-subgroup temperaments =
	== Polaris ==		== Polaris ==
	~~{{Main\| Polaris }}~~		Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]]. It contains [[mos scale]]s of the families {{mos scalesig\|link=1\|2L 5s<3/1>}}, {{mos scalesig\|link=1\|2L 7s<3/1>}}, and {{mos scalesig\|link=1\|2L 9s<3/1>}}. The hendecatonic scale is generally the most usable of these, with the heptatonic and enneatonic scales usually being lopsided. Good tunings for polaris include [[15edt]], [[17edt]] and [[32edt]].

	Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]].

	[[Subgroup]]: 3.5.11		[[Subgroup]]: 3.5.11
Line 654:		Line 652:

	[[Badness]] (Sintel): 1.01		[[Badness]] (Sintel): 1.01

			; Scales
			* [[Polaris9]]
			* [[Polaris11]]

	== Delta centauri ==		== Delta centauri ==
Line 1,160:		Line 1,162:
	Named by {{u\|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup.		Named by {{u\|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup.

	CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone\|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{~~MOS~~ scalesig\|2L 3s<11/5>}}, {{~~MOS~~ scalesig\|2L 5s<11/5>}}, {{~~MOS~~ scalesig\|2L 7s<11/5>}}, and {{~~MOS~~ scalesig\|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.		CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone\|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{mos scalesig\|2L 3s<11/5>}}, {{mos scalesig\|2L 5s<11/5>}}, {{mos scalesig\|2L 7s<11/5>}}, and {{mos scalesig\|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord.

	[[Subgroup]]: 5.11.13		[[Subgroup]]: 5.11.13

FloraC: Merge from miaplacidus

2026-10-17T09:20:36Z

Merge from miaplacidus

@@ Line 12: / Line 12: @@
 * Deneb has a ~11/9 generator, three of which give the ~9/5.
 * Canopus has a ~15/7 generator, five of which give the ~45/1 (5/1 up two tritaves).
 * Alnilam has a ~55/27 generator, ten of which give the ~1215/1 (5/1 up five tritaves).
 * Izar has a ~30375/16807 generator, twelve of which give the ~1215/1 (5/1 up five tritaves).
@@ Line 456: / Line 457: @@
 Badness (Sintel): 0.542
 == Izar ==

FloraC: Merge from electra

2026-10-17T09:07:29Z

Merge from electra

@@ Line 7: / Line 7: @@
 Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
-* Arcturus, antares aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
+* Arcturus, antares, aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
 * BPS has a ~9/7 generator, two of which give the ~5/3.
 * Sirius has a ~25/21 generator, three of which give the ~5/3.
@@ Line 269: / Line 269: @@
 By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with [[mos scale]]s of 18, 25, 32, or 39 steps.
-This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.
+This is essentially [[#Electra|electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament.
 Subgroup: 3.5.7.11
@@ Line 760: / Line 760: @@
 Badness (Sintel): 0.838
 == Alnilam ==