No-fives subgroup temperaments - Revision history

FloraC: Text replacement - "octave" to "octave"

2026-07-14T14:15:00Z

Text replacement - "octave" to "octave"

← Older revision		Revision as of 14:15, 14 July 2026
Line 881:		Line 881:

	=== Alphaxenean ===		=== Alphaxenean ===
	Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8\|Pythagorean whole tone]]. It also divides the [[~~2/1\|~~octave]] into two.		Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8\|Pythagorean whole tone]]. It also divides the [[octave]] into two.

	[[Subgroup]]: 2.3.11		[[Subgroup]]: 2.3.11

FloraC: /* Temperaments with a 2.3.11 gene */ move octatonic here

2026-07-05T06:57:38Z

Temperaments with a 2.3.11 gene: move octatonic here

@@ Line 899: / Line 899: @@
 [[Badness]] (Sintel): 0.395
 === Infraug ===
@@ Line 945: / Line 965: @@
 * CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}
-{{Optimal ET sequence|legend=1| 14, 19, 33 }}
+{{Optimal ET sequence|legend=0| 14, 19, 33 }}
 Badness (Sintel): 1.59

FloraC: /* Temperaments with a higher-limit gene */ move deviaug here

2026-07-02T11:39:10Z

Temperaments with a higher-limit gene: move deviaug here

← Older revision		Revision as of 11:39, 2 July 2026
Line 1,220:		Line 1,220:

	[[Badness]] (Sintel): 0.0294		[[Badness]] (Sintel): 0.0294

			=== Deviaug ===
			Deviaug tempers out [[6912/6859]] in the 2.3.19 subgroup, setting [[24/19]] to 1/3 of an octave.

			[[Subgroup]]: 2.3.19

			[[Comma list]]: 6912/6859

			{{Mapping\|legend=2\| 3 0 8 \| 0 1 1 }}
			: mapping generators: ~24/19, ~3

			[[Optimal tuning]]s:
			* [[WE]]: ~24/19 = 399.8571{{c}}, ~3/2 = 701.9795{{c}}
			: [[error map]]: {{val\| -0.429 -0.404 +2.894 }}
			* [[CWE]]: ~24/19 = 400.0000{{c}}, ~3/2 = 701.8810{{c}}
			: error map: {{val\| 0.000 -0.074 +4.368 }}

			{{Optimal ET sequence\|legend=1\| 9, 12, 75, 87, 99, 111, 123h, 135h }}

			[[Badness]] (Sintel): 0.237

	=== Lipsett ===		=== Lipsett ===

FloraC: /* Superflat */ fix a value

2026-06-30T13:14:43Z

Superflat: fix a value

← Older revision		Revision as of 13:14, 30 June 2026
Line 1,115:		Line 1,115:
	: [[error map]]: {{val\| +3.129 -3.177 -4.349 }}		: [[error map]]: {{val\| +3.129 -3.177 -4.349 }}
	* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
	: error map: {{val\| 0.000 -8.347 +14.960 }}		: error map: {{val\| 0.000 -8.347 -14.960 }}

	{{Optimal ET sequence\|legend=1\| 5f, 7, 12, 19, 45f, 64f, 147bfff }}		{{Optimal ET sequence\|legend=1\| 5f, 7, 12, 19, 45f, 64f, 147bfff }}

FloraC: /* Tridecapyth */ review the description

2026-06-30T13:08:49Z

Tridecapyth: review the description

← Older revision		Revision as of 13:08, 30 June 2026
Line 1,082:		Line 1,082:

	=== Tridecapyth ===		=== Tridecapyth ===
	Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It ~~can be seen as~~ a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping; for 118edo we get the 118f [[val]] that takes the second-~~best~~, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However~~, due to its small note count~~, [[41edo]] ~~technically~~ uses this mapping ~~too~~, so ~~that~~ [[94edo]], the val sum of 41 and 53~~, also uses this mapping, suggesting~~ it is of interest to flatter tunings of [[garibaldi]] with fifths tending close to pure~~; this corresponds to the extension of garibaldi called [[cassandra]]~~. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].		Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It shows a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping – for 118edo we get the 118f [[val]] that takes the second-closest, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, [[41edo]] uses this mapping, and so does [[94edo]], the val sum of 41 and 53. Thus it is of interest to flatter tunings of [[garibaldi]]/[[cassandra]] with fifths tending close to pure. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].

	[[Subgroup]]: 2.3.13		[[Subgroup]]: 2.3.13

FloraC: /* Temperaments with a 2.3.13 gene */ move full data for tridecapyth here

2026-06-30T12:53:07Z

Temperaments with a 2.3.13 gene: move full data for tridecapyth here

@@ Line 1,043: / Line 1,043: @@
 == Temperaments with a 2.3.13 gene ==
 === Threedic ===
 [[Subgroup]]: 2.3.13
@@ Line 1,089: / Line 1,080: @@
 [[Badness]] (Sintel): 0.200
 === Superflat ===

Hkm: /* Dog */

2026-06-16T04:04:53Z

Dog

← Older revision		Revision as of 04:04, 16 June 2026
Line 1,211:		Line 1,211:

	[[Badness]] (Sintel): 0.0294		[[Badness]] (Sintel): 0.0294

	~~=== Dog ===~~
	~~Dog is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19-subgroup analogue of [[mavila]].~~

	~~[[Subgroup]]: 2.3.19~~

	~~[[Comma list]]: 81/76~~

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

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~2 = 1203.3813{{c}}, ~3/2 = 680.5089{{c}}
	~~: [[error map]]: {{val\| +3.381 -18.065 +31.285 }}~~
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 680.5856{{c}}
	~~: error map: {{val\| 0.000 -21.369 +24.8295 }}~~

	~~{{Optimal ET sequence\|legend=1\| 2, 5h, 7, 16, 23 }}~~

	~~[[Badness]] (Sintel): 0.491~~

	=== Lipsett ===		=== Lipsett ===

Hkm: /* 2.3.11.13 */

2026-06-08T23:14:18Z

2.3.11.13

← Older revision		Revision as of 23:14, 8 June 2026
Line 918:		Line 918:
	[[Badness]] (Sintel): 0.734		[[Badness]] (Sintel): 0.734

	==== 2.3.11.13 ====		==== 2.3.11.13 subgroup ====
	Subgroup: 2.3.11.13		Subgroup: 2.3.11.13

Hkm: /* 2.3.13.23 */

2026-06-08T23:12:59Z

2.3.13.23

← Older revision		Revision as of 23:12, 8 June 2026
Line 1,132:		Line 1,132:
	Badness (Sintel): 0.135		Badness (Sintel): 0.135

	==== 2.3.13.23 ====		==== 2.3.13.23 subgroup ====
	Subgroup: 2.3.13.23		Subgroup: 2.3.13.23

Hkm: /* Tridecapyth */

2026-06-08T18:01:45Z

Tridecapyth

← Older revision		Revision as of 18:01, 8 June 2026
Line 1,043:		Line 1,043:

	== Temperaments with a 2.3.13 gene ==		== Temperaments with a 2.3.13 gene ==
	=== [[Tridecapyth ~~comma#Temperaments\|Tridecapyth]]~~ ===		=== Tridecapyth ===
			Sets a stack of 10 (9/8)s equal to 13/4.

	[[Subgroup]]: 2.3.13		[[Subgroup]]: 2.3.13

	[[Comma list]]: 3489660928/3486784401 ~~= (9/8)<sup>10</sup> / (13/4)~~		[[Comma list]]: 3489660928/3486784401

	See: [[Tridecapyth comma#Temperaments]].		See: [[Tridecapyth comma#Temperaments]].