No-fives subgroup temperaments - Revision history

Lucius Chiaraviglio: /* Jackpot */ Fix typo; fix link

2026-08-18T14:10:08Z

Jackpot: Fix typo; fix link

← Older revision		Revision as of 14:10, 18 August 2026
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	=== Jackpot ===		=== Jackpot ===
	Named by {{u\|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1\|3rd harmonic]]. The name refers to the [[EDO ~~oin~~]] of 7 & 77.		Named by {{u\|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1\|3rd harmonic]]. The name refers to the [[Temperament merging\|EDO join]] of 7 & 77.

	[[Subgroup]]: 2.3.29		[[Subgroup]]: 2.3.29

CompactStar: /* Jackpot */

2026-08-18T07:10:00Z

Jackpot

← Older revision		Revision as of 07:10, 18 August 2026
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	=== Jackpot ===		=== Jackpot ===
	Named by {{u\|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1\|3rd harmonic]].		Named by {{u\|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1\|3rd harmonic]]. The name refers to the [[EDO oin]] of 7 & 77.

	[[Subgroup]]: 2.3.29		[[Subgroup]]: 2.3.29

FloraC: /* Temperaments with a higher-limit gene */ restore jackpot

2026-08-18T03:02:10Z

Temperaments with a higher-limit gene: restore jackpot

← Older revision		Revision as of 03:02, 18 August 2026
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	[[Badness]] (Sintel): 0.317		[[Badness]] (Sintel): 0.317

			=== Jackpot ===
			Named by {{u\|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1\|3rd harmonic]].

			[[Subgroup]]: 2.3.29

			[[Comma list]]: 17249876309/17179869184

			{{Mapping\|legend=2\| 7 0 34 \| 0 1 0 }}
			: mapping generators: ~32/29, ~3

			[[Optimal tuning]]s:
			* [[WE]]: ~32/29 = 171.4434{{c}}, ~3/2 = 701.8515{{c}}
			: [[error map]]: {{val\| +0.103 -0.000 -0.503 }}
			* [[CWE]]: ~32/29 = 171.4286{{c}}, ~3/2 = 701.8456{{c}}
			: error map: {{val\| 0.000 -0.109 -1.006 }}

			{{Optimal ET sequence\|legend=1\| 7, 42, 49, 56, 63, 70, 77, 147, 224 }}

			[[Badness]] (Sintel): 1.02

	=== Sematology ===		=== Sematology ===

FloraC: /* Temperaments with a 2.3.11 gene */ correct the ratio

2026-08-17T04:43:05Z

Temperaments with a 2.3.11 gene: correct the ratio

← Older revision		Revision as of 04:43, 17 August 2026
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	[[Subgroup]]: 2.3.11		[[Subgroup]]: 2.3.11

	[[Comma list]]: ~~131769~~/~~131072~~		[[Comma list]]: 264627/262144

	{{Mapping\|legend=2\| 1 0 9 \| 0 2 -7 }}		{{Mapping\|legend=2\| 1 0 9 \| 0 2 -7 }}

FloraC: /* Temperaments with a 2.3.11 gene */ rework the intro to alphaxenean; + betaxenean

2026-08-17T04:42:31Z

Temperaments with a 2.3.11 gene: rework the intro to alphaxenean; + betaxenean

← Older revision		Revision as of 04:42, 17 August 2026
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	Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.		Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.

	The name ''io'' was coined by ~~[[User:CompactStar~~\|CompactStar]] in 2024 based on the [[Kite's color notation\|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.		The name ''io'' was coined by {{u\|CompactStar}} in 2024 based on the [[Kite's color notation\|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.

	[[Subgroup]]: 2.3.11		[[Subgroup]]: 2.3.11
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	=== 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 [[~~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 has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period.

	[[Subgroup]]: 2.3.11		[[Subgroup]]: 2.3.11
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	=== Octatonic ===		=== Octatonic ===
	[[12/11]] is very close to 1 step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.		[[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.

	[[Subgroup]]: 2.3.11		[[Subgroup]]: 2.3.11
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	[[Badness]] (Sintel): 0.515		[[Badness]] (Sintel): 0.515

			=== Betaxenean ===
			Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1\|perfect twelfth]] into two halves of [[512/297]].

			[[Subgroup]]: 2.3.11

			[[Comma list]]: 131769/131072

			{{Mapping\|legend=2\| 1 0 9 \| 0 2 -7 }}
			: mapping generators: ~2, ~512/297

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}}
			: [[error map]]: {{val\| +0.593 -0.581 -0.791 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}}
			: error map: {{val\| 0.000 -1.561 -2.699 }}

			{{Optimal ET sequence\|legend=1\| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }}

			[[Badness]] (Sintel): 0.709

	=== Infraug ===		=== Infraug ===

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
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	=== 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
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	[[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
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	: [[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
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	=== 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