Garischismic family - Revision history

FloraC: Extend androschismic to 2.3.5.7.11.13.19

2026-06-14T10:45:18Z

Extend androschismic to 2.3.5.7.11.13.19

← Older revision		Revision as of 10:45, 14 June 2026
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	The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.		The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.

	Garischismic can be easily notated with [[chain-of-fifths notation]] with two additional ~~set~~ of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].		Garischismic can be easily notated with [[chain-of-fifths notation]] with two additional sets of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	{{Main\| Cassaschismic }}		{{Main\| Cassaschismic }}

	Cassaschismic is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.		Cassaschismic maps [[prime interval\|prime]] [[11/1\|11]] to +23 perfect fifths, so it is an expansion of [[gary]]. It is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: 2080/2079, 43904/43875~~, 154880/154791~~		Comma list: 2080/2079, 10648/10647, 43904/43875

	Mapping: {{mapping\| 1 0 0 25 62 82 \| 0 1 0 -14 -34 -43 \| 0 0 1 0 -2 -3 }}		Mapping: {{mapping\| 1 0 0 25 62 82 \| 0 1 0 -14 -34 -43 \| 0 0 1 0 -2 -3 }}
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	Badness (Sintel): 0.942		Badness (Sintel): 0.942

			=== 2.3.5.7.11.13.19 subgroup ===
			Subgroup: 2.3.5.7.11.13.19

			Comma list: 1216/1215, 2080/2079, 3136/3135, 10648/10647

			Mapping: {{mapping\| 1 0 0 25 62 82 -6 \| 0 1 0 -14 -34 -43 5 \| 0 0 1 0 -2 -3 1 }}

			Optimal tunings:
			* WE: ~2 = 1199.9282{{c}}, ~3/2 = 702.1718{{c}}, ~5/4 = 386.5228{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2181{{c}}, ~5/4 = 386.5012{{c}}

			{{Optimal ET sequence\|legend=0\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 851, 1133, 1403, 1984, 3117bcef, 3387bcef }}

			Badness (Sintel): 0.580

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Garischismic family\| ]] <!-- main article -->		[[Category:Garischismic family\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: Expand

2026-05-18T06:49:49Z

Expand

← Older revision		Revision as of 06:49, 18 May 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''garischismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo\|legend=1\| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).		The '''garischismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo\|legend=1\| 25 -14 0 -1 }}).

	The ~~best extension to the 11~~-~~limit identifies the~~ [[11/8]] ~~at +23 fifths~~. ~~This is also~~ the ~~mapping used in~~ [[~~cassandra~~]], ~~so we call it~~ [[~~#Cassaschismic\|cassaschismic~~]]~~. An alternative~~, ~~supported by~~ [[~~andromeda~~]]~~, is~~ [[~~#Androschismic\|androschismic~~]].		== Garischismic ==
			The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.

			Garischismic can be easily notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].

	~~== Garischismic ==~~
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	[[Badness]] (Sintel): 5.79		[[Badness]] (Sintel): 5.79

			=== Overview to extensions ===
			The best extension to the 11-limit identifies the [[11/8]] at +23 fifths. This is also the mapping used in [[cassandra]], so we call it [[#Cassaschismic\|cassaschismic]]. An alternative, supported by [[andromeda]], is [[#Androschismic\|androschismic]].

	== Cassaschismic ==		== Cassaschismic ==
			{{Main\| Cassaschismic }}

			Cassaschismic is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

FloraC: - redundant category

2026-02-06T11:57:33Z

- redundant category

← Older revision		Revision as of 11:57, 6 February 2026
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Garischismic family\| ]] <!-- main article -->		[[Category:Garischismic family\| ]] <!-- main article -->
	~~[[Category:Garischismic\| ]] <!-- key article -->~~
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: - CTE tunings

2026-02-06T11:57:07Z

- CTE tunings

← Older revision		Revision as of 11:57, 6 February 2026
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}
	: error map: {{val\| 0.000 +0.257 +0.136 +0.201 }}		: error map: {{val\| 0.000 +0.257 +0.136 +0.201 }}
	~~<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}
	: error map: {{val\| 0.000 +0.274 +0.068 -0.032 -0.051 }}		: error map: {{val\| 0.000 +0.274 +0.068 -0.032 -0.051 }}
	~~<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}		{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}
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	* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}		* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
	~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}		{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
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	* WE: ~2 = 1199.9817{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}		* WE: ~2 = 1199.9817{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
	~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851 }}		{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851 }}
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	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}
	: error map: {{val\| 0.000 +0.263 +0.191 +0.125 +0.266 }}		: error map: {{val\| 0.000 +0.263 +0.191 +0.125 +0.266 }}
	~~<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}
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	* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}		* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}
	~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}		{{Optimal ET sequence\|legend=0\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-02-06T11:56:46Z

Switch to Sintel's badness, WE & CWE tunings

← Older revision		Revision as of 11:56, 6 February 2026
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2 = ~~1200~~.~~0000~~{{c}}, ~3/2 = 702.~~2224~~{{c}}, ~5/4 = 386.~~3137~~{{c}}		* [[WE]]: ~2 = 1199.9155{{c}}, ~3/2 = 702.1584{{c}}, ~5/4 = 386.4827{{c}}
			: [[error map]]: {{val\| -0.085 +0.119 -0.000 +0.027 }}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}
			: error map: {{val\| 0.000 +0.257 +0.136 +0.201 }}
			<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}} -->

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}

	[[Badness]] (~~Smith~~): 1.~~31 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 5.79

	== Cassaschismic ==		== Cassaschismic ==
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2 = ~~1200~~.~~0000~~{{c}}, ~3/2 = 702.~~2280~~{{c}}, ~5/4 = 386.~~3137~~{{c}}		* [[WE]]: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}, ~5/4 = 386.3874{{c}}
			: [[error map]]: {{val\| -0.037 +0.216 -0.000 -0.139 -0.173 }}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}
			: error map: {{val\| 0.000 +0.274 +0.068 -0.032 -0.051 }}
			<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}} -->

	{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}		{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}

	[[Badness]] (~~Smith~~): 1.~~41 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 1.69

	=== 13-limit ===		=== 13-limit ===
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	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~2 = ~~1200~~.~~0000~~{{c}}, ~3/2 = 702.~~2289~~{{c}}, ~5/4 = 386.~~2869~~{{c}}		* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
			<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}} -->

	{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}		{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}

	Badness (~~Smith~~): 0.~~872 × 10<sup>-3</sup>~~		Badness (Sintel): 0.815

	=== 2.3.5.7.11.13.19 subgroup ===		=== 2.3.5.7.11.13.19 subgroup ===
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	Subgroup-val mapping: {{mapping\| 1 0 0 25 -33 -13 -6 \| 0 1 0 -14 23 12 5 \| 0 0 1 0 0 -1 1 }}		Subgroup-val mapping: {{mapping\| 1 0 0 25 -33 -13 -6 \| 0 1 0 -14 23 12 5 \| 0 0 1 0 0 -1 1 }}

	Optimal ~~tuning (~~CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}}		Optimal tunings:
			* WE: ~2 = 1199.9817{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
			<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}} -->

	{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851 }}		{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851 }}

	Badness (~~Smith~~): 0.~~496 × 10<sup>-3</sup>~~		Badness (Sintel): 0.486

	== Androschismic ==		== Androschismic ==
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2 = ~~1200~~.~~0000~~{{c}}, ~3/2 = 702.~~2308~~{{c}}, ~5/4 = 386.~~3806~~{{c}}		* [[WE]]: ~2 = 1199.9118{{c}}, ~3/2 = 702.1606{{c}}, ~5/4 = 386.5301{{c}}
			: [[error map]]: {{val\| -0.088 +0.117 +0.040 -0.045 +0.044 }}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}
			: error map: {{val\| 0.000 +0.263 +0.191 +0.125 +0.266 }}
			<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}} -->

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}

	[[Badness]] (~~Smith~~): 1.~~64 × 10<sup>-3</sup>~~		[[Badness]] (Sintel): 1.97

	=== 13-limit ===		=== 13-limit ===
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	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~2 = ~~1200~~.~~0000~~{{c}}, ~3/2 = 702.~~2305~~{{c}}, ~5/4 = 386.~~3748~~{{c}}		* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}
			<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}		{{Optimal ET sequence\|legend=0\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}

	Badness (~~Smith~~): 1.~~01 × 10<sup>-3</sup>~~		Badness (Sintel): 0.942

	[[Category:Temperament families]]		[[Category:Temperament families]]

FloraC: Units & misc. cleanup

2026-02-06T11:15:16Z

Units & misc. cleanup

← Older revision		Revision as of 11:15, 6 February 2026
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	{{Mapping\|legend=1\| 1 0 0 25 \| 0 1 0 -14 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 1 0 0 25 \| 0 1 0 -14 \| 0 0 1 0 }}

	: mapping generators: ~2, ~3, ~5		: mapping generators: ~2, ~3, ~5

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~3/2 = 702.2224, ~5/4 = 386.3137		* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}}
	* [[CWE]]: ~2 = ~~1\1~~, ~3/2 = 702.2124, ~5/4 = 386.4496		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}

	[[Badness]]: 1.31 × 10<sup>-3</sup>		[[Badness]] (Smith): 1.31 × 10<sup>-3</sup>

	== Cassaschismic ==		== Cassaschismic ==
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~3/2 = 702.2280, ~5/4 = 386.3137		* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}}
	* [[CWE]]: ~2 = ~~1\1~~, ~3/2 = 702.2290, ~5/4 = 386.3819		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}

	{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}		{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}

	[[Badness]]: 1.41 × 10<sup>-3</sup>		[[Badness]] (Smith): 1.41 × 10<sup>-3</sup>

	=== 13-limit ===		=== 13-limit ===
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	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~3/2 = 702.2289, ~5/4 = 386.2869		* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}}
	* CWE: ~2 = ~~1\1~~, ~3/2 = 702.2303, ~5/4 = 386.3031		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}

	{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}		{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}

	Badness: 0.872 × 10<sup>-3</sup>		Badness (Smith): 0.872 × 10<sup>-3</sup>

	=== 2.3.5.7.11.13.19 subgroup ===		=== 2.3.5.7.11.13.19 subgroup ===
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	Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079		Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079

	~~Sval~~ mapping: {{mapping\| 1 0 0 25 -33 -13 -6 \| 0 1 0 -14 23 12 5 \| 0 0 1 0 0 -1 1 }}		Subgroup-val mapping: {{mapping\| 1 0 0 25 -33 -13 -6 \| 0 1 0 -14 23 12 5 \| 0 0 1 0 0 -1 1 }}

	Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.2293, ~5/4 = 386.3021		Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}}

	{{Optimal ET sequence\|legend=1\| 41, 53, 94, 176, 217, 270, 581, 851 }}		{{Optimal ET sequence\|legend=0\| 41, 53, 94, 176, 217, 270, 581, 851 }}

	Badness: 0.496 × 10<sup>-3</sup>		Badness (Smith): 0.496 × 10<sup>-3</sup>

	== Androschismic ==		== Androschismic ==
	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

	[[Comma list]]: 151263/151250, ~~[[Reef comma\|~~200704/200475]]		[[Comma list]]: 151263/151250, 200704/200475

	{{Mapping\|legend=1\| 1 0 0 25 62 \| 0 1 0 -14 -34 \| 0 0 1 0 -2 }}		{{Mapping\|legend=1\| 1 0 0 25 62 \| 0 1 0 -14 -34 \| 0 0 1 0 -2 }}

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* CTE: ~2 = ~~1\1~~, ~3/2 = 702.2308, ~5/4 = 386.3806		* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}}
	* CWE: ~2 = ~~1\1~~, ~3/2 = 702.2178, ~5/4 = 386.5048		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}

	[[Badness]]: 1.64 × 10<sup>-3</sup>		[[Badness]] (Smith): 1.64 × 10<sup>-3</sup>

	=== 13-limit ===		=== 13-limit ===
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: 2080/2079, ~~[[Punctisma\|~~43904/43875]], 154880/154791		Comma list: 2080/2079, 43904/43875, 154880/154791

	Mapping: {{mapping\| 1 0 0 25 62 82 \| 0 1 0 -14 -34 -43 \| 0 0 1 0 -2 -3 }}		Mapping: {{mapping\| 1 0 0 25 62 82 \| 0 1 0 -14 -34 -43 \| 0 0 1 0 -2 -3 }}

	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~3/2 = 702.2305, ~5/4 = 386.3748		* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}}
	* CWE: ~2 = ~~1\1~~, ~3/2 = 702.2174, ~5/4 = 386.4968		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}

	~~Optimal ET sequence:~~ {{Optimal ET sequence\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}		{{Optimal ET sequence\|legend=0\| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}

	Badness: 1.01 × 10<sup>-3</sup>		Badness (Smith): 1.01 × 10<sup>-3</sup>

	[[Category:Temperament families]]		[[Category:Temperament families]]

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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''garischismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo\|legend=1\| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).		The '''garischismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo\|legend=1\| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).

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			{{Technical data page}}<br><br>
	The '''garischismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo\|legend=1\| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).		The '''garischismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo\|legend=1\| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2\|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7\|septimal major second (8/7)]]. Equivalently stated, the [[7/4\|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).