Schismatic family - Revision history

Eufalesio: Garibaldi more explanations

2026-10-08T17:21:51Z

Garibaldi more explanations

← Older revision		Revision as of 17:21, 8 October 2026
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	{{Main\| Garibaldi }}		{{Main\| Garibaldi }}

	Garibaldi tempers out the [[~~garischisma~~]], equating the [[64/63\|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103\|S8/S9]], [[225/224\|S15]]}.		Garibaldi tempers out the [[septischisma]], equating the [[64/63\|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103\|S8/S9]], [[225/224\|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper.

	[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).		[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).
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	=== Cassandra ===		=== Cassandra ===
	Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity.		Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity. It finds 11/8 as two commas above 4/3 and 13/8 a minor third above 11/8, or equivalently two commas above 128/81.

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	=== Andromeda ===		=== Andromeda ===
			Andromeda is the second-best extension to garibaldi, which is less complex but also more innacurate. 11/8 is two commas down from [[729/512]], as is 13/8 two commas down from [[27/16]].

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

FloraC: Misc. improvements

2026-10-08T15:51:51Z

Misc. improvements

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	== Countertertiaschis ==		== Countertertiaschis ==
	Named by [[Flora Canou]] in 2021, ~~Countertertiaschis~~ may be described as {{nowrap\| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel\|squirrel]], but tempers out 244140625/243045684 for prime 7.		Named by [[Flora Canou]] in 2021, countertertiaschis may be described as {{nowrap\| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel\|squirrel]], but tempers out 244140625/243045684 for prime 7.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Quadrant ==		== Quadrant ==
	Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap\| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term\|term]] equates the syntonic~Pythagorean comma ~~with~~ three [[marvel comma]]s, quadrant ~~equates~~ the ~~syntonic~Pythagorean~~ comma ~~with~~ four. A [[septimal comma]] is then found as a stack of five marvel commas.		Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described as the {{nowrap\| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term\|term]] equates the syntonic and Pythagorean commas for a generic comma step, and splits it into three [[marvel comma]]s, quadrant splits the same generic comma step into four. A [[septimal comma]] is then found as a stack of five marvel commas.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Sesquiquartififths ==		== Sesquiquartififths ==
	Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap\| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.		Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap\| 41 & 171 }} temperament. It splits the [[3/2\|perfect fifth]] into four; its [[ploidacot]] is thus tetracot.

			The name might have been a portmanteau of ''sesquiquart'' and ''quartififths'', each meaning dividing 3/2 into four.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group \| ''A temperament for Maqam'']</ref>		It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group \| ''A temperament for Maqam'']</ref>

	[[~~Bleu~~]] ~~has, like~~ tsaharuk~~, a generator of 1/5 fifth and makes for an instructive comparison. Another temperament with a 1/5 of a fifth generator is~~ [[quanic]], which ~~has~~ a sharper fifth and works better ~~for~~ higher~~-limit temperaments~~, but which also has mos of size 43, 60, 77 and 94 notes. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.		It restricts to the [[2.3.7 subgroup]] as [[no-fives subgroup temperaments #Navy\|navy]]. This relates tsaharuk to [[aberschismic temperaments #Quanic\|quanic]], which favors a sharper fifth and works better in the higher limits, but which also has mos of size 43, 60, 77 and 94 notes. [[Bleu]] has, like tsaharuk, a generator of 1/5 fifth and makes for an instructive comparison. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Quanharuk ==		== Quanharuk ==
	Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap\| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1\|3rd~~]] [[~~harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.		Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap\| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1\|3rd harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Subgroup extensions ==		== Subgroup extensions ==

	=== Tridecaschismic (2.3.5.13) ===		=== Tridecaschismic (2.3.5.13) ===
	Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324\|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].		Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324\|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the [[2.3.5.13 subgroup\|2.3.5.13-subgroup]] [[restriction]] of 13-limit [[cassandra]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13
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	=== Nestoria (2.3.5.19) ===		=== Nestoria (2.3.5.19) ===
	~~: ''~~See also~~: [[~~No-elevens subgroup temperaments #Garibaldia~~]] and [[~~No-elevens subgroup temperaments #Pontia~~\|#Pontia]]''~~		{{See also\| No-elevens subgroup temperaments #Garibaldia \| No-elevens subgroup temperaments #Pontia }}

	Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1\|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.		Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1\|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val.

FloraC: Make references section

2026-10-07T18:15:35Z

Make references section

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	Badness (Sintel): 1.17		Badness (Sintel): 1.17

			== References ==
			<references/>

	[[Category:Schismatic family\| ]] <!-- main article -->		[[Category:Schismatic family\| ]] <!-- main article -->
	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Catalogs of rank-2 temperaments]]		[[Category:Catalogs of rank-2 temperaments]]

FloraC: /* Tsaharuk */ merge from main article (not much content there)

2026-10-07T18:13:59Z

Tsaharuk: merge from main article (not much content there)

← Older revision		Revision as of 18:13, 7 October 2026
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	== Tsaharuk ==		== Tsaharuk ==
	{{~~Main~~\| ~~Tsaharuk~~ }}		[[File:Tsaharuk.jpg\|alt=Tsaharuk.jpg\|thumb\|Parametric horagram of the tsaharuk temperament.]]
			{{See also\| Tsaharuk24 }}

	Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap\| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2\|perfect fifth]], so its [[ploidacot]] is pentacot.		Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap\| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2\|perfect fifth]], so its [[ploidacot]] is pentacot.

			It was proposed by [[Jacques Dudon]] based on [[Julien Jalaleddine Weiss]] qanun tunings, which has [[mos]] of size 43, 60, 77 and 94 notes.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100718.html Yahoo! Tuning Group \| ''A temperament for Maqam'']</ref>

			[[Bleu]] has, like tsaharuk, a generator of 1/5 fifth and makes for an instructive comparison. Another temperament with a 1/5 of a fifth generator is [[quanic]], which has a sharper fifth and works better for higher-limit temperaments, but which also has mos of size 43, 60, 77 and 94 notes. [[Edo]]s that support tsaharuk include [[77edo]], [[94edo]], [[171edo]] and [[248edo]], and edos supporting quanic are [[94edo]], [[111edo]] and [[205edo]]. Quanic and tsaharuk therefore become identical if 94edo is the tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	Comma list: 325/324, 32805/32768		Comma list: 325/324, 32805/32768

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 15 -28 \| 0 1 -8 20 }}		{{Mapping\|legend=2\| 1 0 15 -28 \| 0 1 -8 20 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 325/324, 361/360, 513/512		Comma list: 325/324, 361/360, 513/512

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 15 -28 9 \| 0 1 -8 20 -3 }}		{{Mapping\|legend=2\| 1 0 15 -28 9 \| 0 1 -8 20 -3 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 171/170, 256/255, 324/323		Comma list: 171/170, 256/255, 324/323

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 15 -7 9 \| 0 1 -8 7 -3 }}		{{Mapping\|legend=2\| 1 0 15 -7 9 \| 0 1 -8 7 -3 }}

	{{Mapping\|legend=4\| 1 0 15 0 0 0 -7 9 \| 0 1 -8 0 0 0 7 -3 }}		{{Mapping\|legend=4\| 1 0 15 0 0 0 -7 9 \| 0 1 -8 0 0 0 7 -3 }}
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	Comma list: 361/360, 513/512, 676/675		Comma list: 361/360, 513/512, 676/675

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 15 14 9 \| 0 2 -16 -13 -6 }}		{{Mapping\|legend=2\| 1 0 15 14 9 \| 0 2 -16 -13 -6 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 361/360, 481/480, 513/512, 676/675		Comma list: 361/360, 481/480, 513/512, 676/675

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 15 14 9 6 \| 0 2 -16 -13 -6 -1 }}		{{Mapping\|legend=2\| 1 0 15 14 9 6 \| 0 2 -16 -13 -6 -1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 4624/4617, 6144/6137, 6885/6859		Comma list: 4624/4617, 6144/6137, 6885/6859

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 2 -1 5 4 \| 0 -5 40 -11 3 }}		{{Mapping\|legend=2\| 1 2 -1 5 4 \| 0 -5 40 -11 3 }}

	{{Mapping\|legend=4\| 1 2 -1 0 0 0 5 4 \| 0 -5 40 0 0 0 -11 3 }}		{{Mapping\|legend=4\| 1 2 -1 0 0 0 5 4 \| 0 -5 40 0 0 0 -11 3 }}

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	Subgroup-val mapping: {{mapping\| 1 0 15 -7 9 \| 0 1 -8 7 -3 }}		Subgroup-val mapping: {{mapping\| 1 0 15 -7 9 \| 0 1 -8 7 -3 }}

	~~Gencom mapping:~~ {{~~mapping~~\| 1 0 15 0 0 0 -7 9 \| 0 1 -8 0 0 0 7 -3 }}		{{Mapping\|legend=4\| 1 0 15 0 0 0 -7 9 \| 0 1 -8 0 0 0 7 -3 }}

	Optimal tunings:		Optimal tunings:
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	Subgroup-val mapping: {{mapping\| 1 2 -1 5 4 \| 0 -5 40 -11 3 }}		Subgroup-val mapping: {{mapping\| 1 2 -1 5 4 \| 0 -5 40 -11 3 }}

	~~Gencom mapping:~~ {{~~mapping~~\| 1 2 -1 0 0 0 5 4 \| 0 -5 40 0 0 0 -11 3 }}		{{Mapping\|legend=4\| 1 2 -1 0 0 0 5 4 \| 0 -5 40 0 0 0 -11 3 }}

	Optimal tunings:		Optimal tunings:

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	[[Comma list]]: 256/255, 1458/1445		[[Comma list]]: 256/255, 1458/1445

	{{Mapping\|legend=2\| 1 0 15 -7 \| 0 1 -8 7 }}		{{Mapping\|legend=3\| 1 0 15 -7 \| 0 1 -8 7 }}

	{{Mapping\|legend=5\| 1 0 15 0 0 0 -7 \| 0 1 -8 0 0 0 7 }}		{{Mapping\|legend=5\| 1 0 15 0 0 0 -7 \| 0 1 -8 0 0 0 7 }}
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	[[Comma list]]: 361/360, 513/512		[[Comma list]]: 361/360, 513/512

	{{Mapping\|legend=2\| 1 0 15 9 \| 0 1 -8 -3 }}		{{Mapping\|legend=3\| 1 0 15 9 \| 0 1 -8 -3 }}

	{{Mapping\|legend=5\| 1 0 15 0 0 0 0 9 \| 0 1 -8 0 0 0 0 -3 }}		{{Mapping\|legend=5\| 1 0 15 0 0 0 0 9 \| 0 1 -8 0 0 0 0 -3 }}
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	[[Comma list]]: 676/675, 32805/32768		[[Comma list]]: 676/675, 32805/32768

	{{Mapping\|legend=2\| 1 0 15 14 \| 0 2 -16 -13 }}		{{Mapping\|legend=3\| 1 0 15 14 \| 0 2 -16 -13 }}

	{{Mapping\|legend=5\| 1 0 15 0 0 14 \| 0 2 -16 0 0 -13 }}		{{Mapping\|legend=5\| 1 0 15 0 0 14 \| 0 2 -16 0 0 -13 }}
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	[[Comma list]]: 32805/32768, 1419857/1417176		[[Comma list]]: 32805/32768, 1419857/1417176

	{{Mapping\|legend=2\| 1 2 -1 5 \| 0 -5 40 -11 }}		{{Mapping\|legend=3\| 1 2 -1 5 \| 0 -5 40 -11 }}

	{{Mapping\|legend=5\| 1 2 -1 0 0 0 5 \| 0 -5 40 0 0 0 -11 }}		{{Mapping\|legend=5\| 1 2 -1 0 0 0 5 \| 0 -5 40 0 0 0 -11 }}

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	{{Mapping\|legend=2\| 1 0 15 -7 \| 0 1 -8 7 }}		{{Mapping\|legend=2\| 1 0 15 -7 \| 0 1 -8 7 }}

	{{Mapping\|legend=3\| 1 0 15 0 0 0 -7 \| 0 1 -8 0 0 0 7 }}		{{Mapping\|legend=5\| 1 0 15 0 0 0 -7 \| 0 1 -8 0 0 0 7 }}
	: mapping generators: ~2, ~3		: mapping generators: ~2, ~3

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	{{Mapping\|legend=2\| 1 0 15 9 \| 0 1 -8 -3 }}		{{Mapping\|legend=2\| 1 0 15 9 \| 0 1 -8 -3 }}

	{{Mapping\|legend=3\| 1 0 15 0 0 0 0 9 \| 0 1 -8 0 0 0 0 -3 }}		{{Mapping\|legend=5\| 1 0 15 0 0 0 0 9 \| 0 1 -8 0 0 0 0 -3 }}
	: mapping generators: ~2, ~3		: mapping generators: ~2, ~3

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	{{Mapping\|legend=3\| 1 0 15 0 0 14 \| 0 2 -16 0 0 -13 }}		{{Mapping\|legend=5\| 1 0 15 0 0 14 \| 0 2 -16 0 0 -13 }}
	: mapping generators: ~2, ~26/15		: mapping generators: ~2, ~26/15

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	{{Mapping\|legend=2\| 1 2 -1 5 \| 0 -5 40 -11 }}		{{Mapping\|legend=2\| 1 2 -1 5 \| 0 -5 40 -11 }}

	{{Mapping\|legend=3\| 1 2 -1 0 0 0 5 \| 0 -5 40 0 0 0 -11 }}		{{Mapping\|legend=5\| 1 2 -1 0 0 0 5 \| 0 -5 40 0 0 0 -11 }}
	: mapping generators: ~2, ~18/17		: mapping generators: ~2, ~18/17

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Show changes

Godtone: unfortunately thats not how optimal ET sequences work. it's a defined thing by fumica's temperament evaluator. the standard ive been abiding by is that missing entries are to be discussed on a case-by-case basis

2026-08-19T22:53:16Z

unfortunately thats not how optimal ET sequences work. it's a defined thing by fumica's temperament evaluator. the standard ive been abiding by is that missing entries are to be discussed on a case-by-case basis

← Older revision		Revision as of 22:53, 19 August 2026
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	Garibaldi tempers out the [[garischisma]], equating the [[64/63\|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103\|S8/S9]], [[225/224\|S15]]}.		Garibaldi tempers out the [[garischisma]], equating the [[64/63\|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103\|S8/S9]], [[225/224\|S15]]}.

			[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]).

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]		* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915]

	{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94~~, 147~~}}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94 }}

	[[Badness]] (Sintel): 0.548		[[Badness]] (Sintel): 0.548