Schismatic family - Revision history

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

Eufalesio: 147 is lower error in 7-limit garibaldi

2026-08-19T16:59:49Z

147 is lower error in 7-limit garibaldi

← Older revision		Revision as of 16:59, 19 August 2026
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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 }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 94, 147}}

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

Eliora: /* Tertiaschis */

2026-08-16T17:19:33Z

Tertiaschis

← Older revision		Revision as of 17:19, 16 August 2026
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	== Tertiaschis ==		== Tertiaschis ==
	Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap\| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel\|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[~~daemotertiachis~~]] which is made using every other generator of tertiaschis.		Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap\| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel\|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis.

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

Eliora: /* Tertiaschis */

2026-08-16T17:19:19Z

Tertiaschis

← Older revision		Revision as of 17:19, 16 August 2026
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	== Tertiaschis ==		== Tertiaschis ==
	Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap\| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel\|squirrel]], but tempers out 1071875/1062882 for prime 7.		Named by [[Xenllium]] in 2021, tertiaschis may be described as {{nowrap\| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel\|squirrel]], but tempers out 1071875/1062882 for prime 7. See also [[daemotertiachis]] which is made using every other generator of tertiaschis.

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

FloraC: - interwiki (moved to schismic)

2026-08-14T12:21:53Z

- interwiki (moved to schismic)

← Older revision		Revision as of 12:21, 14 August 2026
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	~~{{interwiki~~
	~~\| en = Schismatic family~~
	~~\| de = Schismatische Temperaturen~~
	~~\| es =~~
	~~\| ja =~~
	}}
	{{Technical data page}}		{{Technical data page}}
	The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4\|just major third]] and a [[8192/6561\|Pythagorean diminished fourth]].		The [[5-limit]] parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4\|just major third]] and a [[8192/6561\|Pythagorean diminished fourth]].

FloraC: Recategorize

2026-08-14T12:20:47Z

Recategorize

← Older revision		Revision as of 12:20, 14 August 2026
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	Badness (Sintel): 1.17		Badness (Sintel): 1.17

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

FloraC: /* Term */ adopt more specific link; improve phrasing

2026-07-12T13:35:07Z

Term: adopt more specific link; improve phrasing

← Older revision		Revision as of 13:35, 12 July 2026
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	== Term ==		== Term ==
	Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap\| 12 & 171 }}, and is the unique temperament that ~~equates a~~ syntonic~Pythagorean ~~comma~~ with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In ~~some~~ 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.		Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap\| 12 & 171 }}, and is the unique temperament that tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region\|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.

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

Hkm: /* Tridecafifths */

2026-06-30T17:03:12Z

Tridecafifths

← Older revision		Revision as of 17:03, 30 June 2026
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	Badness (Sintel): 1.46		Badness (Sintel): 1.46

	~~== Tridecafifths ==~~
	Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap\| 89 & 200 }} temperament. It divides the [[3/2\|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning.

	~~[[Subgroup]]: 2.3.5.7~~

	~~[[Comma list]]: 32805/32768, {{monzo\| -14 -1 -9 13 }}~~

	~~{{Mapping\|legend=1\| 1 1 7 6 \| 0 13 -104 -71 }}~~
	~~: mapping generators: ~2, ~1323/1280~~

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}}
	~~: [[error map]]: {{val\| +0.143 -0.023 +0.375 -0.816 }}~~
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}}
	~~: error map: {{val\| 0.000 -0.261 -0.221 -0.421 }}~~

	~~{{Optimal ET sequence\|legend=1\| 89, 200, 289 }}~~

	~~[[Badness]] (Sintel): 10.9~~

	~~=== 11-limit ===~~
	~~Subgroup: 2.3.5.7.11~~

	~~Comma list: 441/440, 32805/32768, 55296000/55240493~~

	~~Mapping: {{mapping\| 1 1 7 6 4 \| 0 13 -104 -71 -12 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}}

	~~{{Optimal ET sequence\|legend=0\| 89, 200, 289 }}~~

	~~Badness (Sintel): 4.23~~

	== Subgroup extensions ==		== Subgroup extensions ==

Hkm: /* Subgroup extensions */

2026-06-16T02:24:49Z

Subgroup extensions

← Older revision		Revision as of 02:24, 16 June 2026
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	== Subgroup extensions ==		== Subgroup extensions ==
	~~=== Maqamschismic (2.3.5.11) ===~~
	Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]].

	~~Subgroup: 2.3.5.11~~

	~~Comma list: 2200/2187, 4125/4096~~

	~~Subgroup-val mapping: {{mapping\| 1 0 15 -33 \| 0 1 -8 23 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}}
	* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}}

	~~{{Optimal ET sequence\|legend=0\| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }}~~

	~~Badness (Sintel): 1.34~~

	~~==== 2.3.5.11.13 subgroup ====~~
	~~Subgroup: 2.3.5.11.13~~

	~~Comma list: 325/324, 352/351, 4125/4096~~

	~~Subgroup-val mapping: {{mapping\| 1 0 15 -33 -28 \| 0 1 -8 23 20 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}}
	* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}}

	~~{{Optimal ET sequence\|legend=0\| 12e, …, 41, 53, 94, 147e }}~~

	~~Badness (Sintel): 0.862~~

	=== Tridecaschismic (2.3.5.13) ===		=== Tridecaschismic (2.3.5.13) ===

Godtone: detail

2026-06-06T14:31:57Z

detail

← Older revision		Revision as of 14:31, 6 June 2026
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	: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia\|#Pontia]]''		: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia\|#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, 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.

	[[Subgroup]]: 2.3.5.19		[[Subgroup]]: 2.3.5.19