Schismatic family - Revision history

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

2026-07-12T13:35:07Z

Term: adopt more specific link; improve phrasing

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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

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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

Overthink: /* Nestoria (2.3.5.19) */ 53edo isn't the flattest in the sequence

2026-06-05T17:46:46Z

Nestoria (2.3.5.19): 53edo isn't the flattest in the sequence

← Older revision		Revision as of 17:46, 5 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 [[~~53edo~~]] is already arguably undertempered for it despite being ~~the flattest tuning appearing in~~ the optimal ~~ET sequence~~.		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.

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

Godtone: /* Nestoria (2.3.5.19) */ missing info

2026-06-05T14:58:54Z

Nestoria (2.3.5.19): missing info

← Older revision		Revision as of 14:58, 5 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]].		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 [[53edo]] is already arguably undertempered for it despite being the flattest tuning appearing in the optimal ET sequence.

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

Godtone: /* Tridecaschismic (2.3.5.13) */ "thus also" feels weird to use if theres no explanation for how it follows

2026-05-06T18:05:38Z

Tridecaschismic (2.3.5.13): "thus also" feels weird to use if theres no explanation for how it follows

← Older revision		Revision as of 18:05, 6 May 2026
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	=== 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, ~~thus also tempering out~~ 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 [[restriction]] of 13-limit [[cassandra]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Godtone: /* Tridecaschismic (2.3.5.13) */ grammar

2026-05-06T17:30:19Z

Tridecaschismic (2.3.5.13): grammar

← Older revision		Revision as of 17:30, 6 May 2026
Line 2,814:		Line 2,814:

	=== 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, thus also tempering out the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is ~~has among~~ the lowest badness of 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, thus also tempering out 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]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Hotcrystal0: mention tridecapyth comma for tridecaschismic

2026-04-21T20:25:31Z

mention tridecapyth comma for tridecaschismic

← Older revision		Revision as of 20:25, 21 April 2026
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	=== 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, ~~benefitting~~ from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is has among the lowest badness of 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, thus also tempering out the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is has among the lowest badness of schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Overthink: /* Overview to extensions */ here as well

2026-04-07T21:03:22Z

Overview to extensions: here as well

← Older revision		Revision as of 21:03, 7 April 2026
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	[[#Bischismic\|Bischismic]] adds {{monzo\| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa\|Salsa]] adds [[parahemif comma\|{{monzo\| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis\|Hemischis]] adds {{monzo\| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron\|Guiron]] adds [[1029/1024\|{{monzo\| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term\|Term]] adds {{monzo\| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel\|Squirrel]], [[#Tertiaschis\|tertiaschis]], and [[#Countertertiaschis\|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant\|Quadrant]] adds {{monzo\| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic\|Kleischismic]] adds {{monzo\| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths\|Sesquiquartififths]] adds {{monzo\| -35 15 0 4 }} and slices the fifth in four.		[[#Bischismic\|Bischismic]] adds {{monzo\| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa\|Salsa]] adds [[parahemif comma\|{{monzo\| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis\|Hemischis]] adds {{monzo\| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron\|Guiron]] adds [[1029/1024\|{{monzo\| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term\|Term]] adds {{monzo\| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel\|Squirrel]], [[#Tertiaschis\|tertiaschis]], and [[#Countertertiaschis\|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant\|Quadrant]] adds {{monzo\| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic\|Kleischismic]] adds {{monzo\| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths\|Sesquiquartififths]] adds {{monzo\| -35 15 0 4 }} and slices the fifth in four.

	Temperaments involving larger splits include [[#Tsaharuk\|tsaharuk]], [[#Quanharuk\|quanharuk]], [[#Quintilipyth\|quintilipyth]], [[#Quintaschis\|quintaschis]], [[#Altinex\|altinex]], [[Stearnsmic clan #Pogo\|pogo]], [[#Sextilifourths\|sextilifourths]], [[#~~Octant~~\|~~octant~~]], [[#~~Nonant~~\|~~nonant~~]], [[#~~Septant~~\|~~septant~~]], [[#Septiquarschis\|septiquarschis]], and [[#Tridecafifths\|tridecafifths]]. Those split the schismic structure into five to thirteen parts.		Temperaments involving larger splits include [[#Tsaharuk\|tsaharuk]], [[#Quanharuk\|quanharuk]], [[#Quintilipyth\|quintilipyth]], [[#Quintaschis\|quintaschis]], [[#Altinex\|altinex]], [[Stearnsmic clan #Pogo\|pogo]], [[#Sextilifourths\|sextilifourths]], [[#Septant\|septant]], [[#Octant\|octant]], [[#Nonant\|nonant]], [[#Septiquarschis\|septiquarschis]], and [[#Tridecafifths\|tridecafifths]]. Those split the schismic structure into five to thirteen parts.

	Temperaments discussed elsewhere include:		Temperaments discussed elsewhere include: