Aberschismic family - Revision history

FloraC at 08:36, 1 October 2026

2026-10-01T08:36:20Z

← Older revision		Revision as of 08:36, 1 October 2026
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	[[Category:Aberschismic family\| ]] <!-- main article -->		[[Category:Aberschismic family\| ]] <!-- main article -->
	[[Category:Aberschismic]]		[[Category:Aberschismic\| ]] <!-- key article -->
	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Catalogs of rank-3 temperaments]]		[[Category:Catalogs of rank-3 temperaments]]

FloraC: Restore the category of the corresponding temp

2026-10-01T08:32:39Z

Restore the category of the corresponding temp

← Older revision		Revision as of 08:32, 1 October 2026
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	[[Category:Aberschismic family\| ]] <!-- main article -->		[[Category:Aberschismic family\| ]] <!-- main article -->
			[[Category:Aberschismic]]
	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Catalogs of rank-3 temperaments]]		[[Category:Catalogs of rank-3 temperaments]]

FloraC: Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

2026-09-24T12:33:37Z

Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

← Older revision		Revision as of 12:33, 24 September 2026
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	Comma list: 196/195, 352/351, 364/363		Comma list: 196/195, 352/351, 364/363

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 17 22 \| 0 1 0 -6 -10 -13 \| 0 0 1 1 1 1 }}		{{Mapping\|legend=0\| 1 0 0 10 17 22 \| 0 1 0 -6 -10 -13 \| 0 0 1 1 1 1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 196/195, 256/255, 352/351, 364/363		Comma list: 196/195, 256/255, 352/351, 364/363

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 17 22 8 \| 0 1 0 -6 -10 -13 -1 \| 0 0 1 1 1 1 -1 }}		{{Mapping\|legend=0\| 1 0 0 10 17 22 8 \| 0 1 0 -6 -10 -13 -1 \| 0 0 1 1 1 1 -1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 352/351, 540/539, 729/728		Comma list: 352/351, 540/539, 729/728

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 -18 -13 \| 0 1 0 -6 15 12 \| 0 0 1 1 -1 -1 }}		{{Mapping\|legend=0\| 1 0 0 10 -18 -13 \| 0 1 0 -6 15 12 \| 0 0 1 1 -1 -1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 352/351, 400/399, 456/455, 495/494		Comma list: 352/351, 400/399, 456/455, 495/494

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 -18 -13 -6 \| 0 1 0 -6 15 12 5 \| 0 0 1 1 -1 -1 1 }}		{{Mapping\|legend=0\| 1 0 0 10 -18 -13 -6 \| 0 1 0 -6 15 12 5 \| 0 0 1 1 -1 -1 1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 325/324, 352/351, 385/384		Comma list: 325/324, 352/351, 385/384

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 -3 2 \| 0 1 0 -6 7 4 \| 0 0 1 1 -2 -2 }}		{{Mapping\|legend=0\| 1 0 0 10 -3 2 \| 0 1 0 -6 7 4 \| 0 0 1 1 -2 -2 }}

	Lattice basis:		Lattice basis:
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	Comma list: 176/175, 351/350, 847/845		Comma list: 176/175, 351/350, 847/845

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 6 11 \| 0 1 0 -6 -6 -9 \| 0 0 1 1 3 3 }}		{{Mapping\|legend=0\| 1 0 0 10 6 11 \| 0 1 0 -6 -6 -9 \| 0 0 1 1 3 3 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 352/351, 847/845, 3025/3024		Comma list: 352/351, 847/845, 3025/3024

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 7 12 \| 0 1 1 -5 -2 -5 \| 0 0 2 2 -1 -1 }}		{{Mapping\|legend=0\| 1 0 0 10 7 12 \| 0 1 1 -5 -2 -5 \| 0 0 2 2 -1 -1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 352/351, 676/675, 847/845		Comma list: 352/351, 676/675, 847/845

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 -6 -1 \| 0 2 0 -12 9 3 \| 0 0 1 1 1 1 }}		{{Mapping\|legend=0\| 1 0 0 10 -6 -1 \| 0 2 0 -12 9 3 \| 0 0 1 1 1 1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 400/399, 1216/1215		Comma list: 400/399, 1216/1215

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 10 -6 \| 0 1 0 -6 5 \| 0 0 1 1 1 }}		{{Mapping\|legend=0\| 1 0 0 10 -6 \| 0 1 0 -6 5 \| 0 0 1 1 1 }}

	Optimal tunings:		Optimal tunings:

FloraC: Recategorize

2026-08-14T11:41:43Z

Recategorize

← Older revision		Revision as of 11:41, 14 August 2026
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	== References ==		== References ==

			[[Category:Aberschismic family\| ]] <!-- main article -->
	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:~~Aberschismic family\| ]] <!~~-~~- main article -->~~		[[Category:Catalogs of rank-3 temperaments]]
	~~[[Category:Rank~~ 3]]
	~~[[Category:Listen~~]]

FloraC: /* Overview to extensions */ further explain the mapping for 13

2026-07-26T06:41:58Z

Overview to extensions: further explain the mapping for 13

← Older revision		Revision as of 06:41, 26 July 2026
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	=== Overview to extensions ===		=== Overview to extensions ===
	==== 11- and 13-limit extensions ====		==== 11- and 13-limit extensions ====
	Strong extensions of aberschismic are [[#Pele\|pele]], [[#Laka\|laka]], [[#Akea\|akea]], and [[#Lono\|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F♯). All these extensions follow the trend of tuning the fifth a little sharp~~. Thus~~ a ~~successful~~ mapping of 13 ~~can be found by fixing the~~ [[13/11]] at the minor third (~~C–Eb~~), tempering out [[~~352/351~~]], [[847/845]], and [[2080/2079]].		Strong extensions of aberschismic are [[#Pele\|pele]], [[#Laka\|laka]], [[#Akea\|akea]], and [[#Lono\|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F♯). All these extensions follow the trend of tuning the fifth a little sharp, so a mapping for 13 arises from identifying [[13/11]] with the minor third (C–E♭), which is further facilitated by the fact that the minor third in aberschismic is the exact mean of [[6/5]] and [[7/6]], thus tempering out the [[semiparticular]] [[847/845]] ([[S-expression\|S11/S13]]) and by extension [[352/351]] and [[2080/2079]].

	Temperaments discussed elsewhere include:		Temperaments discussed elsewhere include:
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].		A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].

	== Pele ==		== Pele ==

FloraC: /* Kapo */ the only practical extension for 13 is to map 13/11 to m3, like the other temps, even tho it's not supported by the OPV here

2026-07-26T06:01:09Z

Kapo: the only practical extension for 13 is to map 13/11 to m3, like the other temps, even tho it's not supported by the OPV here

@@ Line 280: / Line 280: @@
 == Kapo ==
 [[Subgroup]]: 2.3.5.7.11
@@ Line 301: / Line 303: @@
 [[Badness]] (Sintel): 1.19
 == Namaka ==

FloraC: Make counterpyth a redirect as there's not much too add in a dedicated article

2026-06-08T11:16:58Z

Make counterpyth a redirect as there's not much too add in a dedicated article

← Older revision		Revision as of 11:16, 8 June 2026
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	Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.		Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

	It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (~~C–vBb~~).		It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vB♭).

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

	Laka can be described as the {{nowrap\| 41 & 53 & 58 }} temperament, tempering out [[540/539]], and finds the interval class of 11 at the up augmented third (C–^E#). [[Gene Ward Smith]] considered it a [[17-limit]] temperament, assigning the vanishing of [[442/441]] ({{nowrap\| 41g & 53 & 58 }}) as the main extension, but {{nowrap\| 41 & 53g & 58 }} also makes for a competitive extension.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group \| ''Laka 17-limit minimax planar temperament'']</ref> Indeed, laka makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, skipping prime 17, as the 19 is accurate and easily available in a 24-tone scale. [[152edo]] makes for an excellent tuning, using the 152f val for prime 13.		Laka can be described as the {{nowrap\| 41 & 53 & 58 }} temperament, tempering out [[540/539]], and finds the interval class of 11 at the up augmented third (C–^E♯). [[Gene Ward Smith]] considered it a [[17-limit]] temperament, assigning the vanishing of [[442/441]] ({{nowrap\| 41g & 53 & 58 }}) as the main extension, but {{nowrap\| 41 & 53g & 58 }} also makes for a competitive extension.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group \| ''Laka 17-limit minimax planar temperament'']</ref> Indeed, laka makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, skipping prime 17, as the 19 is accurate and easily available in a 24-tone scale. [[152edo]] makes for an excellent tuning, using the 152f val for prime 13.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	== Lono ==		== Lono ==
	Lono tempers out [[176/175]] and may be described as the {{nowrap\| 46 & 53 & 58 }} temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v<sup>3</sup>F#). It notably also tempers out [[8019/8000]], thus setting 11/10, 10/9, 9/8, and 8/7 a comma apart from each other. [[111edo]] is a great tuning for it. [[157edo]] is a viable alternative, which is almost as good.		Lono tempers out [[176/175]] and may be described as the {{nowrap\| 46 & 53 & 58 }} temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v<sup>3</sup>F♯). It notably also tempers out [[8019/8000]], thus setting 11/10, 10/9, 9/8, and 8/7 a comma apart from each other. [[111edo]] is a great tuning for it. [[157edo]] is a viable alternative, which is almost as good.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	== Subgroup extensions ==		== Subgroup extensions ==
	=== Counterpyth (2.3.5.7.19) ===		=== Counterpyth (2.3.5.7.19) ===
	~~{{Main~~\| ~~Counterpyth }}~~		[[File:Lattice Counterpyth RTT.png\|thumb\|Lattice for counterpyth.]]

	~~Developed analogous to~~ [[parapyth]], counterpyth is an extension of aberschismic with an even milder fifth, as it finds [[19/15]] at the major third (C–E) and [[19/10]] at the major seventh (C–B)~~. Notice~~ the factorization {{nowrap\| 5120/5103 {{=}} ([[400/399]])⋅([[1216/1215]]) }}. Other important ratios are [[21/19]] at the diminished third (~~C–Ebb~~) and [[19/14]] at the augmented third (~~C–E#~~).		Inspired by [[Margo Schulter]]'s [[parapyth]], counterpyth was named and first explored by [[Flora Canou]] in 2024. It is an extension of aberschismic with an even milder fifth, as it finds [[19/15]] at the major third (C–E) and [[19/10]] at the major seventh (C–B), taking advantage of the factorization {{nowrap\| 5120/5103 {{=}} ([[400/399]])⋅([[1216/1215]]) }}. Other important ratios are [[21/19]] at the diminished third (C–E𝄫) and [[19/14]] at the augmented third (C–E♯).

	It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.		It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.

FloraC: Update for the birth of the main aberschismic article

2026-06-08T11:10:43Z

Update for the birth of the main aberschismic article

← Older revision		Revision as of 11:10, 8 June 2026
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	~~{{Interwiki~~
	~~\| en = Aberschismic family~~
	~~\| de = Aberrschismisch~~
	~~\| es =~~
	~~\| ja =~~
	~~\| ro =~~
	~~\| ko = 헤미패미티 (음률)~~
	}}
	{{Technical data page}}		{{Technical data page}}
	The '''aberschismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[aberschisma]] ({{monzo\|legend=1\| 10 -6 1 -1 }}, [[ratio]]: [[5120/5103]]).		The '''aberschismic family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the [[aberschisma]] ({{monzo\|legend=1\| 10 -6 1 -1 }}, [[ratio]]: [[5120/5103]]).

	== Aberschismic ==		== Aberschismic ==
	Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (~~C–F#~~) and [[50/49]] with the [[Pythagorean comma]].		{{Main\| Aberschismic }}

			Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (C–F♯) and [[50/49]] with the [[Pythagorean comma]] (C–B♯′).

	Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.		Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.
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	[[Projection pair]]s: 7 5120/729		[[Projection pair]]s: 7 5120/729

	~~; Music~~
	* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
	* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]

	=== Overview to extensions ===		=== Overview to extensions ===
	==== 11- and 13-limit extensions ====		==== 11- and 13-limit extensions ====
	Strong extensions of aberschismic are [[#Pele\|pele]], [[#Laka\|laka]], [[#Akea\|akea]], and [[#Lono\|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (~~C–vGb~~); laka, up augmented third (C–^E#); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third (C–Eb), tempering out [[352/351]], [[847/845]], and [[2080/2079]].		Strong extensions of aberschismic are [[#Pele\|pele]], [[#Laka\|laka]], [[#Akea\|akea]], and [[#Lono\|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F♯). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third (C–Eb), tempering out [[352/351]], [[847/845]], and [[2080/2079]].

	Temperaments discussed elsewhere include:		Temperaments discussed elsewhere include:

Inthar at 13:11, 6 June 2026

2026-06-06T13:11:39Z

← Older revision		Revision as of 13:11, 6 June 2026
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	{{Interwiki		{{Interwiki
	\| en = Aberschismic family		\| en = Aberschismic family
	\| de = ~~Hemifamity~~		\| de = Aberrschismisch
	\| es =		\| es =
	\| ja =		\| ja =

FloraC: Recategorize (name change followup)

2026-06-06T12:46:16Z

Recategorize (name change followup)

← Older revision		Revision as of 12:46, 6 June 2026
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:~~Hemifamity~~ family\| ]] <!-- main article -->		[[Category:Aberschismic family\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]
	[[Category:Listen]]		[[Category:Listen]]