Aberschismic family - Revision history

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]]

FloraC: FloraC moved page Hemifamity family to Aberschismic family

2026-06-06T12:33:12Z

FloraC moved page Hemifamity family to Aberschismic family

← Older revision	Revision as of 12:33, 6 June 2026
(No difference)

FloraC: Officialize the name aberschismic (see Talk: 5120/5103)

2026-06-06T12:29:03Z

Officialize the name aberschismic (see Talk: 5120/5103)

← Older revision		Revision as of 12:29, 6 June 2026
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	{{Interwiki		{{Interwiki
	\| en =		\| en = Aberschismic family
	\| de = Hemifamity		\| de = Hemifamity
	\| es =		\| es =
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	}}		}}
	{{Technical data page}}		{{Technical data page}}
	The '''~~hemifamity~~ family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] [[~~5120/5103~~]] ({{monzo\|legend=1\| 10 -6 1 -1 }}~~), the hemifamity comma. These temperaments divide 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]]~~.		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]]).

	~~Hemifamity~~ 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, ~~hemifamity~~ can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.		== 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]].

			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–vBb).

	~~== Hemifamity ==~~
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	=== Overview to extensions ===		=== Overview to extensions ===
	==== 11- and 13-limit extensions ====		==== 11- and 13-limit extensions ====
	Strong extensions of ~~hemifamity~~ 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]] ~~at the~~ 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–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]].

	Temperaments discussed elsewhere include:		Temperaments discussed elsewhere include:
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	{{Main\| Counterpyth }}		{{Main\| Counterpyth }}

	Developed analogous to [[parapyth]], counterpyth is an extension of ~~hemifamity~~ 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#).		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#).

	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: Move counterpyth to subgroup extensions section as it skips multiple primes

2026-05-29T09:23:38Z

Move counterpyth to subgroup extensions section as it skips multiple primes

@@ Line 1: / Line 1: @@
-{{interwiki
+{{Interwiki
 | en =
 | de = Hemifamity
@@ Line 20: / Line 20: @@
 {{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
 : mapping generators: ~2, ~3, ~5
@@ Line 61: / Line 60: @@
 ==== Subgroup extensions ====
-A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.
+A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].
 == Pele ==
@@ Line 364: / Line 342: @@
 Badness (Sintel): 0.731
 == References ==

Overthink: /* Lono */ important part of its structure

2026-05-20T01:20:57Z

Lono: important part of its structure

← Older revision		Revision as of 01:20, 20 May 2026
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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#). [[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

FloraC: /* Laka */ delete 17-limit laka, as the current structure doesn't make sense

2026-04-25T03:34:09Z

Laka: delete 17-limit laka, as the current structure doesn't make sense

@@ Line 156: / Line 156: @@
 {{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. It should be noted that {{nowrap| 41 & 53g & 58 }} also makes for a possible extension.
+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
@@ Line 186: / Line 179: @@
 [[Badness]] (Sintel): 0.992
-[[Projection pair]]s: 5120/729 11 14348907/1310720
+[[Projection pair]]s: <code>7 5120/729 11 14348907/1310720</code>
 === 13-limit ===
@@ Line 226: / Line 219: @@
 Badness (Sintel): 0.647
 == Akea ==

Inthar: grammar

2026-04-13T12:36:56Z

grammar

← Older revision		Revision as of 12:36, 13 April 2026
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	}}		}}
	{{Technical data page}}		{{Technical data page}}
	The '''hemifamity family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] [[5120/5103]] ({{monzo\|legend=1\| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide 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]] by the augmented fourth (C–F#) and [[50/49]] by the [[Pythagorean comma]].		The '''hemifamity family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperaments]] [[tempering out\|tempers out]] [[5120/5103]] ({{monzo\|legend=1\| 10 -6 1 -1 }}), the hemifamity comma. These temperaments divide 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]].

	Hemifamity 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, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.		Hemifamity 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, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.