Aberschismic family: Difference between revisions

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{{Interwiki

| en =

| de = Hemifamity

| es =

| ja =

| ro =

| ko = 헤미패미티 (음률)

}}

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.

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: mapping generators: ~2, ~3, ~5

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}, ~5/4 = 386.5465{{c}}

: error map: {{val| 0.000 +0.862 +0.233 +0.821 }}

~~~~

[[Minimax tuning]]: c = 5120/5103

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==== 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]].

Temperaments discussed elsewhere include:

* ''[[Kahoupokane]]'' (+121/120) → [[Biyatismic clan #Kahoupokane|Biyatismic clan]]

==== 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]].

~~=== 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#).~~

~~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.~~

~~Subgroup: 2.3.5.7.19~~

~~Comma list: 400/399, 1216/1215~~

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

~~Optimal tunings:~~

* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}

~~~~

~~{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}~~

~~Badness (Sintel): 0~~.~~347~~

== Pele ==

Pele tempers out [[441/440]] as well as [[896/891]] and may be described as the {{nowrap| 41 & 46 & 58 }} temperament, finding the interval class of 11 at the down diminished fifth (C–vGb). It also extends [[parapyth]]. [[145edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7.11

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2804{{c}}, ~5/4 = 387.3911{{c}}

: error map: {{val| 0.000 +1.325 +1.077 -1.117 +3.269 }}

~~~~

[[Minimax tuning]]:

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* WE: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}, ~5/4 = 388.0342{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}, ~5/4 = 387.7761{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.3960{{c}}, ~3/2 = 703.0725{{c}}, ~5/4 = 388.4246{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4518{{c}}, ~5/4 = 388.4909{{c}}

~~~~

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Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]]. [[Gene Ward Smith]] considered it ~~to be~~ a [[17-limit]] temperament, assigning ~~†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.

~~<blockquote>~~

~~It's the way the numbers fall. The Laka geometry happens to work reasonably well in the 13-limit but not so well in the 17-limit. There isn't one obvious 17-limit~~ extension ~~and none of them are competitive with other 17-limit temperaments~~.

~~</blockquote>~~

~~—[[Graham Breed]]~~<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref>

It makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, ~~omitting harmonic~~ 17, as the 19 is accurate and easily available in a 24-tone scale.

[[Subgroup]]: 2.3.5.7.11

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6175{{c}}, ~5/4 = 386.4170{{c}}

: error map: {{val| 0.000 +0.663 +0.103 +1.886 +1.528 }}

~~~~

[[Minimax tuning]]

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[[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 ===

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* WE: ~2 = 1199.4742{{c}}, ~3/2 = 702.3385{{c}}, ~5/4 = 387.0965{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5780{{c}}, ~5/4 = 386.7718{{c}}

~~~~

Minimax tuning:

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* WE: ~2 = 1199.4881{{c}}, ~3/2 = 702.3224{{c}}, ~5/4 = 386.8881{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5613{{c}}, ~5/4 = 386.6230{{c}}

~~~~

{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *

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Badness (Sintel): 0.647

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 352/351, 442/441, 540/539, 561/560~~

~~Mapping: {{mapping| 1 0 0 10 -18 -13 32 | 0 1 0 -6 15 12 -22 | 0 0 1 1 -1 -1 3 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.6101{{c}}, ~3/2 = 702.3391{{c}}, ~5/4 = 387.1022{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5253{{c}}, ~5/4 = 386.8488{{c}}

~~Minimax tuning:~~

* 17-odd-limit

: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]

~~: unchanged-interval (eigenmonzo) basis: 2.11.13/7~~

~~{{Optimal ET sequence|legend=0| 58, 94, 111, 152f, 205, 263df }}~~

~~Badness (Sintel): 1.13~~

== Akea ==

[[File:Lattice Akea.png|thumb|Lattice for 13-limit akea.]]

[[File:Lattice Akea-commatic.png|thumb|Ditto, but rearranged to basis {~2, ~3, ~81/80}.]]

Akea tempers out [[385/384]] and may be described as the {{nowrap| 41 & 46 & 53 }} temperament, finding the interval class of 11 at the double-up fourth (C–^^F). [[140edo]], [[181edo]] and especially [[321edo]] can be used as tunings. Note that [[94edo]] is a notable tuning not appearing on the optimal ET sequence.

[[Subgroup]]: 2.3.5.7.11

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8511{{c}}, ~5/4 = 385.1712{{c}}

: error map: {{val| 0.000 +0.896 -1.143 -0.761 -1.703 }}

~~~~

[[Minimax tuning]]:

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* WE: ~2 = 1200.0943{{c}}, ~3/2 = 702.9377{{c}}, ~5/4 = 385.4278{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8853{{c}}, ~5/4 = 385.4002{{c}}

~~~~

Minimax tuning:

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== 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.

[[Subgroup]]: 2.3.5.7.11

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9356{{c}}, ~5/4 = 389.4076{{c}}

: error map: {{val| 0.000 +0.981 +3.094 +2.968 -0.708 }}

~~~~

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* WE: ~2 = 1199.3329{{c}}, ~3/2 = 702.5519{{c}}, ~5/4 = 389.5508{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9205{{c}}, ~5/4 = 389.4341{{c}}

~~~~

{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8413{{c}}, ~128/99 = 441.9493{{c}}

: error map: {{val| 0.000 +0.886 +0.426 +0.866 +1.050 }}

~~~~

[[Minimax tuning]]:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.5081{{c}}, ~5/4 = 387.3182{{c}}

: error map: {{val| 0.000 +1.061 +1.004 +0.395 -0.426 }}

~~~~

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* WE: ~2 = 1199.7072{{c}}, ~26/15 = 951.2767{{c}}, ~5/4 = 387.4314{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}, ~5/4 = 387.2360{{c}}

~~~~

{{Optimal ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198, 536f }}

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Badness (Sintel): 0.731

== ~~Kahoupokane~~ ==

== Subgroup extensions ==

~~[[Subgroup]]:~~ 2.3.5.7.11

=== Counterpyth (2.3.5.7.19) ===

[[~~Comma list~~]]~~: 121~~/~~120,~~ 5120/5103

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#).

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

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.

~~: mapping generators: ~2~~, ~~~3, ~11/10~~

~~[[Optimal tuning]]s~~:

Subgroup: 2.3.5.7.19

* [[WE]]: ~2 ~~= 1200~~.~~1911{{c}}, ~~~3~~/2 = 703.1412{{c}}, ~11/10 = 158~~.~~1068{{c}}~~

~~: [[error map]]: {{val| +0.191 +1.377 +0.996 +0.401 -~~5.~~710 }}~~

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.0417{{c}}, ~11/10 = 157.9917{{c}}

~~: error map: {{val| 0.000 +1.087 +0.744 -0.018 -6~~.~~268 }}~~

~~{{Optimal ET sequence|legend=1| 7~~, ~~17c, 24d, 29, 46, 53, 82e, 99 }}~~

Comma list: 400/399, 1216/1215

~~[[Badness]] (Sintel): 2.73~~

Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 121/120, 169/168, 352/351~~

Mapping: {{mapping| 1 0 ~~1 11 2 7~~ | 0 1 1 -5 ~~1 -2~~ | 0 0 ~~-2 -2 -~~1 -1 }}

Optimal tunings:

* WE: ~2 = ~~1200~~.~~4435~~{{c}}, ~3/2 = ~~703~~.~~1443~~{{c}}, ~11/10 = ~~158~~.~~4176~~{{c}}

* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.~~9013~~{{c}}, ~11/10 = ~~158~~.~~1657~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}

{{Optimal ET sequence|legend=0| ~~7, 17c, 24d~~, 29, 46, 53, ~~82e~~, 99, ~~181eef~~ }}

{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}

Badness (Sintel): 1.27

Badness (Sintel): 0.347

== References ==

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Hemifamity family| ]]

~~[[Category:Hemifamity| ]] ~~

[[Category:Rank 3]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
+{{Interwiki
+| en =
+| de = Hemifamity
+| es =
+| ja =
+| ro =
+| ko = 헤미패미티 (음률)
+}}
 {{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.
@@ Line 12: / Line 20: @@
 {{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
 : mapping generators: ~2, ~3, ~5
@@ Line 26: / Line 33: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}, ~5/4 = 386.5465{{c}}
 : error map: {{val| 0.000 +0.862 +0.233 +0.821 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7918{{c}}, ~5/4 = 386.0144{{c}} -->
 [[Minimax tuning]]: c = 5120/5103
@@ Line 49: / Line 55: @@
 ==== 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]].
+Temperaments discussed elsewhere include:
+* ''[[Kahoupokane]]'' (+121/120) → [[Biyatismic clan #Kahoupokane|Biyatismic clan]]
 ==== 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]].
-=== 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#).
-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.
-Subgroup: 2.3.5.7.19
-Comma list: 400/399, 1216/1215
-Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
-Optimal tunings:
-* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6411{{c}}, ~5/4 = 385.4452{{c}} -->
-{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
-Badness (Sintel): 0.347
 == Pele ==
 {{Main| Pele }}
 {{See also| Pentacircle clan }}
+Pele tempers out [[441/440]] as well as [[896/891]] and may be described as the {{nowrap| 41 & 46 & 58 }} temperament, finding the interval class of 11 at the down diminished fifth (C–vGb). It also extends [[parapyth]]. [[145edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 96: / Line 85: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2804{{c}}, ~5/4 = 387.3911{{c}}
 : error map: {{val| 0.000 +1.325 +1.077 -1.117 +3.269 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2829{{c}}, ~5/4 = 386.5647{{c}} -->
 [[Minimax tuning]]:
@@ Line 119: / Line 107: @@
 * WE: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}, ~5/4 = 388.0342{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}, ~5/4 = 387.7761{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4398{{c}}, ~5/4 = 386.8933{{c}} -->
 Minimax tuning:
@@ Line 139: / Line 126: @@
 * WE: ~2 = 1199.3960{{c}}, ~3/2 = 703.0725{{c}}, ~5/4 = 388.4246{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4518{{c}}, ~5/4 = 388.4909{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 703.5544{{c}}, ~5/4 = 387.9654{{c}} -->
 {{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}
@@ Line 148: / Line 134: @@
 {{Main| Laka }}
-Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]]. [[Gene Ward Smith]] considered it to be a [[17-limit]] temperament, assigning †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.
-<blockquote>
-It's the way the numbers fall. The Laka geometry happens to work reasonably well in the 13-limit but not so well in the 17-limit. There isn't one obvious 17-limit extension and none of them are competitive with other 17-limit temperaments.
-</blockquote>
-—[[Graham Breed]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref>
-It makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, omitting harmonic 17, as the 19 is accurate and easily available in a 24-tone scale.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 168: / Line 147: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6175{{c}}, ~5/4 = 386.4170{{c}}
 : error map: {{val| 0.000 +0.663 +0.103 +1.886 +1.528 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5133{{c}}, ~5/4 = 385.5563{{c}} -->
 [[Minimax tuning]]
@@ Line 179: / Line 157: @@
 [[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 191: / Line 169: @@
 * WE: ~2 = 1199.4742{{c}}, ~3/2 = 702.3385{{c}}, ~5/4 = 387.0965{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5780{{c}}, ~5/4 = 386.7718{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4078{{c}}, ~5/4 = 385.5405{{c}} -->
 Minimax tuning:
@@ Line 214: / Line 191: @@
 * WE: ~2 = 1199.4881{{c}}, ~3/2 = 702.3224{{c}}, ~5/4 = 386.8881{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5613{{c}}, ~5/4 = 386.6230{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4062{{c}}, ~5/4 = 385.5254{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *
@@ Line 221: / Line 197: @@
 Badness (Sintel): 0.647
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 352/351, 442/441, 540/539, 561/560
-Mapping: {{mapping| 1 0 0 10 -18 -13 32 | 0 1 0 -6 15 12 -22 | 0 0 1 1 -1 -1 3 }}
-Optimal tunings:
-* WE: ~2 = 1199.6101{{c}}, ~3/2 = 702.3391{{c}}, ~5/4 = 387.1022{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5253{{c}}, ~5/4 = 386.8488{{c}}
-Minimax tuning:
-* 17-odd-limit
-: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]
-: unchanged-interval (eigenmonzo) basis: 2.11.13/7
-{{Optimal ET sequence|legend=0| 58, 94, 111, 152f, 205, 263df }}
-Badness (Sintel): 1.13
 == Akea ==
 [[File:Lattice Akea.png|thumb|Lattice for 13-limit akea.]]
 [[File:Lattice Akea-commatic.png|thumb|Ditto, but rearranged to basis {~2, ~3, ~81/80}.]]
+Akea tempers out [[385/384]] and may be described as the {{nowrap| 41 & 46 & 53 }} temperament, finding the interval class of 11 at the double-up fourth (C–^^F). [[140edo]], [[181edo]] and especially [[321edo]] can be used as tunings. Note that [[94edo]] is a notable tuning not appearing on the optimal ET sequence.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 257: / Line 215: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8511{{c}}, ~5/4 = 385.1712{{c}}
 : error map: {{val| 0.000 +0.896 -1.143 -0.761 -1.703 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8909{{c}}, ~5/4 = 385.3273{{c}} -->
 [[Minimax tuning]]:
@@ Line 284: / Line 241: @@
 * WE: ~2 = 1200.0943{{c}}, ~3/2 = 702.9377{{c}}, ~5/4 = 385.4278{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8853{{c}}, ~5/4 = 385.4002{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9018{{c}}, ~5/4 = 385.4158{{c}} -->
 Minimax tuning:
@@ Line 298: / Line 254: @@
 == 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.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 309: / Line 267: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9356{{c}}, ~5/4 = 389.4076{{c}}
 : error map: {{val| 0.000 +0.981 +3.094 +2.968 -0.708 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8941{{c}}, ~5/4 = 388.5932{{c}} -->
 {{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}
@@ Line 325: / Line 282: @@
 * WE: ~2 = 1199.3329{{c}}, ~3/2 = 702.5519{{c}}, ~5/4 = 389.5508{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9205{{c}}, ~5/4 = 389.4341{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8670{{c}}, ~5/4 = 388.6277{{c}} -->
 {{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}
@@ Line 344: / Line 300: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8413{{c}}, ~128/99 = 441.9493{{c}}
 : error map: {{val| 0.000 +0.886 +0.426 +0.866 +1.050 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8776{{c}}, ~128/99 = 441.7516{{c}} -->
 [[Minimax tuning]]:
@@ Line 368: / Line 323: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.5081{{c}}, ~5/4 = 387.3182{{c}}
 : error map: {{val| 0.000 +1.061 +1.004 +0.395 -0.426 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.4956{{c}}, ~5/4 = 386.7868{{c}} -->
 {{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
@@ Line 384: / Line 338: @@
 * WE: ~2 = 1199.7072{{c}}, ~26/15 = 951.2767{{c}}, ~5/4 = 387.4314{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}, ~5/4 = 387.2360{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4871{{c}}, ~5/4 = 386.6606{{c}} -->
 {{Optimal ET sequence|legend=0| 29, 53, 58, 87, 111, 140, 198, 536f }}
@@ Line 390: / Line 343: @@
 Badness (Sintel): 0.731
-== Kahoupokane ==
+== Subgroup extensions ==
-[[Subgroup]]: 2.3.5.7.11
+=== Counterpyth (2.3.5.7.19) ===
+{{Main| Counterpyth }}
-[[Comma list]]: 121/120, 5120/5103
+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#).
-{{Mapping|legend=1| 1 0 1 11 2 | 0 1 1 -5 1 | 0 0 -2 -2 -1 }}
+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.
-: mapping generators: ~2, ~3, ~11/10
-[[Optimal tuning]]s:
+Subgroup: 2.3.5.7.19
-* [[WE]]: ~2 = 1200.1911{{c}}, ~3/2 = 703.1412{{c}}, ~11/10 = 158.1068{{c}}
-: [[error map]]: {{val| +0.191 +1.377 +0.996 +0.401 -5.710 }}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.0417{{c}}, ~11/10 = 157.9917{{c}}
-: error map: {{val| 0.000 +1.087 +0.744 -0.018 -6.268 }}
-{{Optimal ET sequence|legend=1| 7, 17c, 24d, 29, 46, 53, 82e, 99 }}
+Comma list: 400/399, 1216/1215
-[[Badness]] (Sintel): 2.73
+Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 169/168, 352/351
-Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}
 Optimal tunings:
-* WE: ~2 = 1200.4435{{c}}, ~3/2 = 703.1443{{c}}, ~11/10 = 158.4176{{c}}
+* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9013{{c}}, ~11/10 = 158.1657{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}
-{{Optimal ET sequence|legend=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}
+{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
-Badness (Sintel): 1.27
+Badness (Sintel): 0.347
 == References ==
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Hemifamity family| ]] <!-- main article -->
-[[Category:Hemifamity| ]] <!-- key article -->
 [[Category:Rank 3]]
 [[Category:Listen]]