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{{Technical data page}}
=Vital statistics=
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]]).
[[Comma|Comma]] c = 5120/5103


7-limit minimax: 3 and 7 1/7c sharp, 5 just
== Aberschismic ==
{{Main| Aberschismic }}


[|1 0 0 0>, |10/7 1/7 1/7 -1/7>,  
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♯′).
|0 0 1 0>, |10/7 -6/7 1/7 6/7>]


[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 7/6
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.


9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
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♭).


[|1 0 0 0>, |5/4 1/4 1/8 -1/8>,
[[Subgroup]]: 2.3.5.7
|0 0 1 0>, |5/2 -3/2 1/4 3/4>]


[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 9/7
[[Comma list]]: [[5120/5103]]


Lattice basis: 3/2 length 0.5670, 10/9 length 1.8063
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
: mapping generators: ~2, ~3, ~5


Angle(3/2, 10/9) = 82.112 degrees
[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]


Map to lattice: [<0 1 2 -4|, <0 0 1 1|]
Lattice basis:  
: 3/2 length = 0.5670, 10/9 length = 1.8063
: Angle (3/2, 10/9) = 82.112 degrees


Map: [<1 0 0 10|, <0 1 0 -6|, <0 0 1 1|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}, ~5/4 = 386.7266{{c}}
: [[error map]]: {{val| -0.283 +0.426 -0.153 +0.222 }}
* [[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 }}


Generators: 2, 3, 5
[[Minimax tuning]]: c = 5120/5103
* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just
: {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp
: {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7


[[EDO|EDOs]]: [[99edo|99]], [[140edo|140]], [[239edo|239]], [[292edo|292]], [[391edo|391]], [[490edo|490]], [[881edo|881bd]]
{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}


Badness: 0.000153
[[Badness]] (Sintel): 0.675


[[Projection_pair|Projection pair]]s: 7 5120/729
[[Projection pair]]s: 7 5120/729


==Music==
=== Overview to extensions ===
By Gene Ward Smith
==== 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, 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]].


[http://www.archive.org/details/Choraled Choraled] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play]
Temperaments discussed elsewhere include:
* ''[[Kahoupokane]]'' (+121/120) → [[Biyatismic clan #Kahoupokane|Biyatismic clan]]


By Chris Vaisvil
==== Subgroup extensions ====
A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].


[http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 Hemifamity27]
== Pele ==
{{Main| Pele }}
{{See also| Pentacircle clan }}


=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.
[[Comma|Commas]]: 441/440, 896/891


[[Minimax_tuning|Minimax tuning]]
[[Subgroup]]: 2.3.5.7.11


[|1 0 0 0 0&gt;, |17/10 0 1/10 0 -1/10&gt;,
[[Comma list]]: 441/440, 896/891
|17/5 -2 6/5 0 -1/5&gt;,
|16/5 -2 3/5 0 2/5&gt;, |17/5 -2 1/5 0 4/5&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9
{{Mapping|legend=1| 1 0 0 10 17 | 0 1 0 -6 -10 | 0 0 1 1 1 }}


Lattice basis: 3/2 length 0.3812 56/55 length 1.5893
[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]


Angle(3/2, 56/55) = 90.4578 degrees
Lattice basis:
: 3/2 length = 0.3812, 56/55 length = 1.5893
: Angle(3/2, 56/55) = 90.4578 degrees


Map to lattice: [&lt;0 1 4 -2 -6|, &lt;0 0 -1 -1 -1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5424{{c}}, ~3/2 = 703.0109{{c}}, ~5/4 = 387.6427{{c}}
: [[error map]]: {{val| -0.458 +0.598 +0.414 -1.995 +2.097 }}
* [[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 }}


Map: [&lt;1 0 0 10 17|, &lt;0 1 0 -6 -10|, &lt;0 0 1 1 1|]
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


[[generator|Generators]]: 2, 3, 5
{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}


[[EDO|Edos]]: [[41edo|41]], [[46edo|46]], [[77edo|77e]], [[87edo|87]], [[99edo|99e]], [[128edo|128]], [[145edo|145]], [[186edo|186e]], [[232edo|232]], [[285edo|285e]], [[331edo|331e]]
[[Badness]] (Sintel): 0.779


Badness: 0.000648
[[Projection pair]]s: 7 5120/729 11 655360/59049


Projection pairs: 7 5120/729 11 655360/59049
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 196/195, 352/351, 364/363
Commas: 196/195, 352/351, 364/363


13-limit minimax
Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}


Eigenmonzos: 2, 10/9, 13/10
Optimal tunings:  
* 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}}


15-limit minimax
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9


Eigenmonzos: 2, 15/13, 6/5
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}


Map: [&lt;1 0 0 10 17 22|, &lt;0 1 0 -6 -10 -13|, &lt;0 0 1 1 1 1|]
Badness (Sintel): 0.658


EDOs: 29, 41, 46, 58, 87, 145, 232, 377cef
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.000703
Comma list: 196/195, 256/255, 352/351, 364/363


=Laka=
Mapping: {{mapping| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }}
[[Comma|Commas]]: 5120/5103, 540/539


[[Minimax_tuning|Minimax tuning]]
Optimal tunings:
* 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}}


[|1 0 0 0 0&gt;, |4/3 0 2/21 -1/21 1/21&gt;,  
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}
|0 0 1 0 0&gt;, |2 0 3/7 2/7 -2/7&gt;,  
|2 0 3/7 -5/7 5/7&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 14/11
Badness (Sintel): 0.884


Projection pairs: 5120/729 11 14348907/1310720
== Laka ==
{{Main| Laka }}


Map: [&lt;1 0 0 10 -18|, &lt;0 1 0 -6 15|, &lt;0 0 1 1 -1|]
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.


EDOs: 84, 97, 99e, 111, 121, 130, 152, 555de, 707de, 859bde
[[Subgroup]]: 2.3.5.7.11


==13 limit==
[[Comma list]]: 540/539, 5120/5103
[[Comma|Commas]]: 352/351, 540/539, 847/845


13 and 15 limit minimax tuning
{{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}


[|1 0 0 0 0 0&gt;, |13/8 -1/2 1/8 0 0 1/8&gt;,
[[Optimal tuning]]s:
|13/4 -3 5/4 0 0 1/4&gt;, |7/2 0 1/2 0 0 -1/2&gt;,  
* [[WE]]: ~2 = 1199.6201{{c}}, ~3/2 = 702.4416{{c}}, ~5/4 = 386.6781{{c}}
|25/8 -9/2 5/8 0 0 13/8&gt;,
: [[error map]]: {{val| -0.380 +0.107 -0.395 +0.924 +0.527 }}
|13/4 -3 1/4 0 0 5/4&gt;]
* [[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 }}


[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 14/13
[[Minimax tuning]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7


==17 limit==
{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
[[Comma|Commas]]: 352/351, 540/539, 847/845, 442/441


17-limit minimax
[[Badness]] (Sintel): 0.992


[|1 0 0 0 0 0 0&gt;,
[[Projection pair]]s: <code>7 5120/729 11 14348907/1310720</code>
|13/12 0 0 1/12 1/6 -1/12 0&gt;,
|-7/4 0 0 5/4 3/2 -5/4 0&gt;,
|7/4 0 0 3/4 1/2 -3/4 0&gt;,
|0 0 0 0 1 0 0&gt;,
|7/4 0 0 -1/4 1/2 1/4 0&gt;,
|35/12 0 0 23/12 5/6 -23/12 0&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 14/13
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 10 -18 -13 32|,  
Comma list: 352/351, 540/539, 729/728


&lt;0 1 0 -6 15 12 -22|, &lt;0 0 1 1 -1 -1 3|]
Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}


[[generator|Generators]]: 2, 3, 5
Optimal tunings:  
* 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}}


[[EDO|Edos]]: [[94edo|94]], [[111edo|111]], [[152edo|152]], [[205edo|205]], [[345edo|345]]
Minimax tuning:
* 13- and 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]
: unchanged-interval (eigenmonzo) basis: 2.11.13/7


=Akea=
{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }} *
[[Comma|Commas]]: 385/384, 2200/2187


[[Minimax_tuning|Minimax tuning]]
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]


[|1 0 0 0 0&gt;, |5/3 0 1/6 -1/6 0&gt;,
Badness (Sintel): 0.769
|26/9 0 13/18 -7/18 -1/3&gt;,
|26/9 0 -5/18 11/18 -1/3&gt;,
|26/9 0 -5/18 -7/18 2/3&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 14/11, 7/5
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19


Map: [&lt;1 0 0 10 -3|, &lt;0 1 0 -6 7|, &lt;0 0 1 1 -2|]
Comma list: 352/351, 400/399, 456/455, 495/494


[[generator|Generators]]: 2, 3, 5
Mapping: {{mapping| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}


EDOs: 34, 41, 53, 87, 140, 181, 321
Optimal tunings:  
* 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}}


Badness: 0.000998
{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *


==13-limit==
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Commas: 325/324, 352/351, 385/384


13 and 15 limit minimax
Badness (Sintel): 0.647


[|1 0 0 0 0 0&gt;, |5/3 0 1/6 -1/6 0 0&gt;,
== Akea ==
|26/9 0 13/18 -7/18 -1/3 0&gt;,
[[File:Lattice Akea.png|thumb|Lattice for 13-limit akea.]]
|26/9 0 -5/18 11/18 -1/3 0&gt;,  
[[File:Lattice Akea-commatic.png|thumb|Ditto, but rearranged to basis {~2, ~3, ~81/80}.]]
|26/9 0 -5/18 -7/18 2/3 0&gt;,  
|26/9 0 -7/9 1/9 2/3 0&gt;]


Eigenmonzos: 2, 14/11, 7/5
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.


Lattice basis: 3/2 length 0.5354 27/20 length 1.0463
[[Subgroup]]: 2.3.5.7.11


Angle(3/2, 27/20) = 80.5628 degrees
[[Comma list]]: 385/384, 2200/2187


Map to lattice: [&lt;0 1 3 -3 1 -2|, &lt;0 0 -1 -1 2 2|]
{{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -2 }}


Map: [&lt;1 0 0 10 -3 2|, &lt;0 1 0 -6 7 4|, &lt;0 0 1 1 -2 -2|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1396{{c}}, ~3/2 = 702.9241{{c}}, ~5/4 = 385.1817{{c}}
: [[error map]]: {{val| +0.140 +1.109 -0.853 -0.351 -1.213 }}
* [[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 }}


Generators: 2, 3, 5
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


[[EDO|EDOs]]: 7, 41, 46, 53, 87, [[94edo|94]], [[140edo|140]], [[181edo|181]], [[321edo|321]], [[408edo|408]]
{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}


Badness: 0.000822
[[Badness]] (Sintel): 1.20


Scales: [[akea46_13|akea46_13]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Kapo=
Comma list: 325/324, 352/351, 385/384
[[Comma|Commas]]: 5120/5103, 3025/3024


[[Minimax_tuning|Minimax tuning]]
Mapping: {{mapping| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}


[|1 0 0 0 0&gt;, |8/5 2/5 0 -1/15 -2/15&gt;,  
Lattice basis:
|14/5 6/5 0 7/15 -16/15&gt;,
: 3/2 length = 0.5354, 27/20 length = 1.0463
|16/5 -6/5 0 13/15 -4/15&gt;,  
: Angle (3/2, 27/20) = 80.5628 degrees
|16/5 -6/5 0 -2/15 11/15&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 9/7
Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]


Map: [&lt;1 0 0 10 7|, &lt;0 1 1 -5 -2|, &lt;0 0 2 2 -1|]
Optimal tunings:  
* 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}}


[[generator|Generators]]: 2, 3, 128/99
Minimax tuning:
* 13- and 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5


[[EDO|EDOs]]: [[152edo|152]], [[198edo|198]], [[239edo|239]], [[350edo|350]], [[391edo|391]], [[478edo|478]], [[589edo|589]]
{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}


EDOs: 41, 87, 111, 152, 239, 391
Badness (Sintel): 0.769


Badness: 0.000994
Scales: [[akea46_13]]


=Lono=
== Lono ==
Commas: 176/175, 5120/5103
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.


Map: [&lt;1 0 0 10 6|, &lt;0 1 0 -6 -6|, &lt;0 0 1 1 3|]
[[Subgroup]]: 2.3.5.7.11


EDOs: 7, 12, 39, 46, 53, 58, 99, 111
[[Comma list]]: 176/175, 5120/5103


Badness: 0.00118
{{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}


=Namaka=
[[Optimal tuning]]s:
Commas: 3388/3375, 5120/5103
* [[WE]]: ~2 = 1199.3368{{c}}, ~3/2 = 702.5643{{c}}, ~5/4 = 389.5319{{c}}
: [[error map]]: {{val| -0.663 -0.054 +1.892 +1.341 -2.088 }}
* [[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 }}


Map: [&lt;1 0 0 10 -6|, &lt;0 2 0 -12 9|, &lt;0 0 1 1 1|]
{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}


EDOs: 29, 53, 58, 87, 111, 140, 198
[[Badness]] (Sintel): 1.41


Badness: 0.00174
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 176/175, 351/350, 847/845
Commas: 352/351, 676/675, 847/845


Map: [&lt;1 0 0 10 -6 -1|, &lt;0 2 0 -12 9 3|, &lt;0 0 1 1 1 1|]
Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}


EDOs: 29, 53, 58, 87, 111, 140, 198
Optimal tunings:  
* 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}}


Badness: 0.000781
{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}
[[Category:family]]
 
[[Category:hemifamity]]
Badness (Sintel): 0.850
[[Category:planar]]
 
[[Category:theory]]
== Kapo ==
Kapo tempers out 3025/3024, the [[lehmerisma]], as well as 16384/16335, the [[semiporwellisma]], thus splitting the [[~]][[5/3]] to two equal parts of ~[[128/99]] each. The only practical 13-limit extension maps [[13/11]] to the diatonic minor third, like the other temperaments in this family.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 5120/5103
 
{{Mapping|legend=1| 1 0 0 10 7 | 0 1 1 -5 -2 | 0 0 2 2 -1 }}
: mapping generators: ~2, ~3, ~128/99
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7125{{c}}, ~3/2 = 702.6631{{c}}, ~128/99 = 441.8973{{c}}
: [[error map]]: {{val| -0.287 +0.421 -0.143 +0.216 +0.021 }}
* [[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]]:
* [[11-odd-limit]]:
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9
 
{{Optimal ET sequence|legend=1| 41, 65d, 87, 111, 152, 239, 391, 980bcde, 1132bcdde, 1371bbcddee }}
 
[[Badness]] (Sintel): 1.19
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.6256{{c}}, ~3/2 = 702.7250{{c}}, ~84/65 = 442.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9733{{c}}, ~84/65 = 442.1684{{c}}
 
{{Optimal ET sequence|legend=0| 41, 46, 65d, 87, 111, 152f, 198, 350f, 437f, 635bcff }}
 
Badness (Sintel): 0.938
 
== Namaka ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3388/3375, 5120/5103
 
{{Mapping|legend=1| 1 0 0 10 -6 | 0 2 0 -12 9 | 0 0 1 1 1 }}
: mapping generators: ~2, ~400/231, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7179{{c}}, ~400/231 = 951.2909{{c}}, ~5/4 = 387.4982{{c}}
: [[error map]]: {{val| -0.282 +0.627 +0.620 -0.203 -1.074 }}
* [[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 }}
 
{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
 
[[Badness]] (Sintel): 2.09
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tunings:
* 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 }}
 
Badness (Sintel): 0.731
 
== Subgroup extensions ==
=== Counterpyth (2.3.5.7.19) ===
[[File:Lattice Counterpyth RTT.png|thumb|Lattice for counterpyth.]]
 
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.
 
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
 
== References ==
 
[[Category:Temperament families]]
[[Category:Aberschismic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Aberschismic_family"