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The '''hemifamity family''' of temperaments are rank-3 temperaments tempering out [[5120/5103]].  
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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.
 
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 ==
== Hemifamity ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[5120/5103]]
[[Comma list]]: [[5120/5103]]


[[Mapping]]: [{{val| 1 0 0 10 }}, {{val| 0 1 0 -6 }}, {{val| 0 0 1 1 }}]
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
 
: mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]
[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]
Line 15: Line 27:
: 3/2 length = 0.5670, 10/9 length = 1.8063
: 3/2 length = 0.5670, 10/9 length = 1.8063
: Angle (3/2, 10/9) = 82.112 degrees
: Angle (3/2, 10/9) = 82.112 degrees
[[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 }}


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


{{Val list|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}
{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}


[[Badness]]: 0.153 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.675


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


; Music
; Music
* [http://www.archive.org/details/Choraled Choraled] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
* [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]]
* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]
 
=== Overview to 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]].
 
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 considered in [[#Subgroup extensions]].


== Pele ==
== Pele ==
Subgroup: 2.3.5.7.11
{{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


[[Comma list]]: 441/440, 896/891
[[Comma list]]: 441/440, 896/891


[[Mapping]]: [{{val| 1 0 0 10 17 }}, {{val| 0 1 0 -6 -10 }}, {{val| 0 0 1 1 1 }}]
{{Mapping|legend=1| 1 0 0 10 17 | 0 1 0 -6 -10 | 0 0 1 1 1 }}


[[Minimax tuning]]:
[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]
* [[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]]s: 2, 10/9, 11/9
 
Mapping to lattice: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]


Lattice basis:  
Lattice basis:  
Line 52: Line 80:
: Angle(3/2, 56/55) = 90.4578 degrees
: Angle(3/2, 56/55) = 90.4578 degrees


{{Val list|legend=1| 29, 41, 58, 87, 99e, 145, 186e, 331e }}<nowiki>*</nowiki>
[[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 }}
 
[[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


<nowiki>*</nowiki> [[optimal patent val]]: [[232edo|232]]
{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}


[[Badness]]: 0.648 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.779


[[Projection pair]]s: 7 5120/729 11 655360/59049
[[Projection pair]]s: 7 5120/729 11 655360/59049
Line 65: Line 102:
Comma list: 196/195, 352/351, 364/363
Comma list: 196/195, 352/351, 364/363


Mapping: [{{val| 1 0 0 10 17 22 }}, {{val| 0 1 0 -6 -10 -13 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}
 
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}}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzos (unchanged intervals): 2, 10/9, 13/10
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
* 15-odd-limit eigenmonzos (unchanged intervals): 2, 15/13, 6/5
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9
 
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 145, 232 }}
 
Badness (Sintel): 0.658
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
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}}


Vals: {{val list| 29, 41, 46, 58, 87, 145, 232, 377cef }}
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}


Badness: 0.703 × 10<sup>-3</sup>
Badness (Sintel): 0.884


== Laka ==
== Laka ==
Subgroup: 2.3.5.7.11
{{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.
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 5120/5103
[[Comma list]]: 540/539, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 -18 }}, {{val| 0 1 0 -6 15 }}, {{val| 0 0 1 1 -1 }}]
{{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6201{{c}}, ~3/2 = 702.4416{{c}}, ~5/4 = 386.6781{{c}}
: [[error map]]: {{val| -0.380 +0.107 -0.395 +0.924 +0.527 }}
* [[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]]
[[Minimax tuning]]
* [[11-odd-limit]]
* [[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 }}]
: [{{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]]s (unchanged intervals): 2, 5/4, 14/11
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
 
{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}


{{Val list|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
[[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 ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 847/845
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 }}
 
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}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit
* 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 }}]
: [{{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 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/8, 14/13
: unchanged-interval (eigenmonzo) basis: 2.11.13/7


Vals: {{val list| 41, 53, 58, 94, 111, 152f, 415de }}*
{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }} *


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


=== 17-limit ===
Badness (Sintel): 0.769
Subgroup: 2.3.5.7.11.13.17
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
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 }}


Comma list: 352/351, 442/441, 540/539, 847/845
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}}


Mapping: [{{val| 1 0 0 10 -18 -13 32 }}, {{val| 0 1 0 -6 15 12 -22 }}, {{val| 0 0 1 1 -1 -1 3 }}]
{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *


Minimax tuning:
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
* 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 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/8, 14/13


Vals: {{Val list| 58, 94, 111, 152f, 205, 263df }}
Badness (Sintel): 0.647


== Akea ==
== Akea ==
Subgroup: 2.3.5.7.11
[[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


[[Comma list]]: 385/384, 2200/2187
[[Comma list]]: 385/384, 2200/2187


[[Mapping]]: [{{val| 1 0 0 10 -3 }}, {{val| 0 1 0 -6 7 }}, {{val| 0 0 1 1 -2 }}]
{{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -2 }}


Mapping generators: ~2, ~3, ~5
[[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[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&gt;, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
: [{{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]]s (unchanged intervals): ~2, ~14/11, ~7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


{{Val list|legend=1| 34, 41, 53, 87, 140, 181, 321 }}
{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}


Badness: 0.998 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.20


=== 13-limit ===
=== 13-limit ===
Line 142: Line 230:
Comma list: 325/324, 352/351, 385/384
Comma list: 325/324, 352/351, 385/384


Mapping: [{{val| 1 0 0 10 -3 2 }}, {{val| 0 1 0 -6 7 4 }}, {{val| 0 0 1 1 -2 -2 }}]
Mapping: {{mapping| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}
 
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 }}]
: Eigenmonzos (unchanged intervals): 2, 14/11, 7/5


Lattice basis:  
Lattice basis:  
Line 155: Line 238:
Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]
Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]


Vals: {{val list| 34, 41, 46, 53, 87, 140, 321, 461e }}
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}}
 
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
 
{{Optimal ET sequence|legend=0| 34, 41, 46, 53, 87, 140, 321, 461e }}


Badness: 0.822 × 10<sup>-3</sup>
Badness (Sintel): 0.769


Scales: [[akea46_13]]
Scales: [[akea46_13]]


== Lono ==
== Lono ==
Subgroup: 2.3.5.7.11
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


[[Comma list]]: 176/175, 5120/5103
[[Comma list]]: 176/175, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 6 }}, {{val| 0 1 0 -6 -6 }}, {{val| 0 0 1 1 3 }}]
{{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}
 
[[Optimal tuning]]s:  
* [[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 }}


{{Val list|legend=1| 46, 53, 58, 99, 111, 268cd }}
{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}


[[Badness]]: 1.18 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Line 177: Line 277:
Comma list: 176/175, 351/350, 847/845
Comma list: 176/175, 351/350, 847/845


Mapping: [{{val| 1 0 0 10 6 11 }}, {{val| 0 1 0 -6 -6 -9 }}, {{val| 0 0 1 1 3 3 }}]
Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}
 
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}}


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


Badness: 0.908 × 10<sup>-3</sup>
Badness (Sintel): 0.850


== Kapo ==
== Kapo ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 5120/5103
[[Comma list]]: 3025/3024, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 7 }}, {{val| 0 1 1 -5 -2 }}, {{val| 0 0 2 2 -1 }}]
{{Mapping|legend=1| 1 0 0 10 7 | 0 1 1 -5 -2 | 0 0 2 2 -1 }}
: mapping generators: ~2, ~3, ~128/99


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]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]:  
* [[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 }}]
: [{{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]]s (unchanged intervals): 2, 11/9, 9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/9


{{Val list|legend=1| 41, 87, 111, 152, 239, 391 }}
{{Optimal ET sequence|legend=1| 41, 65d, 87, 111, 152, 239, 391, 980bcde, 1132bcdde, 1371bbcddee }}


[[Badness]]: 0.994 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.19


== Namaka ==
== Namaka ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3388/3375, 5120/5103
[[Comma list]]: 3388/3375, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 -6 }}, {{val| 0 2 0 -12 9 }}, {{val| 0 0 1 1 1 }}]
{{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 }}


{{Val list|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}


[[Badness]]: 1.74 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 217: Line 333:
Comma list: 352/351, 676/675, 847/845
Comma list: 352/351, 676/675, 847/845


Mapping: [{{val| 1 0 0 10 -6 -1 }}, {{val| 0 2 0 -12 9 3 }}, {{val| 0 0 1 1 1 1 }}]
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) ===
{{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 }}


Vals: {{val list| 29, 53, 58, 87, 111, 140, 198 }}
Badness (Sintel): 0.347


Badness: 0.781 × 10<sup>-3</sup>
== References ==


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Aberschismic_family"