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**Imported revision 164978201 - Original comment: links added, some of them should describe the terminology**
Move counterpyth to subgroup extensions section as it skips multiple primes
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en =
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-09-23 16:01:44 UTC</tt>.<br>
| de = Hemifamity
: The original revision id was <tt>164978201</tt>.<br>
| es =
: The revision comment was: <tt>links added, some of them should describe the terminology</tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| ro =
<h4>Original Wikitext content:</h4>
| ko = 헤미패미티 (음률)
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">===Vital statistics===
}}
[[Comma]] c = 5120/5103
{{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]].


7-limit minimax: 3 and 7 1/7c sharp, 5 just
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.
[|1 0 0 0&gt;, |10/7 1/7 1/7 -1/7&gt;,  
|0 0 1 0&gt;, |10/7 -6/7 1/7 6/7&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 7/6


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–vBb).
[|1 0 0 0&gt;, |5/4 1/4 1/8 -1/8&gt;,
|0 0 1 0&gt;, |5/2 -3/2 1/4 3/4&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 9/7


Lattice basis: 3/2 length 0.5670, 10/9 length 1.8063
== Hemifamity ==
Angle(3/2, 10/9) = 82.112 degrees
[[Subgroup]]: 2.3.5.7
Map to lattice: [&lt;0 1 2 -4|, &lt;0 0 1 1|]
[[edo|EDOs]]: [[99edo|99]], [[140edo|140]], [[239edo|239]], [[292edo|292]], [[490edo|490]], [[531edo|531]]


==11-limit children==
[[Comma list]]: [[5120/5103]]


===Pele===
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}
[[Comma|Commas]]: 441/440, 896/891
: mapping generators: ~2, ~3, ~5
 
[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]
 
Lattice basis:
: 3/2 length = 0.5670, 10/9 length = 1.8063
: 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
* [[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
 
{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}
 
[[Badness]] (Sintel): 0.675
 
[[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 ===
==== 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 ==
{{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
 
{{Mapping|legend=1| 1 0 0 10 17 | 0 1 0 -6 -10 | 0 0 1 1 1 }}
 
[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]
 
Lattice basis:
: 3/2 length = 0.3812, 56/55 length = 1.5893
: Angle(3/2, 56/55) = 90.4578 degrees
 
[[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
 
{{Optimal ET sequence|legend=1| 29, 41, 58, 87, 99e, 145, 186e }}
 
[[Badness]] (Sintel): 0.779
 
[[Projection pair]]s: 7 5120/729 11 655360/59049
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 364/363
 
Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}
 
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:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
* 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}}
 
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}
 
Badness (Sintel): 0.884
 
== 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.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 5120/5103
 
{{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]]
[|1 0 0 0 0&gt;, |17/10 0 1/10 0 -1/10&gt;,  
* [[11-odd-limit]]
|17/5 -2 6/5 0 -1/5&gt;,
: [{{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 }}]
|16/5 -2 3/5 0 2/5&gt;, |17/5 -2 1/5 0 4/5&gt;]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9
 
{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
 
[[Badness]] (Sintel): 0.992
 
[[Projection pair]]s: <code>7 5120/729 11 14348907/1310720</code>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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:
* 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
 
{{Optimal ET sequence|legend=0| 41, 53, 58, 94, 111, 152f, 415dff }} *
 
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]


Map: [&lt;1 0 0 10 17|, &lt;0 1 0 -6 -10|, &lt;0 0 1 1 1|]
Badness (Sintel): 0.769
[[Generator|Generators]]: 2, 3, 5
[[edo|Edos]]: [[128edo|128]], [[145edo|145]], [[232edo|232]]


===Laka===
=== 2.3.5.7.11.13.19 subgroup ===
[[Comma|Commas]]: 5120/5103, 540/539
Subgroup: 2.3.5.7.11.13.19


[[Minimax tuning]]
Comma list: 352/351, 400/399, 456/455, 495/494
[|1 0 0 0 0&gt;, |4/3 0 2/21 -1/21 1/21&gt;,  
 
|0 0 1 0 0&gt;, |2 0 3/7 2/7 -2/7&gt;,  
Mapping: {{mapping| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }}
|2 0 3/7 -5/7 5/7&gt;]
 
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 14/11
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}}
 
{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }} *
 
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
 
Badness (Sintel): 0.647
 
== 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
 
[[Comma list]]: 385/384, 2200/2187
 
{{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -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 }}
 
[[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
 
{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}
 
[[Badness]] (Sintel): 1.20
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384
 
Mapping: {{mapping| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}
 
Lattice basis:
: 3/2 length = 0.5354, 27/20 length = 1.0463
: Angle (3/2, 27/20) = 80.5628 degrees
 
Mapping to lattice: [{{val| 0 1 3 -3 1 -2 }}, {{val| 0 0 -1 -1 2 2 }}]
 
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 (Sintel): 0.769
 
Scales: [[akea46_13]]
 
== 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
 
[[Comma list]]: 176/175, 5120/5103
 
{{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 }}
 
{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}
 
[[Badness]] (Sintel): 1.41
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 847/845
 
Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}
 
Badness (Sintel): 0.850
 
== Kapo ==
[[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
 
== 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


13 limit
== Subgroup extensions ==
[[Comma|Commas]]: 352/351, 540/539, 847/845
=== Counterpyth (2.3.5.7.19) ===
{{Main| Counterpyth }}


13 and 15 limit minimax tuning
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#).
[|1 0 0 0 0 0&gt;, |13/8 -1/2 1/8 0 0 1/8&gt;,  
|13/4 -3 5/4 0 0 1/4&gt;, |7/2 0 1/2 0 0 -1/2&gt;,
|25/8 -9/2 5/8 0 0 13/8&gt;,
|13/4 -3 1/4 0 0 5/4&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 14/13


17 limit
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.
[[Comma|Commas]]: 352/351, 540/539, 847/845, 442/441


17-limit minimax
Subgroup: 2.3.5.7.19
[|1 0 0 0 0 0 0&gt;,
|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


Map: [&lt;1 0 0 10 -18 -13 32|,  
Comma list: 400/399, 1216/1215
&lt;0 1 0 -6 15 12 -22|, &lt;0 0 1 1 -1 -1 3|]
[[Generator|Generators]]: 2, 3, 5
[[edo|Edos]]: [[94edo|94]], [[111edo|111], [[152edo|152]], [[205edo|205]], [[345edo|345]]


===Akea===
Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }}
[[Comma|Commas]]: 385/384, 2200/2187


[[Minimax tuning]]
Optimal tunings:
[|1 0 0 0 0&gt;, |4/3 0 2/21 -1/21 1/21&gt;,
* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}}
|0 0 1 0 0&gt;, |2 0 3/7 2/7 -2/7&gt;,  
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}}
|2 0 3/7 -5/7 5/7&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 14/11, 7/5


Map: [&lt;1 0 0 10 -3|, &lt;0 1 0 -6 7|, &lt;0 0 1 1 -2|]
{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
[[Generator|Generators]]: 2, 3, 5
[[edo|EDOs]]: [[94edo|94]], [[140edo|140]], [[181edo|181]], [[408edo|408]]


===Kapo===
Badness (Sintel): 0.347
[[Comma|Commas]]: 5120/5103, 3025/3024


[[Minimax tuning]]
== References ==
[|1 0 0 0 0&gt;, |8/5 2/5 0 -1/15 -2/15&gt;,
|14/5 6/5 0 7/15 -16/15&gt;,
|16/5 -6/5 0 13/15 -4/15&gt;,
|16/5 -6/5 0 -2/15 11/15&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 9/7


Map: [&lt;1 0 0 10 7|, &lt;0 1 1 -5 -2|, &lt;0 0 2 2 -1|]
[[Category:Temperament families]]
[[Generator|Generators]]: 2, 3, 128/99
[[Category:Hemifamity family| ]] <!-- main article -->
[[edo|EDOs]]: [[152edo|152]], [[198edo|198]], [[239edo|239]], [[350edo|350]], [[391edo|391]], [[478edo|478]], [[589edo|589]]</pre></div>
[[Category:Rank 3]]
<h4>Original HTML content:</h4>
[[Category:Listen]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Hemifamity family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 5120/5103&lt;br /&gt;
&lt;br /&gt;
7-limit minimax: 3 and 7 1/7c sharp, 5 just&lt;br /&gt;
[|1 0 0 0&amp;gt;, |10/7 1/7 1/7 -1/7&amp;gt;, &lt;br /&gt;
|0 0 1 0&amp;gt;, |10/7 -6/7 1/7 6/7&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 7/6&lt;br /&gt;
&lt;br /&gt;
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp&lt;br /&gt;
[|1 0 0 0&amp;gt;, |5/4 1/4 1/8 -1/8&amp;gt;, &lt;br /&gt;
|0 0 1 0&amp;gt;, |5/2 -3/2 1/4 3/4&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 9/7&lt;br /&gt;
&lt;br /&gt;
Lattice basis: 3/2 length 0.5670, 10/9 length 1.8063&lt;br /&gt;
Angle(3/2, 10/9) = 82.112 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 2 -4|, &amp;lt;0 0 1 1|]&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/239edo"&gt;239&lt;/a&gt;, &lt;a class="wiki_link" href="/292edo"&gt;292&lt;/a&gt;, &lt;a class="wiki_link" href="/490edo"&gt;490&lt;/a&gt;, &lt;a class="wiki_link" href="/531edo"&gt;531&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-11-limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit children&lt;/h2&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11-limit children-Pele"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Pele&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 441/440, 896/891&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |17/10 0 1/10 0 -1/10&amp;gt;, &lt;br /&gt;
|17/5 -2 6/5 0 -1/5&amp;gt;,&lt;br /&gt;
|16/5 -2 3/5 0 2/5&amp;gt;, |17/5 -2 1/5 0 4/5&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 11/9&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 10 17|, &amp;lt;0 1 0 -6 -10|, &amp;lt;0 0 1 1 1|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/128edo"&gt;128&lt;/a&gt;, &lt;a class="wiki_link" href="/145edo"&gt;145&lt;/a&gt;, &lt;a class="wiki_link" href="/232edo"&gt;232&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-11-limit children-Laka"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Laka&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 5120/5103, 540/539&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |4/3 0 2/21 -1/21 1/21&amp;gt;, &lt;br /&gt;
|0 0 1 0 0&amp;gt;, |2 0 3/7 2/7 -2/7&amp;gt;, &lt;br /&gt;
|2 0 3/7 -5/7 5/7&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 14/11&lt;br /&gt;
&lt;br /&gt;
13 limit&lt;br /&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 352/351, 540/539, 847/845&lt;br /&gt;
&lt;br /&gt;
13 and 15 limit minimax tuning&lt;br /&gt;
[|1 0 0 0 0 0&amp;gt;, |13/8 -1/2 1/8 0 0 1/8&amp;gt;, &lt;br /&gt;
|13/4 -3 5/4 0 0 1/4&amp;gt;, |7/2 0 1/2 0 0 -1/2&amp;gt;, &lt;br /&gt;
|25/8 -9/2 5/8 0 0 13/8&amp;gt;, &lt;br /&gt;
|13/4 -3 1/4 0 0 5/4&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/8, 14/13&lt;br /&gt;
&lt;br /&gt;
17 limit&lt;br /&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 352/351, 540/539, 847/845, 442/441&lt;br /&gt;
&lt;br /&gt;
17-limit minimax&lt;br /&gt;
[|1 0 0 0 0 0 0&amp;gt;, &lt;br /&gt;
|13/12 0 0 1/12 1/6 -1/12 0&amp;gt;, &lt;br /&gt;
|-7/4 0 0 5/4 3/2 -5/4 0&amp;gt;, &lt;br /&gt;
|7/4 0 0 3/4 1/2 -3/4 0&amp;gt;, &lt;br /&gt;
|0 0 0 0 1 0 0&amp;gt;, &lt;br /&gt;
|7/4 0 0 -1/4 1/2 1/4 0&amp;gt;, &lt;br /&gt;
|35/12 0 0 23/12 5/6 -23/12 0&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/8, 14/13&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 10 -18 -13 32|, &lt;br /&gt;
&amp;lt;0 1 0 -6 15 12 -22|, &amp;lt;0 0 1 1 -1 -1 3|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/111edo"&gt;111], [[152edo|152&lt;/a&gt;, &lt;a class="wiki_link" href="/205edo"&gt;205&lt;/a&gt;, &lt;a class="wiki_link" href="/345edo"&gt;345&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-11-limit children-Akea"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Akea&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 385/384, 2200/2187&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |4/3 0 2/21 -1/21 1/21&amp;gt;, &lt;br /&gt;
|0 0 1 0 0&amp;gt;, |2 0 3/7 2/7 -2/7&amp;gt;, &lt;br /&gt;
|2 0 3/7 -5/7 5/7&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 14/11, 7/5&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 10 -3|, &amp;lt;0 1 0 -6 7|, &amp;lt;0 0 1 1 -2|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&lt;/a&gt;, &lt;a class="wiki_link" href="/408edo"&gt;408&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-11-limit children-Kapo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Kapo&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 5120/5103, 3025/3024&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |8/5 2/5 0 -1/15 -2/15&amp;gt;, &lt;br /&gt;
|14/5 6/5 0 7/15 -16/15&amp;gt;, &lt;br /&gt;
|16/5 -6/5 0 13/15 -4/15&amp;gt;, &lt;br /&gt;
|16/5 -6/5 0 -2/15 11/15&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 9/7&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 10 7|, &amp;lt;0 1 1 -5 -2|, &amp;lt;0 0 2 2 -1|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 128/99&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/198edo"&gt;198&lt;/a&gt;, &lt;a class="wiki_link" href="/239edo"&gt;239&lt;/a&gt;, &lt;a class="wiki_link" href="/350edo"&gt;350&lt;/a&gt;, &lt;a class="wiki_link" href="/391edo"&gt;391&lt;/a&gt;, &lt;a class="wiki_link" href="/478edo"&gt;478&lt;/a&gt;, &lt;a class="wiki_link" href="/589edo"&gt;589&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 09:23, 29 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The hemifamity family of rank-3 temperaments tempers out 5120/5103 (monzo: [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

Subgroup: 2.3.5.7

Comma list: 5120/5103

Mapping: [⟨1 0 0 10], ⟨0 1 0 -6], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~5

Mapping 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

Optimal tunings:

  • WE: ~2 = 1199.7172 ¢, ~3/2 = 702.6636 ¢, ~5/4 = 386.7266 ¢
error map: ⟨-0.283 +0.426 -0.153 +0.222]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8166 ¢, ~5/4 = 386.5465 ¢
error map: ⟨0.000 +0.862 +0.233 +0.821]

Minimax tuning: c = 5120/5103

[[1 0 0 0⟩, [10/7 1/7 1/7 -1/7⟩, [0 0 1 0⟩, [10/7 -6/7 1/7 6/7⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7/3
[[1 0 0 0⟩, [5/4 1/4 1/8 -1/8⟩, [0 0 1 0⟩, [5/2 -3/2 1/4 3/4⟩]
unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness (Sintel): 0.675

Projection pairs: 7 5120/729

Music

Overview to extensions

11- and 13-limit extensions

Strong extensions of hemifamity are pele, laka, akea, and 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–v3F#). 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:

Subgroup extensions

A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in #Subgroup extensions.

Pele

Main article: Pele
See also: Pentacircle clan

Pele tempers out 441/440 as well as 896/891 and may be described as the 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

Mapping: [⟨1 0 0 10 17], ⟨0 1 0 -6 -10], ⟨0 0 1 1 1]]

Mapping to lattice: [⟨0 1 4 -2 -6], ⟨0 0 -1 -1 -1]]

Lattice basis:

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

Optimal tunings:

  • WE: ~2 = 1199.5424 ¢, ~3/2 = 703.0109 ¢, ~5/4 = 387.6427 ¢
error map: ⟨-0.458 +0.598 +0.414 -1.995 +2.097]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2804 ¢, ~5/4 = 387.3911 ¢
error map: ⟨0.000 +1.325 +1.077 -1.117 +3.269]

Minimax tuning:

[[1 0 0 0 0⟩, [17/10 0 1/10 0 -1/10⟩, [17/5 -2 6/5 0 -1/5⟩, [16/5 -2 3/5 0 2/5⟩, [17/5 -2 1/5 0 4/5⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 29, 41, 58, 87, 99e, 145, 186e

Badness (Sintel): 0.779

Projection pairs: 7 5120/729 11 655360/59049

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

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 ¢, ~3/2 = 703.1192 ¢, ~5/4 = 388.0342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4225 ¢, ~5/4 = 387.7761 ¢

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

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 703.0725 ¢, ~5/4 = 388.4246 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4518 ¢, ~5/4 = 388.4909 ¢

Optimal ET sequence: 29, 41, 46, 58, 87, 99ef, 145

Badness (Sintel): 0.884

Laka

Main article: Laka

Laka can be described as the 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 (41g & 53 & 58) as the main extension, but 41 & 53g & 58 also makes for a competitive extension.[1] 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

Mapping: [⟨1 0 0 10 -18], ⟨0 1 0 -6 15], ⟨0 0 1 1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.6201 ¢, ~3/2 = 702.4416 ¢, ~5/4 = 386.6781 ¢
error map: ⟨-0.380 +0.107 -0.395 +0.924 +0.527]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6175 ¢, ~5/4 = 386.4170 ¢
error map: ⟨0.000 +0.663 +0.103 +1.886 +1.528]

Minimax tuning

[[1 0 0 0 0⟩, [4/3 0 2/21 -1/21 1/21⟩, [0 0 1 0 0⟩, [2 0 3/7 2/7 -2/7⟩, [2 0 3/7 -5/7 5/7⟩]
unchanged-interval (eigenmonzo) basis: 2.5.11/7

Optimal ET sequence: 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee

Badness (Sintel): 0.992

Projection pairs: 7 5120/729 11 14348907/1310720

13-limit

Subgroup: 2.3.5.7.11.13

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

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 ¢, ~3/2 = 702.3385 ¢, ~5/4 = 387.0965 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5780 ¢, ~5/4 = 386.7718 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [13/8 -1/2 1/8 0 0 1/8⟩, [13/4 -3 5/4 0 0 1/4⟩, [7/2 0 1/2 0 0 -1/2⟩, [25/8 -9/2 5/8 0 0 13/8⟩, [13/4 -3 1/4 0 0 5/4⟩]
unchanged-interval (eigenmonzo) basis: 2.11.13/7

Optimal ET sequence: 41, 53, 58, 94, 111, 152f, 415dff *

* optimal patent val: 205

Badness (Sintel): 0.769

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: [⟨1 0 0 10 -18 -13 -6], ⟨0 1 0 -6 15 12 5], ⟨0 0 1 1 -1 -1 1]]

Optimal tunings:

  • WE: ~2 = 1199.4881 ¢, ~3/2 = 702.3224 ¢, ~5/4 = 386.8881 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5613 ¢, ~5/4 = 386.6230 ¢

Optimal ET sequence: 41, 53, 58h, 94, 111, 152f, 415dffhh *

* optimal patent val: 205

Badness (Sintel): 0.647

Akea

Lattice for 13-limit akea.
Ditto, but rearranged to basis {~2, ~3, ~81/80}.

Akea tempers out 385/384 and may be described as the 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

Mapping: [⟨1 0 0 10 -3], ⟨0 1 0 -6 7], ⟨0 0 1 1 -2]]

Optimal tunings:

  • WE: ~2 = 1200.1396 ¢, ~3/2 = 702.9241 ¢, ~5/4 = 385.1817 ¢
error map: ⟨+0.140 +1.109 -0.853 -0.351 -1.213]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8511 ¢, ~5/4 = 385.1712 ¢
error map: ⟨0.000 +0.896 -1.143 -0.761 -1.703]

Minimax tuning:

[[1 0 0 0 0⟩, [5/3 0 1/6 -1/6 0⟩, [26/9 0 13/18 -7/18 -1/3⟩, [26/9 0 -5/18 11/18 -1/3⟩, [26/9 0 -5/18 -7/18 2/3⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 53, 87, 140, 181, 321

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 10 -3 2], ⟨0 1 0 -6 7 4], ⟨0 0 1 1 -2 -2]]

Lattice basis:

3/2 length = 0.5354, 27/20 length = 1.0463
Angle (3/2, 27/20) = 80.5628 degrees

Mapping to lattice: [⟨0 1 3 -3 1 -2], ⟨0 0 -1 -1 2 2]]

Optimal tunings:

  • WE: ~2 = 1200.0943 ¢, ~3/2 = 702.9377 ¢, ~5/4 = 385.4278 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8853 ¢, ~5/4 = 385.4002 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [5/3 0 1/6 -1/6 0 0⟩, [26/9 0 13/18 -7/18 -1/3 0⟩, [26/9 0 -5/18 11/18 -1/3 0⟩, [26/9 0 -5/18 -7/18 2/3 0⟩, [26/9 0 -7/9 1/9 2/3 0⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 46, 53, 87, 140, 321, 461e

Badness (Sintel): 0.769

Scales: akea46_13

Lono

Lono tempers out 176/175 and may be described as the 46 & 53 & 58 temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v3F#). 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

Mapping: [⟨1 0 0 10 6], ⟨0 1 0 -6 -6], ⟨0 0 1 1 3]]

Optimal tunings:

  • WE: ~2 = 1199.3368 ¢, ~3/2 = 702.5643 ¢, ~5/4 = 389.5319 ¢
error map: ⟨-0.663 -0.054 +1.892 +1.341 -2.088]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9356 ¢, ~5/4 = 389.4076 ¢
error map: ⟨0.000 +0.981 +3.094 +2.968 -0.708]

Optimal ET sequence: 46, 53, 58, 99, 111, 268cd

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

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

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 ¢, ~3/2 = 702.5519 ¢, ~5/4 = 389.5508 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9205 ¢, ~5/4 = 389.4341 ¢

Optimal ET sequence: 46, 53, 58, 99, 104c, 111, 268cd

Badness (Sintel): 0.850

Kapo

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5120/5103

Mapping: [⟨1 0 0 10 7], ⟨0 1 1 -5 -2], ⟨0 0 2 2 -1]]

mapping generators: ~2, ~3, ~128/99

Optimal tunings:

  • WE: ~2 = 1199.7125 ¢, ~3/2 = 702.6631 ¢, ~128/99 = 441.8973 ¢
error map: ⟨-0.287 +0.421 -0.143 +0.216 +0.021]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8413 ¢, ~128/99 = 441.9493 ¢
error map: ⟨0.000 +0.886 +0.426 +0.866 +1.050]

Minimax tuning:

[[1 0 0 0 0⟩, [8/5 2/5 0 -1/15 -2/15⟩, [14/5 6/5 0 7/15 -16/15⟩, [16/5 -6/5 0 13/15 -4/15⟩, [16/5 -6/5 0 -2/15 11/15⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7.11/9

Optimal ET sequence: 41, 65d, 87, 111, 152, 239, 391, 980bcde, 1132bcdde, 1371bbcddee

Badness (Sintel): 1.19

Namaka

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 5120/5103

Mapping: [⟨1 0 0 10 -6], ⟨0 2 0 -12 9], ⟨0 0 1 1 1]]

mapping generators: ~2, ~400/231, ~5

Optimal tunings:

  • WE: ~2 = 1199.7179 ¢, ~400/231 = 951.2909 ¢, ~5/4 = 387.4982 ¢
error map: ⟨-0.282 +0.627 +0.620 -0.203 -1.074]
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.5081 ¢, ~5/4 = 387.3182 ¢
error map: ⟨0.000 +1.061 +1.004 +0.395 -0.426]

Optimal ET sequence: 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: [⟨1 0 0 10 -6 -1], ⟨0 2 0 -12 9 3], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.7072 ¢, ~26/15 = 951.2767 ¢, ~5/4 = 387.4314 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.5016 ¢, ~5/4 = 387.2360 ¢

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198, 536f

Badness (Sintel): 0.731

Subgroup extensions

Counterpyth (2.3.5.7.19)

Main article: 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 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: [⟨1 0 0 10 -6], ⟨0 1 0 -6 5], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.6953 ¢, ~3/2 = 702.5169 ¢, ~5/4 = 386.2648 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6771 ¢, ~5/4 = 386.0544 ¢

Optimal ET sequence: 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh

Badness (Sintel): 0.347

References

  1. ↑ Yahoo! Tuning Group | Laka 17-limit minimax planar temperament
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