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Hemifamity family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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]]. Hemifamity can be compared to [[garibaldi]], with garibaldi expanding 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.
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-09-23 16:01:44 UTC</tt>.<br>
: The original revision id was <tt>164978201</tt>.<br>
: The revision comment was: <tt>links added, some of them should describe the terminology</tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<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


7-limit minimax: 3 and 7 1/7c sharp, 5 just
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;, |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
== Hemifamity ==
[|1 0 0 0&gt;, |5/4 1/4 1/8 -1/8&gt;,
[[Subgroup]]: 2.3.5.7
|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
[[Comma list]]: [[5120/5103]]
Angle(3/2, 10/9) = 82.112 degrees
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==
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}


===Pele===
: mapping generators: ~2, ~3, ~5
[[Comma|Commas]]: 441/440, 896/891
 
[[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]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.7918, ~5/4 = 386.0144
 
[[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]] (Smith): 0.153 × 10<sup>-3</sup>
 
[[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]].
 
==== Subgroup extensions ====
A notable 2.3.5.7.19 subgroup extension, counterpyth, is given right below.
 
=== 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 tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.6411, ~5/4 = 385.4452
 
{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }}
 
Badness (Smith): 0.212 × 10<sup>-3</sup>
 
== Pele ==
{{Main| Pele }}
{{See also| Pentacircle clan }}
 
[[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]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 703.2829, ~5/4 = 386.5647
 
[[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]] (Smith): 0.648 × 10<sup>-3</sup>
 
[[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 tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.4398, ~5/4 = 386.8933
 
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 (Smith): 0.703 × 10<sup>-3</sup>
 
=== 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 tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.5544, ~5/4 = 387.9654
 
{{Optimal ET sequence|legend=0| 29, 41, 46, 58, 87, 99ef, 145 }}
 
Badness (Smith): 0.930 × 10<sup>-3</sup>
 
== Laka ==
{{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.
 
<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
 
[[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]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.5133, ~5/4 = 385.5563


[[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]] (Smith): 0.825 × 10<sup>-3</sup>
 
[[Projection pair]]s: 5120/729 11 14348907/1310720
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 10 17|, &lt;0 1 0 -6 -10|, &lt;0 0 1 1 1|]
Comma list: 352/351, 540/539, 729/728
[[Generator|Generators]]: 2, 3, 5
[[edo|Edos]]: [[128edo|128]], [[145edo|145]], [[232edo|232]]


===Laka===
Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}
[[Comma|Commas]]: 5120/5103, 540/539


[[Minimax tuning]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4078, ~5/4 = 385.5405
[|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;,  
Minimax tuning:
|2 0 3/7 -5/7 5/7&gt;]
* 13- and 15-odd-limit
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 14/11
: [{{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]]
 
Badness (Smith): 0.822 × 10<sup>-3</sup>
 
=== 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 }}
 
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.4062, ~5/4 = 385.5254
 
{{Optimal ET sequence|legend=0| 41, 53, 58h, 94, 111, 152f, 415dffhh }}*
 
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
 
Badness (Smith): 0.661 × 10<sup>-3</sup>
 
=== 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 }}
 
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 (Smith): 1.19 × 10<sup>-3</sup>
 
== 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}.]]
 
[[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]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8909, ~5/4 = 385.3273
 
[[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]] (Smith): 0.998 × 10<sup>-3</sup>
 
=== 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 tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.9018, ~5/4 = 385.4158
 
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 (Smith): 0.822 × 10<sup>-3</sup>
 
Scales: [[akea46_13]]
 
== Lono ==
[[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]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8941, ~5/4 = 388.5932
 
{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}
 
[[Badness]] (Smith): 1.18 × 10<sup>-3</sup>
 
=== 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 tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.8670, ~5/4 = 388.6277
 
{{Optimal ET sequence|legend=0| 46, 53, 58, 99, 104c, 111, 268cd }}
 
Badness (Smith): 0.908 × 10<sup>-3</sup>
 
== 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]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8776, ~128/99 = 441.7516
 
[[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, 87, 111, 152, 239, 391 }}
 
[[Badness]] (Smith): 0.994 × 10<sup>-3</sup>
 
== 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


13 limit
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~400/231 = 951.4956, ~5/4 = 386.7868
[[Comma|Commas]]: 352/351, 540/539, 847/845


13 and 15 limit minimax tuning
{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
[|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
[[Badness]] (Smith): 1.74 × 10<sup>-3</sup>
[[Comma|Commas]]: 352/351, 540/539, 847/845, 442/441


17-limit minimax
=== 13-limit ===
[|1 0 0 0 0 0 0&gt;,
Subgroup: 2.3.5.7.11.13
|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: 352/351, 676/675, 847/845
&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 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}
[[Comma|Commas]]: 385/384, 2200/2187


[[Minimax tuning]]
Optimal tuning (CTE): ~2 = 1200.0000, ~26/15 = 951.4871, ~5/4 = 386.6606
[|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;,  
|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| 29, 53, 58, 87, 111, 140, 198 }}
[[Generator|Generators]]: 2, 3, 5
[[edo|EDOs]]: [[94edo|94]], [[140edo|140]], [[181edo|181]], [[408edo|408]]


===Kapo===
Badness (Smith): 0.781 × 10<sup>-3</sup>
[[Comma|Commas]]: 5120/5103, 3025/3024


[[Minimax tuning]]
== Notes ==
[|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:Pages with mostly numerical content]]
[[edo|EDOs]]: [[152edo|152]], [[198edo|198]], [[239edo|239]], [[350edo|350]], [[391edo|391]], [[478edo|478]], [[589edo|589]]</pre></div>
[[Category:Hemifamity family| ]] <!-- main article -->
<h4>Original HTML content:</h4>
[[Category:Hemifamity| ]] <!-- key article -->
<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;
[[Category:Rank 3]]
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 5120/5103&lt;br /&gt;
[[Category:Listen]]
&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>
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