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Augmented family: Difference between revisions

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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 = Augmented family
: This revision was by author [[User:clamengh|clamengh]] and made on <tt>2015-11-01 06:24:59 UTC</tt>.<br>
| de = Übermässige Temperaturen
: The original revision id was <tt>564624523</tt>.<br>
}}
: The revision comment was: <tt></tt><br>
{{Technical data page}}
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The '''augmented family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[2/1|octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].  
<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">[[toc]]
The 5-limit parent comma for the **augmented family** is 128/125, the [[128_125|diesis]]. Its [[monzo]] is |7 0 -3&gt;, and flipping that yields &lt;&lt;3 0 -7|| for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for a major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.


=Seven limit children=  
== Augmented ==
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and hemis 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quarter-tones instead of semitones.
{{Main| Augmented (temperament) }}


=August=
The [[period]] is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third. Its [[ploidacot]] is triploid monocot.
[[Comma]]s: 36/35, 128/125


[[POTE tuning|POTE generator]]: 696.011
[[Subgroup]]: 2.3.5


[[Map]]: [&lt;3 0 7 -1|, &lt;0 1 0 2|]
[[Comma list]]: 128/125
[[Wedgie]]: &lt;&lt;3 0 6 -7 1 14||
[[edo|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]
[[Badness]]: 0.0265


==11-limit==
{{Mapping|legend=1| 3 0 7 | 0 1 0 }}
Commas: 36/35, 45/44, 56/55


POTE generator: ~3/2 = 692.514
: mapping generators: ~5/4, ~3


Map: [&lt;3 0 7 -1 1|, &lt;0 1 0 2 2|]
[[Optimal tuning]]s:
EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
* [[WE]]: ~5/4 = 399.0128{{c}}, ~3/2 = 704.8937{{c}} (~16/15 = 93.1320{{c}})
Badness: 0.0202
: [[error map]]: {{val| -2.962 -0.023 +6.776 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.0691{{c}} (~16/15 = 94.9309{{c}})
: error map: {{val| 0.000 +3.114 +13.686 }}


==13-limit==
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}
Commas: 27/26, 36/35, 45/44, 56/55


POTE generator: ~3/2 = 688.783
[[Badness]] (Sintel): 0.523


Map: [&lt;3 0 7 -1 1 -3|, &lt;0 1 0 2 2 3|]
=== Overview to extensions ===
EDOs: 9, 12, [[21edo|21]], [[33edo|33ef]], [[54edo|54bcef]]
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal augmented adds [[64/63]], august [[36/35]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones. All of these can be extended to the [[11-limit]] by adding [[56/55]] and [[176/175]], which sum to 128/125, equating the 400{{c}} major third to [[14/11]] as well as 5/4.
Badness: 0.0184


==Augustus==  
== Septimal augmented (augene) ==
Commas: 26/25, 36/35, 45/44, 56/55
{{Main| Augmented (temperament) }}


POTE generator: ~3/2 = 685.356
Septimal augmented, a.k.a. augene, tempers out [[64/63]] and [[126/125]]. It may be described as the {{nowrap| 12 & 15 }} temperament. [[27edo]] and [[39edo]] in the 39d val make for good tunings. In the 11-limit, it tempers together [[7/5]] and [[11/8]], and 27edo in the 27e val may be recommended as a tuning.  


Map: [&lt;3 0 7 -1 1 11|, &lt;0 1 0 2 2 0|]
[[Subgroup]]: 2.3.5.7
EDOs: 9, 12, 21f
Badness: 0.0222


=Augene=
[[Comma list]]: 64/63, 126/125
Commas: 64/63, 126/125


[[POTE tuning|POTE generator]]: 709.257
{{Mapping|legend=1| 3 0 7 18 | 0 1 0 -2 }}


Map: [&lt;3 0 7 18|, &lt;0 1 0 -2|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;3 0 -6 -7 -18 -14||
* [[WE]]: ~5/4 = 398.7461{{c}}, ~3/2 = 707.0335{{c}} (~21/20 = 90.4587{{c}})
EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]
: [[error map]]: {{val| -3.762 +1.317 +4.909 +2.060 }}
Badness: 0.0248
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 709.3249{{c}} (~21/20 = 90.6751{{c}})
: error map: {{val| 0.000 +7.370 +13.686 +12.524 }}


==11-limit==
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}
Commas: 56/55, 64/63, 100/99


POTE generator: ~3/2 = 711.177
[[Badness]] (Sintel): 0.628


Map: [&lt;3 0 7 18 20|, &lt;0 1 0 -2 -2|]
=== 11-limit ===
EDOs: 12, 15, 27e
Subgroup: 2.3.5.7.11
Badness: 0.0196


===13-limit===
Comma list: 56/55, 64/63, 100/99
Commas: 40/39, 56/55, 64/63, 66/65


POTE generator: ~3/2 = 712.013
Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}


Map: [&lt;3 0 7 18 20 16|, &lt;0 1 0 -2 -2 -1|]
Optimal tunings:  
EDOs: 12f, 15, 27ef
* WE: ~5/4 = 398.4962{{c}}, ~3/2 = 708.5030{{c}} (~21/20 = 88.4895{{c}})
Badness: 0.0208
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 711.6031{{c}} (~21/20 = 88.3969{{c}})


===Ogene===
{{Optimal ET sequence|legend=0| 12, 15, 27e }}
Commas: 56/55, 64/63, 91/90, 100/99


POTE generator: ~3/2 = 712.609
Badness (Sintel): 0.648


Map: [&lt;3 0 7 18 20 -8|, &lt;0 1 0 -2 -2 4|]
==== 13-limit ====
EDOs: 12, 15, 27e, 69bcef
Subgroup: 2.3.5.7.11.13
Badness: 0.0229


===Agene===
Comma list: 40/39, 56/55, 64/63, 66/65
Commas: 56/55, 64/63, 78/77, 100/99


POTE generator: ~3/2 = 709.677
Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}


Map: [&lt;3 0 7 18 20 35|, &lt;0 1 0 -2 -2 -5|]
Optimal tunings:  
EDOs: 12f, 27e
* WE: ~5/4 = 398.0488{{c}}, ~3/2 = 708.5402{{c}} (~21/20 = 87.5574{{c}})
Badness: 0.0231
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.6704{{c}} (~21/20 = 87.3296{{c}})


==Eugene==
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}
Commas: 55/54, 64/63, 77/75


POTE generator: ~3/2 = 714.150
Badness (Sintel): 0.859


Map: [&lt;3 0 7 18 -4|, &lt;0 1 0 -2 3|]
==== Ogene ====
EDOs: 15, 27, 42
Subgroup: 2.3.5.7.11.13
Badness: 0.0356


[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3|Sad Like Winter Leaves]] by [[http://soundcloud.com/cityoftheasleep/sad-like-winter-trees|Igliashon Jones]]
Comma list: 56/55, 64/63, 91/90, 100/99
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3|Galticeran Sonatina]] by [[http://soundcloud.com/joelgranttaylor/galticeran_sonatina|Joel Taylor]]


=Inflated=
Mapping: {{mapping| 3 0 7 18 20 -8 | 0 1 0 -2 -2 4 }}
Commas: 28/27, 128/125


POTE generator: ~3/2 = 722.719
Optimal tunings:  
* WE: ~5/4 = 398.6473{{c}}, ~3/2 = 710.1987{{c}} (~21/20 = 87.0959{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.5057{{c}} (~21/20 = 87.4943{{c}})


Map: [&lt;3 0 7 -6|, &lt;0 1 0 3|]
{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}
Wedgie: &lt;&lt;3 0 9 -7 6 21||
EDOs: 15, 48bc, 63bc, 78bc
Badness: 0.0547


==11-limit==
Badness (Sintel): 0.946
Commas: 28/27, 55/54, 128/125


POTE generator: ~3/2 = 722.663
==== Agene ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;3 0 7 -6|, &lt;0 1 0 3|]
Comma list: 56/55, 64/63, 78/77, 100/99
EDOs: 15, 48bce, 63bce, 78bce
Badness: 0.0312


=Deflated=
Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}
Commas: 21/20, 128/125


POTE generator: ~3/2 = 681.629
Optimal tunings:  
* WE: ~5/4 = 398.5229{{c}}, ~3/2 = 707.0562{{c}} (~21/20 = 89.9897{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 710.1903{{c}} (~21/20 = 89.8097{{c}})


Map: [&lt;3 0 7 13|, &lt;0 1 0 -1|]
{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}
Wedgie: &lt;&lt;3 0 -3 -7 -13 -7||
EDOs: 3, 9, 18bd, 21d, 30bd
Badness: 0.0591


=Hexe=
Badness (Sintel): 0.955
Commas: 50/49, 128/125


[[POTE tuning|POTE generator]]: 710.963
=== Eugene ===
Subgroup: 2.3.5.7.11


Map: [&lt;6 0 14 17|, &lt;0 1 0 0|]
Comma list: 55/54, 64/63, 77/75
Wedgie: &lt;&lt;6 0 0 -14 -17 0||
EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[[42edo|42d]], [[54edo|54cd]]
Badness: 0.0577


==11-limit==
Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}
Commas: 50/49, 56/55, 125/121


POTE generator: ~3/2 = 714.304
Optimal tunings:  
* WE: ~5/4 = 399.1743{{c}}, ~3/2 = 712.6763{{c}} (~21/20 = 85.6723{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 713.9414{{c}} (~21/20 = 86.0586{{c}})


Map: [&lt;6 0 14 17 21|, &lt;0 1 0 0 0|]
{{Optimal ET sequence|legend=0| 12e, 15, 27, 42 }}
EDOs: 6, 12, 30de
Badness: 0.0384


==13-limit==
Badness (Sintel): 1.18
Commas: 50/49, 56/55, 66/65, 105/104


POTE generator: ~3/2 = 710.005
== August ==
August tempers out 36/35 and 225/224. It may be described as the {{nowrap| 9 & 12 }} temperament. Unlike augene, august calls for a flat tuning of the fifth, and besides [[12edo]], [[21edo]] is among the possible tunings.  


Map: [&lt;6 0 14 17 21 13|, &lt;0 1 0 0 0 1|]
[[Subgroup]]: 2.3.5.7
EDOs: 12f
Badness: 0.0359


=Hemiaug=
[[Comma list]]: 36/35, 128/125
Commas: 128/125, 245/243


[[POTE tuning|POTE generator]]: ~28/27 = 45.118
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}


Map: [&lt;3 1 7 -1|, &lt;0 2 0 5|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;6 0 15 -14 7 35||
* [[WE]]: ~5/4 = 399.1036{{c}}, ~3/2 = 694.4509{{c}} (~16/15 = 103.7564{{c}})
EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
: [[error map]]: {{val| -2.689 -10.193 +7.412 +15.594 }}
Badness: 0.0705
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 694.6812{{c}} (~16/15 = 105.3188{{c}})
: error map: {{val| 0.000 -7.274 +13.686 +20.537 }}


==11-limit==
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}
Commas: 56/55, 128/125, 245/243


POTE generator: ~28/27 = 45.788
[[Badness]] (Sintel): 0.670


Map: [&lt;3 1 7 -1 1|, &lt;0 2 0 5 5|]
=== 11-limit ===
EDOs: 24, 27e, 51ce, 78ce
Subgroup: 2.3.5.7.11
Badness: 0.0382


==13-limit==
Comma list: 36/35, 45/44, 56/55
Commas: 56/55, 91/90, 128/125, 245/243


POTE generator: ~28/27 = 46.250
Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}


Map: [&lt;3 1 7 -1 1 13|, &lt;0 2 0 5 5 -1|]
Optimal tunings:  
EDOs: 24, 27e, 51ce, 78ce
* WE: ~5/4 = 398.9225{{c}}, ~3/2 = 690.6486{{c}} (~16/15 = 107.1966{{c}})
Badness: 0.0302
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 690.8519{{c}} (~16/15 = 109.1481{{c}})


=Triforce=
{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}
Commas: 49/48, 128/125


[[POTE tuning|POTE generator]]: ~7/6 = 247.049
Badness (Sintel): 0.668


Map: [&lt;3 0 7 6|, &lt;0 2 0 1|]
==== 13-limit ====
Wedgie: &lt;&lt;6 0 3 -14 -12 7||
Subgroup: 2.3.5.7.11.13
EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
Badness: 0.0202


[[image:triforce9.jpg]]
Comma list: 27/26, 36/35, 45/44, 56/55


==11-limit==
Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
Commas: 56/55, 77/75, 128/125


[[POTE tuning|POTE generator]]: ~7/6 = 247.068
Optimal tunings:  
* WE: ~5/4 = 399.0956{{c}}, ~3/2 = 687.2261{{c}} (~16/15 = 110.9651{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 687.5057{{c}} (~16/15 = 112.4943{{c}})


Map: [&lt;3 0 7 6 8|, &lt;0 2 0 1 1|]
{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}
EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
Badness: 0.0262


===Musical example:===
Badness (Sintel): 0.762
[[http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15|The Triforce of Courage (tuned in 15edo)]] by Igliashon Jones
==13-limit==
Commas: 49/48, 56/55, 66/65, 77/75


[[POTE tuning|POTE generator]]: ~7/6 = 248.313
==== Augustus ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;3 0 7 6 8 4|, &lt;0 2 0 1 1 3|]
Comma list: 26/25, 36/35, 45/44, 56/55
EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
Badness: 0.0202


[[triphi|Triforce(9) with L:s = phi]]
Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}


==Semitriforce==  
Optimal tunings:
Commas: 49/48, 56/55, 77/75, 507/500
* WE: ~5/4 = 400.4230{{c}}, ~3/2 = 686.0809{{c}} (~16/15 = 114.7650{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 685.8446{{c}} (~16/15 = 114.1554{{c}})


POTE generator: ~7/6 = 246.642
{{Optimal ET sequence|legend=0| 9, 12 }}


Map: [&lt;6 0 14 12 16 27|, &lt;0 2 0 1 1 -1|]
Badness (Sintel): 0.919
EDOs: 6, 24, 54cd, 78cd, 102cdf
Badness: 0.0592


=== ===
== Inflated ==
=Hemiug=  
[[Subgroup]]: 2.3.5.7
Commas: 128/125, 1323/1250


POTE generator: ~21/20 = 52.093
[[Comma list]]: 28/27, 128/125


Map: [&lt;3 1 7 14|, &lt;0 2 0 -3|]
{{Mapping|legend=1| 3 0 7 -6 | 0 1 0 3 }}
Wedgie: &lt;&lt;6 0 -9 -14 -31 -21||
EDOs: 21, 24, 45c, 69cd
Badness: 0.1378


==11-limit==
[[Optimal tuning]]s:
Commas: 56/55, 128/125, 1323/1250
* [[WE]]: ~5/4 = 398.4023{{c}}, ~3/2 = 719.8327{{c}} (~25/24 = 76.9719{{c}})
: [[error map]]: {{val| -3.762 +1.317 +4.909 +2.060 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 721.0196{{c}} (~25/24 = 78.9804{{c}})
: error map: {{val| 0.000 +19.065 +13.686 -5.767 }}


POTE generator: ~21/20 = 51.655
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}


Map: [&lt;3 1 7 14 16|, &lt;0 2 0 -3 -3|]
[[Badness]] (Sintel): 1.39
EDOs: 21, 24, 69cd, 93cd
Badness: 0.0681


==13-limit==  
=== 11-limit ===
Commas: 56/55, 66/65, 105/104, 507/500
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 51.548
Comma list: 28/27, 55/54, 128/125


Map: [&lt;3 1 7 14 16 13|, &lt;0 2 0 -3 -3 -1|]
Mapping: {{mapping| 3 0 7 -6 -4 | 0 1 0 3 3 }}
EDOs: 21, 24, 69cdf, 93cdf
Badness: 0.0423


=Trug=  
Optimal tunings:
Commas: 128/125, 360/343
* WE: ~5/4 = 398.4016{{c}}, ~3/2 = 719.7758{{c}} (~25/24 = 77.0275{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 720.9386{{c}} (~25/24 = 79.0614{{c}})


POTE generator: ~48/35 = 501.980
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}


Map: [&lt;3 1 7 6|, &lt;0 3 0 2|]
Badness (Sintel): 1.03
EDOs: 9bd, 12
Badness: 0.1383


=Oodako=  
== Deflated ==
Commas: 128/125, 2401/2400
[[Subgroup]]: 2.3.5.7


POTE generator: ~8/7 = 223.3540873874507317
[[Comma list]]: 21/20, 128/125


Map: [&lt;3 3 7 8|, &lt;0 4 0 1|]
{{Mapping|legend=1| 3 0 7 13 | 0 1 0 -1 }}
Wedgie: &lt;&lt;12 0 3 -28 -29 7||
EDOs: 6, 21, 27, 75c, 102cd, 129bcd
Badness: 0.1132


==11-limit==
[[Optimal tuning]]s:
Commas: 56/55, 128/125, 3773/3750
* [[WE]]: ~5/4 = 401.9566{{c}}, ~3/2 = 684.9634{{c}} (~16/15 = 118.9497{{c}})
: [[error map]]: {{val| +5.870 -11.122 +27.382 -34.224 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 682.2587{{c}} (~16/15 = 117.7413{{c}})
: error map: {{val| 0.000 -19.696 +13.686 -51.085 }}


POTE generator: ~8/7 = 223.019
{{Optimal ET sequence|legend=1| 3, 6b, 9 }}


Map: [&lt;3 3 7 8 10|, &lt;0 4 0 1 1|]
[[Badness]] (Sintel): 1.50
EDOs: 6, 21, 27e
Badness: 0.0592


==13-limit==  
=== 11-limit ===
Commas: 56/55, 78/77, 128/125, 507/500
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 223.449
Comma list: 21/20, 33/32, 128/125


Map: [&lt;3 3 7 8 10 12|, &lt;0 4 0 1 1 -2|]
Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}
EDOs: 6, 21, 27e
Badness: 0.0423


=Hemisemiaug=  
Optimal tunings:
Commas: 128/125, 12005/11664
* WE: ~5/4 = 402.1799{{c}}, ~3/2 = 683.7477{{c}} (~16/15 = 120.6120{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 680.0162{{c}} (~16/15 = 119.9838{{c}})


POTE generator: ~15/14 = 144.515
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}


Map: [&lt;6 1 14 4|, &lt;0 2 0 3|]
Badness (Sintel): 1.23
Wedgie: &lt;&lt;12 0 18 -28 -5 42||
Edos: 18, 24, 42, 66c, 108bc
Badness: 0.2110


==11-limit==  
== Hexe ==
Commas: 56/55, 128/125, 3773/3645
Hexe tempers out 50/49 and may be described as {{nowrap| 6 & 12 }}, viewed as [[6edo|6et]] with an independent generator for prime 3. Its ploidacot is hexaploid monocot.


POTE generator: ~15/14 = 144.780
[[Subgroup]]: 2.3.5.7


Map: [&lt;6 1 14 4 8|, &lt;0 2 0 3 3|]
[[Comma list]]: 50/49, 128/125
EDOs: 24, 42e, 66ce, 108bce
Badness: 0.0807


=Niner=
{{Mapping|legend=1| 6 0 14 17 | 0 1 0 0 }}
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.


Commas: 128/125, 686/675
: mapping generators: ~28/25, ~3


POTE generator: ~3/2 = 707.167
[[Optimal tuning]]s:
* [[WE]]: ~28/25 = 199.0488{{c}}, ~3/2 = 707.5815{{c}} (~25/24 = 88.6137{{c}})
: [[error map]]: {{val| +5.870 -11.122 +27.382 -34.224 }}
* [[CWE]]: ~28/25 = 200.0000{{c}}, ~3/2 = 708.6907{{c}} (~25/24 = 91.3093{{c}})
: error map: {{val| 0.000 +6.735 +13.686 +31.174 }}


Map: [&lt;9 0 21 11|, &lt;0 1 0 1|]
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}
Wedgie: &lt;&lt;9 0 9 -21 -11 21||
EDOs: 9, 18, 27, 63c, 90c
Badness: 0.0672


==11-limit==
[[Badness]] (Sintel): 1.46
Commas: 56/55, 128/125, 540/539


POTE generator: ~3/2 = 706.726
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 0 21 11 17|, &lt;0 1 0 1 1|]
Comma list: 50/49, 56/55, 125/121
EDOs: 9, 27e, 36, 63ce
Badness: 0.0349


==13-limit==
Mapping: {{mapping| 6 0 14 17 21 | 0 1 0 0 0 }}
Commas: 56/55, 78/77, 91/90, 128/125


POTE generator: ~3/2 = 706.889
Optimal tunings:  
* WE: ~28/25 = 198.6942{{c}}, ~3/2 = 709.6404{{c}} (~25/24 = 85.1362{{c}})
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 711.8043{{c}} (~25/24 = 88.1957{{c}})


Map: [&lt;9 0 21 11 17 19|, &lt;0 1 0 1 1 1|]
{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}
EDOs: 9, 27e, 36, 63ce
Badness: 0.0241


==Music==
Badness (Sintel): 1.27
[[https://www.prismnet.com/~hmiller/music/temp-augmented.html|Herman's Miller page about augmented temperament]]</pre></div>
 
<h4>Original HTML content:</h4>
=== 13-limit ===
<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;Augmented family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:78:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;div style="margin-left
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 56/55, 66/65, 105/104
 
Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}
 
Optimal tunings:
* WE: ~28/25 = 198.4492{{c}}, ~3/2 = 704.4994{{c}} (~25/24 = 89.2973{{c}})
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 706.6050{{c}} (~16/15 = 93.3950{{c}})
 
{{Optimal ET sequence|legend=0| 6f, 12f }}
 
Badness (Sintel): 1.49
 
== Triforce ==
[[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of triforce]]
 
Triforce tempers out 49/48 and may be described as {{nowrap| 9 & 15 }}. Its ploidacot is triploid alpha-dicot. [[24edo]] and [[39edo]] are among the possible tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 128/125
 
{{Mapping|legend=1| 3 0 7 6 | 0 2 0 1 }}
 
: mapping generators: ~5/4, ~7/4
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.7480{{c}}, ~7/4 = 952.3507{{c}} (~35/32 = 152.8547{{c}})
: [[error map]]: {{val| -0.756 +2.746 +11.922 -17.987 }}
* [[CWE]]: ~5/4 = 400.0000{{c
Retrieved from "https://en.xen.wiki/w/Augmented_family"