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The 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo]] is {{Monzo| 7 0 -3 }}, and flipping that yields <<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.
The 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo]] is {{Monzo|7 0 -3}}, and flipping that yields {{multival|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|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.
 
== Augmented ==
Subgroup: 2.3.5


== Augmented  ==
[[Comma]]: 128/125
[[Comma]]: 128/125


[[POTE tuning|POTE generator]]: ~3/2 = 706.638
[[Mapping]]: [{{val|3 0 7}}, {{val|0 1 0}}]


[[Map]]: [<3 0 7|, <0 1 0|]
[[POTE generator]]: ~3/2 = 706.638


{{Vals|legend=1| 3, 12, 27, 39, 51c, 90cc }}
{{Val list|legend=1| 3, 12, 27, 39, 51c, 90cc }}


[[Badness]]: 0.022315
[[Badness]]: 0.022315


=== Seven-limit children ===
=== Seven-limit children ===
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 triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.
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 triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.


== August ==
== August ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 36/35, 128/125
[[Comma list]]: 36/35, 128/125


[[POTE tuning|POTE generator]]: ~3/2 = 696.011
[[Mapping]]: [{{val|3 0 7 -1}}, {{val|0 1 0 2}}]


[[Map]]: [<3 0 7 -1|, <0 1 0 2|]
{{Multival|legend=1|3 0 6 -7 1 14}}


[[Wedgie]]: <<3 0 6 -7 1 14||
[[POTE generator]]: ~3/2 = 696.011


[[EDO|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]
{{Val list|legend=1| 9, 12, 45cd, 57cd, 69cdd }}


[[Badness]]: 0.026459
[[Badness]]: 0.026459


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 56/55
Comma list: 36/35, 45/44, 56/55
Mapping: [{{val|3 0 7 -1 1}}, {{val|0 1 0 2 2}}]


POTE generator: ~3/2 = 692.514
POTE generator: ~3/2 = 692.514


Map: [<3 0 7 -1 1|, <0 1 0 2 2|]
Vals: {{Val list| 9, 12, 21, 33e, 45cde }}


EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
Badness: 0.020191


Badness: 0.020191
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 27/26, 36/35, 45/44, 56/55
Comma list: 27/26, 36/35, 45/44, 56/55
Mapping: [{{val|3 0 7 -1 1 -3}}, {{val|0 1 0 2 2 3}}]


POTE generator: ~3/2 = 688.783
POTE generator: ~3/2 = 688.783


Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]
Vals: {{Val list| 9, 12f, 21, 33ef, 54bceeff }}


EDOs: 9, 12f, [[21edo|21]], [[33edo|33ef]], [[54edo|54bcef]]
Badness: 0.018448


Badness: 0.018448
=== Augustus ===
Subgroup: 2.3.5.7.11.13


=== Augustus  ===
Comma list: 26/25, 36/35, 45/44, 56/55
Comma list: 26/25, 36/35, 45/44, 56/55
Mapping: [{{val|3 0 7 -1 1 11}}, {{val|0 1 0 2 2 0}}]


POTE generator: ~3/2 = 685.356
POTE generator: ~3/2 = 685.356


Map: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]
Vals: {{Val list| 9, 12, 21f }}


EDOs: 9, 12, 21f
Badness: 0.022236


Badness: 0.022236
== Augene ==
Subgroup: 2.3.5.7


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


[[POTE_tuning|POTE generator]]: ~3/2 = 709.257
[[Mapping]]: [{{val|3 0 7 18}}, {{val|0 1 0 -2}}]


Map: [<3 0 7 18|, <0 1 0 -2|]
{{Multival|legend=1|3 0 -6 -7 -18 -14}}


Wedgie: <<3 0 -6 -7 -18 -14||
[[POTE generator]]: ~3/2 = 709.257


EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]
{{Val list|legend=1| 12, 27, 39d, 66cd }}


Badness: 0.024816
[[Badness]]: 0.024816
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 64/63, 100/99
Comma list: 56/55, 64/63, 100/99
Mapping: [{{val|3 0 7 18 20}}, {{val|0 1 0 -2 -2}}]


POTE generator: ~3/2 = 711.177
POTE generator: ~3/2 = 711.177


Map: [<3 0 7 18 20|, <0 1 0 -2 -2|]
Vals: {{Val list| 12, 15, 27e }}


EDOs: 12, 15, 27e
Badness: 0.019613


Badness: 0.019613
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 40/39, 56/55, 64/63, 66/65
Comma list: 40/39, 56/55, 64/63, 66/65
Mapping: [{{val|3 0 7 18 20 16}}, {{val|0 1 0 -2 -2 -1}}]


POTE generator: ~3/2 = 712.013
POTE generator: ~3/2 = 712.013


Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]
Vals: {{Val list| 12f, 15, 27ef }}


EDOs: 12f, 15, 27ef
Badness: 0.020785


Badness: 0.020785
==== Ogene ====
Subgroup: 2.3.5.7.11.13


=== Ogene  ===
Comma list: 56/55, 64/63, 91/90, 100/99
Comma list: 56/55, 64/63, 91/90, 100/99
Mapping: [{{val|3 0 7 18 20 -8}}, {{val|0 1 0 -2 -2 4}}]


POTE generator: ~3/2 = 712.609
POTE generator: ~3/2 = 712.609


Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]
Vals: {{Val list| 12, 15, 27e, 69bceef }}


EDOs: 12, 15, 27e, 69bcef
Badness: 0.022890


Badness: 0.022890
==== Agene ====
Subgroup: 2.3.5.7.11.13


=== Agene  ===
Comma list: 56/55, 64/63, 78/77, 100/99
Comma list: 56/55, 64/63, 78/77, 100/99
Mapping: [{{val|3 0 7 18 20 35}}, {{val|0 1 0 -2 -2 -5}}]


POTE generator: ~3/2 = 709.677
POTE generator: ~3/2 = 709.677


Map: [<3 0 7 18 20 35|, <0 1 0 -2 -2 -5|]
Vals: {{Val list| 12f, 27e }}


EDOs: 12f, 27e
Badness: 0.023113


Badness: 0.023113
=== Eugene ===
Subgroup: 2.3.5.7.11


=== Eugene  ===
Comma list: 55/54, 64/63, 77/75
Comma list: 55/54, 64/63, 77/75
Mapping: [{{val|3 0 7 18 -4}}, {{val|0 1 0 -2 3}}]


POTE generator: ~3/2 = 714.150
POTE generator: ~3/2 = 714.150


Map: [<3 0 7 18 -4|, <0 1 0 -2 3|]
Vals: {{Val list| 15, 27, 42 }}
 
EDOs: 15, 27, 42


Badness: 0.035554
Badness: 0.035554
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* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor]


== Inflated ==
== Inflated ==
Comma list: 28/27, 128/125
Comma list: 28/27, 128/125


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Badness: 0.054729
Badness: 0.054729


=== 11-limit ===
=== 11-limit ===
Comma list: 28/27, 55/54, 128/125
Comma list: 28/27, 55/54, 128/125


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