Augmented family: Difference between revisions

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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


Line 146: Line 168:
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


Revision as of 10:17, 6 June 2021

The 5-limit parent comma for the augmented family is 128/125, the diesis. Its monzo is [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.

Augmented

Subgroup: 2.3.5

Comma: 128/125

Mapping: [⟨3 0 7], ⟨0 1 0]]

POTE generator: ~3/2 = 706.638

Template:Val list

Badness: 0.022315

Seven-limit children

The second comma of the 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

Subgroup: 2.3.5.7

Comma list: 36/35, 128/125

Mapping: [⟨3 0 7 -1], ⟨0 1 0 2]]

Wedgie: ⟨⟨ 3 0 6 -7 1 14 ]]

POTE generator: ~3/2 = 696.011

Template:Val list

Badness: 0.026459

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 56/55

Mapping: [⟨3 0 7 -1 1], ⟨0 1 0 2 2]]

POTE generator: ~3/2 = 692.514

Vals: Template:Val list

Badness: 0.020191

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 56/55

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

POTE generator: ~3/2 = 688.783

Vals: Template:Val list

Badness: 0.018448

Augustus

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 45/44, 56/55

Mapping: [⟨3 0 7 -1 1 11], ⟨0 1 0 2 2 0]]

POTE generator: ~3/2 = 685.356

Vals: Template:Val list

Badness: 0.022236

Augene

Subgroup: 2.3.5.7

Comma list: 64/63, 126/125

Mapping: [⟨3 0 7 18], ⟨0 1 0 -2]]

Wedgie: ⟨⟨ 3 0 -6 -7 -18 -14 ]]

POTE generator: ~3/2 = 709.257

Template:Val list

Badness: 0.024816

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 100/99

Mapping: [⟨3 0 7 18 20], ⟨0 1 0 -2 -2]]

POTE generator: ~3/2 = 711.177

Vals: Template:Val list

Badness: 0.019613

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 64/63, 66/65

Mapping: [⟨3 0 7 18 20 16], ⟨0 1 0 -2 -2 -1]]

POTE generator: ~3/2 = 712.013

Vals: Template:Val list

Badness: 0.020785

Ogene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 91/90, 100/99

Mapping: [⟨3 0 7 18 20 -8], ⟨0 1 0 -2 -2 4]]

POTE generator: ~3/2 = 712.609

Vals: Template:Val list

Badness: 0.022890

Agene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 100/99

Mapping: [⟨3 0 7 18 20 35], ⟨0 1 0 -2 -2 -5]]

POTE generator: ~3/2 = 709.677

Vals: Template:Val list

Badness: 0.023113

Eugene

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 77/75

Mapping: [⟨3 0 7 18 -4], ⟨0 1 0 -2 3]]

POTE generator: ~3/2 = 714.150

Vals: Template:Val list

Badness: 0.035554

Inflated

Comma list: 28/27, 128/125

POTE generator: ~3/2 = 722.719

Map: [<3 0 7 -6|, <0 1 0 3|]

Wedgie: <<3 0 9 -7 6 21||

EDOs: 3d, 15, 48bc, 63bc, 78bc

Badness: 0.054729

11-limit

Comma list: 28/27, 55/54, 128/125

POTE generator: ~3/2 = 722.663

Map: [<3 0 7 -6 -4|, <0 1 0 3 3|]

EDOs: 3de, 15, 48bce, 63bce, 78bce

Badness: 0.031171

Deflated

Comma list: 21/20, 128/125

POTE generator: ~3/2 = 681.629

Map: [<3 0 7 13|, <0 1 0 -1|]

Wedgie: <<3 0 -3 -7 -13 -7||

EDOs: 3, 9, 21d, 30bd

Badness: 0.059079

11-limit

Comma list: 21/20, 33/32, 128/125

POTE generator: ~3/2 = 680.042

Map: [<3 0 7 13 15|, <0 1 0 -1 -1|]

EDOs: 3, 9, 21de, 30bde

Badness: 0.037183

Hexe

Comma list: 50/49, 128/125

POTE generator: ~3/2 = 710.963

Map: [<6 0 14 17|, <0 1 0 0|]

Wedgie: <<6 0 0 -14 -17 0||

EDOs: 6, 12, 30d, 42d, 54cd

Badness: 0.057730

11-limit

Comma list: 50/49, 56/55, 125/121

POTE generator: ~3/2 = 714.304

Map: [<6 0 14 17 21|, <0 1 0 0 0|]

EDOs: 6, 12, 30de

Badness: 0.038412

13-limit

Comma list: 50/49, 56/55, 66/65, 105/104

POTE generator: ~3/2 = 710.005

Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]

EDOs: 6f, 12f

Badness: 0.035946

Hemiaug

Comma list: 128/125, 245/243

POTE generator: ~28/27 = 45.118

Map: [<3 1 7 -1|, <0 2 0 5|]

Wedgie: <<6 0 15 -14 7 35||

EDOs: 24, 27, 105bc

Badness: 0.070463

11-limit

Comma list: 56/55, 128/125, 245/243

POTE generator: ~28/27 = 45.788

Map: [<3 1 7 -1 1|, <0 2 0 5 5|]

EDOs: 24, 27e, 51ce, 78ce

Badness: 0.038232

13-limit

Comma list: 56/55, 91/90, 128/125, 245/243

POTE generator: ~28/27 = 46.250

Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]

EDOs: 24, 27e, 51ce, 78ce

Badness: 0.030159

Triforce

Comma list: 49/48, 128/125

POTE generator: ~7/6 = 247.049

Map: [<3 0 7 6|, <0 2 0 1|]

Wedgie: <<6 0 3 -14 -12 7||

EDOs: 6, 9, 15, 24, 39, 63cd, 102cd

Badness: 0.054988

triforce9.jpg

11-limit

Comma list: 56/55, 77/75, 128/125

POTE generator: ~7/6 = 247.068

Map: [<3 0 7 6 8|, <0 2 0 1 1|]

EDOs: 6, 9, 15, 24, 39, 63cd, 102cd

Badness: 0.026152

Musical example

13-limit

Comma list: 49/48, 56/55, 66/65, 77/75

POTE generator: ~7/6 = 248.313

Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]

EDOs: 6, 9, 15, 24, 63cdf, 87cdf

Badness: 0.020248

Triforce(9) with L:s = phi

Semitriforce

Comma list: 49/48, 56/55, 77/75, 507/500

POTE generator: ~7/6 = 246.642

Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]

EDOs: 6, 24, 54cd, 78cd, 102cdf

Badness: 0.059160


Hemiug

Comma list: 128/125, 1323/1250

POTE generator: ~21/20 = 52.093

Map: [<3 1 7 14|, <0 2 0 -3|]

Wedgie: <<6 0 -9 -14 -31 -21||

EDOs: 21, 24, 45c, 69cd

Badness: 0.137764

11-limit

Comma list: 56/55, 128/125, 1323/1250

POTE generator: ~21/20 = 51.655

Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]

EDOs: 21, 24, 69cd, 93cd

Badness: 0.068062

13-limit

Comma list: 56/55, 66/65, 105/104, 507/500

POTE generator: ~21/20 = 51.548

Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]

EDOs: 21, 24, 69cdf, 93cdf

Badness: 0.042289

Trug

Comma list: 128/125, 360/343

POTE generator: ~48/35 = 501.980

Map: [<3 1 7 6|, <0 3 0 2|]

EDOs: 9bd, 12

Badness: 0.138279

Oodako

Comma list: 128/125, 2401/2400

POTE generator: ~8/7 = 223.3540873874507317

Map: [<3 3 7 8|, <0 4 0 1|]

Wedgie: <<12 0 3 -28 -29 7||

EDOs: 6, 21, 27, 75c, 102cd, 129bcd

Badness: 0.113209

11-limit

Comma list: 56/55, 128/125, 3773/3750

POTE generator: ~8/7 = 223.019

Map: [<3 3 7 8 10|, <0 4 0 1 1|]

EDOs: 6, 21, 27e

Badness: 0.059192

13-limit

Comma list: 56/55, 78/77, 128/125, 507/500

POTE generator: ~8/7 = 223.449

Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]

EDOs: 6, 21, 27e

Badness: 0.042323

Hemisemiaug

Comma list: 128/125, 12005/11664

POTE generator: ~15/14 = 144.515

Map: [<6 1 14 4|, <0 2 0 3|]

Wedgie: <<12 0 18 -28 -5 42||

Edos: 18, 24, 42, 66c, 108bc

Badness: 0.210984

11-limit

Comma list: 56/55, 128/125, 3773/3645

POTE generator: ~15/14 = 144.780

Map: [<6 1 14 4 8|, <0 2 0 3 3|]

EDOs: 24, 42e, 66ce, 108bce

Badness: 0.080738

Niner

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.

Comma list: 128/125, 686/675

POTE generator: ~3/2 = 707.167

Map: [<9 0 21 11|, <0 1 0 1|]

Wedgie: <<9 0 9 -21 -11 21||

EDOs: 9, 18, 27, 63c, 90c

Badness: 0.067157

11-limit

Comma list: 56/55, 128/125, 540/539

POTE generator: ~3/2 = 706.726

Map: [<9 0 21 11 17|, <0 1 0 1 1|]

EDOs: 9, 27e, 36, 63ce

Badness: 0.034861

13-limit

Comma list: 56/55, 78/77, 91/90, 128/125

POTE generator: ~3/2 = 706.889

Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]

EDOs: 9, 27e, 36, 63ce

Badness: 0.024143

Music

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