Augmented family: Difference between revisions

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Badness: 0.035554
Badness: 0.035554


; Music
* [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]
* [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]
* [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
Subgroup: 2.3.5.7
 
[[Comma list]]: 28/27, 128/125


POTE generator: ~3/2 = 722.719
[[Mapping]]: [{{val|3 0 7 -6}}, {{val|0 1 0 3}}]


Map: [<3 0 7 -6|, <0 1 0 3|]
{{Multival|legend=1|3 0 9 -7 6 21}}


Wedgie: <<3 0 9 -7 6 21||
[[POTE generator]]: ~3/2 = 722.719


EDOs: 3d, 15, 48bc, 63bc, 78bc
{{Val list|legend=1| 3d, 15, 48bc, 63bbc, 78bbc }}


Badness: 0.054729
[[Badness]]: 0.054729


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 28/27, 55/54, 128/125
Comma list: 28/27, 55/54, 128/125
Mapping: [{{val|3 0 7 -6 -4}}, {{val|0 1 0 3 3}}]


POTE generator: ~3/2 = 722.663
POTE generator: ~3/2 = 722.663


Map: [<3 0 7 -6 -4|, <0 1 0 3 3|]
Vals: {{Val list| 3de, 15, 48bce, 63bbcee, 78bbcee }}


EDOs: 3de, 15, 48bce, 63bce, 78bce
Badness: 0.031171


Badness: 0.031171
== Deflated ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 21/20, 128/125


== Deflated  ==
[[Mapping]]: [{{val|3 0 7 13}}, {{val|0 1 0 -1}}]
Comma list: 21/20, 128/125


POTE generator: ~3/2 = 681.629
{{Multival|legend=1|3 0 -3 -7 -13 -7}}


Map: [<3 0 7 13|, <0 1 0 -1|]
[[POTE generator]]: ~3/2 = 681.629


Wedgie: <<3 0 -3 -7 -13 -7||
{{Val list|legend=1| 3, 9, 21d, 30bdd }}


EDOs: 3, 9, 21d, 30bd
[[Badness]]: 0.059079


Badness: 0.059079
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 21/20, 33/32, 128/125
Comma list: 21/20, 33/32, 128/125
Mapping: [{{val|3 0 7 13 15}}, {{val|0 1 0 -1 -1}}]


POTE generator: ~3/2 = 680.042
POTE generator: ~3/2 = 680.042


Map: [<3 0 7 13 15|, <0 1 0 -1 -1|]
Vals: {{Val list| 3, 9, 21de, 30bdde }}
 
Badness: 0.037183


EDOs: 3, 9, 21de, 30bde
== Hexe ==
Subgroup: 2.3.5.7


Badness: 0.037183
[[Comma list]]: 50/49, 128/125


== Hexe  ==
[[Mapping]]: [{{val|6 0 14 17}}, {{val|0 1 0 0}}]
Comma list: 50/49, 128/125


[[POTE_tuning|POTE generator]]: ~3/2 = 710.963
{{Multival|legend=1|6 0 0 -14 -17 0}}


Map: [<6 0 14 17|, <0 1 0 0|]
[[POTE generator]]: ~3/2 = 710.963


Wedgie: <<6 0 0 -14 -17 0||
{{Val list|legend=1| 6, 12, 30d, 42dd, 54cdd }}


EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[42edo|42d]], [[54edo|54cd]]
[[Badness]]: 0.057730


Badness: 0.057730
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 50/49, 56/55, 125/121
Comma list: 50/49, 56/55, 125/121
Mapping: [{{val|6 0 14 17 21}}, {{val|0 1 0 0 0}}]


POTE generator: ~3/2 = 714.304
POTE generator: ~3/2 = 714.304


Map: [<6 0 14 17 21|, <0 1 0 0 0|]
Vals: {{Val list| 6, 12, 30de }}


EDOs: 6, 12, 30de
Badness: 0.038412


Badness: 0.038412
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 50/49, 56/55, 66/65, 105/104
Comma list: 50/49, 56/55, 66/65, 105/104
Mapping: [{{val|6 0 14 17 21 13}}, {{val|0 1 0 0 0 1}}]


POTE generator: ~3/2 = 710.005
POTE generator: ~3/2 = 710.005


Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]
Vals: {{Val list| 6f, 12f }}


EDOs: 6f, 12f
Badness: 0.035946


Badness: 0.035946
== Hemiaug ==
Subgroup: 2.3.5.7


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


[[POTE_tuning|POTE generator]]: ~28/27 = 45.118
[[Mapping]]: [{{val|3 1 7 -1}}, {{val|0 2 0 5}}]


Map: [<3 1 7 -1|, <0 2 0 5|]
{{Multival|legend=1|6 0 15 -14 7 35}}


Wedgie: <<6 0 15 -14 7 35||
[[POTE generator]]: ~28/27 = 45.118


EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
{{Val list|legend=1| 24, 27, 105bcc }}


Badness: 0.070463
[[Badness]]: 0.070463
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 128/125, 245/243
Comma list: 56/55, 128/125, 245/243
Mapping: [{{val|3 1 7 -1 1}}, {{val|0 2 0 5 5}}]


POTE generator: ~28/27 = 45.788
POTE generator: ~28/27 = 45.788


Map: [<3 1 7 -1 1|, <0 2 0 5 5|]
Vals: {{Val list| 24, 27e, 51ce, 78cee }}


EDOs: 24, 27e, 51ce, 78ce
Badness: 0.038232


Badness: 0.038232
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 56/55, 91/90, 128/125, 245/243
Comma list: 56/55, 91/90, 128/125, 245/243
Mapping: [{{val|3 1 7 -1 1 13}}, {{val|0 2 0 5 5 -1}}]


POTE generator: ~28/27 = 46.250
POTE generator: ~28/27 = 46.250


Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]
Vals: {{Val list| 24, 27e, 51ce, 78cee }}
 
EDOs: 24, 27e, 51ce, 78ce


Badness: 0.030159
Badness: 0.030159


== Triforce ==
== Triforce ==
Comma list: 49/48, 128/125
[[Comma list]]: 49/48, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.049
[[POTE generator]]: ~7/6 = 247.049


Map: [<3 0 7 6|, <0 2 0 1|]
Map: [<3 0 7 6|, <0 2 0 1|]
Line 288: Line 308:
[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]


=== 11-limit ===
=== 11-limit ===
Comma list: 56/55, 77/75, 128/125
Comma list: 56/55, 77/75, 128/125


Line 299: Line 319:
Badness: 0.026152
Badness: 0.026152


==== Musical example ====
; Musical example
* [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones
* [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones


=== 13-limit ===
=== 13-limit ===
Comma list: 49/48, 56/55, 66/65, 77/75
Comma list: 49/48, 56/55, 66/65, 77/75


Line 315: Line 335:
[[triphi|Triforce(9) with L:s = phi]]
[[triphi|Triforce(9) with L:s = phi]]


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


Revision as of 11:25, 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

Music

Inflated

Subgroup: 2.3.5.7

Comma list: 28/27, 128/125

Mapping: [⟨3 0 7 -6], ⟨0 1 0 3]]

Wedgie: ⟨⟨ 3 0 9 -7 6 21 ]]

POTE generator: ~3/2 = 722.719

Template:Val list

Badness: 0.054729

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 7 -6 -4], ⟨0 1 0 3 3]]

POTE generator: ~3/2 = 722.663

Vals: Template:Val list

Badness: 0.031171

Deflated

Subgroup: 2.3.5.7

Comma list: 21/20, 128/125

Mapping: [⟨3 0 7 13], ⟨0 1 0 -1]]

Wedgie: ⟨⟨ 3 0 -3 -7 -13 -7 ]]

POTE generator: ~3/2 = 681.629

Template:Val list

Badness: 0.059079

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 7 13 15], ⟨0 1 0 -1 -1]]

POTE generator: ~3/2 = 680.042

Vals: Template:Val list

Badness: 0.037183

Hexe

Subgroup: 2.3.5.7

Comma list: 50/49, 128/125

Mapping: [⟨6 0 14 17], ⟨0 1 0 0]]

Wedgie: ⟨⟨ 6 0 0 -14 -17 0 ]]

POTE generator: ~3/2 = 710.963

Template:Val list

Badness: 0.057730

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨6 0 14 17 21], ⟨0 1 0 0 0]]

POTE generator: ~3/2 = 714.304

Vals: Template:Val list

Badness: 0.038412

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨6 0 14 17 21 13], ⟨0 1 0 0 0 1]]

POTE generator: ~3/2 = 710.005

Vals: Template:Val list

Badness: 0.035946

Hemiaug

Subgroup: 2.3.5.7

Comma list: 128/125, 245/243

Mapping: [⟨3 1 7 -1], ⟨0 2 0 5]]

Wedgie: ⟨⟨ 6 0 15 -14 7 35 ]]

POTE generator: ~28/27 = 45.118

Template:Val list

Badness: 0.070463

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 1 7 -1 1], ⟨0 2 0 5 5]]

POTE generator: ~28/27 = 45.788

Vals: Template:Val list

Badness: 0.038232

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 1 7 -1 1 13], ⟨0 2 0 5 5 -1]]

POTE generator: ~28/27 = 46.250

Vals: Template:Val list

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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