Augmented family: Difference between revisions

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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 =
[[Comma]]s: 36/35, 128/125
[[Comma list]]: 36/35, 128/125


[[POTE tuning|POTE generator]]: 696.011
[[POTE tuning|POTE generator]]: 696.011
Line 17: Line 17:
[[EDO|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]
[[EDO|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]


[[Badness]]: 0.0265
[[Badness]]: 0.026459


=== 11-limit ===
== 11-limit ==
Commas: 36/35, 45/44, 56/55
Comma list: 36/35, 45/44, 56/55


POTE generator: ~3/2 = 692.514
POTE generator: ~3/2 = 692.514
Line 28: Line 28:
EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]


Badness: 0.0202
Badness: 0.020191


=== 13-limit ===
== 13-limit ==
Commas: 27/26, 36/35, 45/44, 56/55
Comma list: 27/26, 36/35, 45/44, 56/55


POTE generator: ~3/2 = 688.783
POTE generator: ~3/2 = 688.783
Line 37: Line 37:
Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]
Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]


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


Badness: 0.0184
Badness: 0.018448


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


POTE generator: ~3/2 = 685.356
POTE generator: ~3/2 = 685.356
Line 50: Line 50:
EDOs: 9, 12, 21f
EDOs: 9, 12, 21f


Badness: 0.0222
Badness: 0.022236


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


[[POTE_tuning|POTE generator]]: 709.257
[[POTE_tuning|POTE generator]]: 709.257
Line 63: Line 63:
EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]
EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]


Badness: 0.0248
Badness: 0.024816


=== 11-limit ===
== 11-limit ==
Commas: 56/55, 64/63, 100/99
Comma list: 56/55, 64/63, 100/99


POTE generator: ~3/2 = 711.177
POTE generator: ~3/2 = 711.177
Line 74: Line 74:
EDOs: 12, 15, 27e
EDOs: 12, 15, 27e


Badness: 0.0196
Badness: 0.019613


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


POTE generator: ~3/2 = 712.013
POTE generator: ~3/2 = 712.013
Line 85: Line 85:
EDOs: 12f, 15, 27ef
EDOs: 12f, 15, 27ef


Badness: 0.0208
Badness: 0.020785


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


POTE generator: ~3/2 = 712.609
POTE generator: ~3/2 = 712.609
Line 96: Line 96:
EDOs: 12, 15, 27e, 69bcef
EDOs: 12, 15, 27e, 69bcef


Badness: 0.0229
Badness: 0.022890


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


POTE generator: ~3/2 = 709.677
POTE generator: ~3/2 = 709.677
Line 107: Line 107:
EDOs: 12f, 27e
EDOs: 12f, 27e


Badness: 0.0231
Badness: 0.023113


=== Eugene ===
== Eugene ==
Commas: 55/54, 64/63, 77/75
Comma list: 55/54, 64/63, 77/75


POTE generator: ~3/2 = 714.150
POTE generator: ~3/2 = 714.150
Line 118: Line 118:
EDOs: 15, 27, 42
EDOs: 15, 27, 42


Badness: 0.0356
Badness: 0.035554


* [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]
Line 124: Line 124:
* [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 =
Commas: 28/27, 128/125
Comma list: 28/27, 128/125


POTE generator: ~3/2 = 722.719
POTE generator: ~3/2 = 722.719
Line 133: Line 133:
Wedgie: <<3 0 9 -7 6 21||
Wedgie: <<3 0 9 -7 6 21||


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


Badness: 0.0547
Badness: 0.054729


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


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


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


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


Badness: 0.0312
Badness: 0.031171


== Deflated ==
= Deflated =
Commas: 21/20, 128/125
Comma list: 21/20, 128/125


POTE generator: ~3/2 = 681.629
POTE generator: ~3/2 = 681.629
Line 157: Line 157:
Wedgie: <<3 0 -3 -7 -13 -7||
Wedgie: <<3 0 -3 -7 -13 -7||


EDOs: 3, 9, 18bd, 21d, 30bd
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: 0.0591
Badness: 0.037183


== Hexe ==
= Hexe =
Commas: 50/49, 128/125
Comma list: 50/49, 128/125


[[POTE_tuning|POTE generator]]: 710.963
[[POTE_tuning|POTE generator]]: 710.963
Line 172: Line 183:
EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[42edo|42d]], [[54edo|54cd]]
EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[42edo|42d]], [[54edo|54cd]]


Badness: 0.0577
Badness: 0.057730


=== 11-limit ===
== 11-limit ==
Commas: 50/49, 56/55, 125/121
Comma list: 50/49, 56/55, 125/121


POTE generator: ~3/2 = 714.304
POTE generator: ~3/2 = 714.304
Line 183: Line 194:
EDOs: 6, 12, 30de
EDOs: 6, 12, 30de


Badness: 0.0384
Badness: 0.038412


=== 13-limit ===
== 13-limit ==
Commas: 50/49, 56/55, 66/65, 105/104
Comma list: 50/49, 56/55, 66/65, 105/104


POTE generator: ~3/2 = 710.005
POTE generator: ~3/2 = 710.005
Line 192: Line 203:
Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]
Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]


EDOs: 12f
EDOs: 6f, 12f


Badness: 0.0359
Badness: 0.035946


== Hemiaug ==
= Hemiaug =
Commas: 128/125, 245/243
Comma list: 128/125, 245/243


[[POTE_tuning|POTE generator]]: ~28/27 = 45.118
[[POTE_tuning|POTE generator]]: ~28/27 = 45.118
Line 207: Line 218:
EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]


Badness: 0.0705
Badness: 0.070463


=== 11-limit ===
== 11-limit ==
Commas: 56/55, 128/125, 245/243
Comma list: 56/55, 128/125, 245/243


POTE generator: ~28/27 = 45.788
POTE generator: ~28/27 = 45.788
Line 218: Line 229:
EDOs: 24, 27e, 51ce, 78ce
EDOs: 24, 27e, 51ce, 78ce


Badness: 0.0382
Badness: 0.038232


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


POTE generator: ~28/27 = 46.250
POTE generator: ~28/27 = 46.250
Line 229: Line 240:
EDOs: 24, 27e, 51ce, 78ce
EDOs: 24, 27e, 51ce, 78ce


Badness: 0.0302
Badness: 0.030159


== Triforce ==
= Triforce =
Commas: 49/48, 128/125
Comma list: 49/48, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.049
[[POTE_tuning|POTE generator]]: ~7/6 = 247.049
Line 242: Line 253:
EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]


Badness: 0.0202
Badness: 0.054988


[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]


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


[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
Line 255: Line 266:
EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]


Badness: 0.0262
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 ==
Commas: 49/48, 56/55, 66/65, 77/75
Comma list: 49/48, 56/55, 66/65, 77/75


[[POTE_tuning|POTE generator]]: ~7/6 = 248.313
[[POTE_tuning|POTE generator]]: ~7/6 = 248.313
Line 269: Line 280:
EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]


Badness: 0.0202
Badness: 0.020248


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


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


POTE generator: ~7/6 = 246.642
POTE generator: ~7/6 = 246.642
Line 282: Line 293:
EDOs: 6, 24, 54cd, 78cd, 102cdf
EDOs: 6, 24, 54cd, 78cd, 102cdf


Badness: 0.0592
Badness: 0.059160




== Hemiug ==
= Hemiug =
Commas: 128/125, 1323/1250
Comma list: 128/125, 1323/1250


POTE generator: ~21/20 = 52.093
POTE generator: ~21/20 = 52.093
Line 296: Line 307:
EDOs: 21, 24, 45c, 69cd
EDOs: 21, 24, 45c, 69cd


Badness: 0.1378
Badness: 0.137764


=== 11-limit ===
== 11-limit ==
Commas: 56/55, 128/125, 1323/1250
Comma list: 56/55, 128/125, 1323/1250


POTE generator: ~21/20 = 51.655
POTE generator: ~21/20 = 51.655
Line 307: Line 318:
EDOs: 21, 24, 69cd, 93cd
EDOs: 21, 24, 69cd, 93cd


Badness: 0.0681
Badness: 0.068062


=== 13-limit ===
== 13-limit ==
Commas: 56/55, 66/65, 105/104, 507/500
Comma list: 56/55, 66/65, 105/104, 507/500


POTE generator: ~21/20 = 51.548
POTE generator: ~21/20 = 51.548
Line 318: Line 329:
EDOs: 21, 24, 69cdf, 93cdf
EDOs: 21, 24, 69cdf, 93cdf


Badness: 0.0423
Badness: 0.042289


== Trug ==
= Trug =
Commas: 128/125, 360/343
Comma list: 128/125, 360/343


POTE generator: ~48/35 = 501.980
POTE generator: ~48/35 = 501.980
Line 329: Line 340:
EDOs: 9bd, 12
EDOs: 9bd, 12


Badness: 0.1383
Badness: 0.138279


== Oodako ==
= Oodako =
Commas: 128/125, 2401/2400
Comma list: 128/125, 2401/2400


POTE generator: ~8/7 = 223.3540873874507317
POTE generator: ~8/7 = 223.3540873874507317
Line 342: Line 353:
EDOs: 6, 21, 27, 75c, 102cd, 129bcd
EDOs: 6, 21, 27, 75c, 102cd, 129bcd


Badness: 0.1132
Badness: 0.113209


=== 11-limit ===
== 11-limit ==
Commas: 56/55, 128/125, 3773/3750
Comma list: 56/55, 128/125, 3773/3750


POTE generator: ~8/7 = 223.019
POTE generator: ~8/7 = 223.019
Line 353: Line 364:
EDOs: 6, 21, 27e
EDOs: 6, 21, 27e


Badness: 0.0592
Badness: 0.059192


=== 13-limit ===
== 13-limit ==
Commas: 56/55, 78/77, 128/125, 507/500
Comma list: 56/55, 78/77, 128/125, 507/500


POTE generator: ~8/7 = 223.449
POTE generator: ~8/7 = 223.449
Line 364: Line 375:
EDOs: 6, 21, 27e
EDOs: 6, 21, 27e


Badness: 0.0423
Badness: 0.042323


== Hemisemiaug ==
= Hemisemiaug =
Commas: 128/125, 12005/11664
Comma list: 128/125, 12005/11664


POTE generator: ~15/14 = 144.515
POTE generator: ~15/14 = 144.515
Line 377: Line 388:
Edos: 18, 24, 42, 66c, 108bc
Edos: 18, 24, 42, 66c, 108bc


Badness: 0.2110
Badness: 0.210984


=== 11-limit ===
== 11-limit ==
Commas: 56/55, 128/125, 3773/3645
Comma list: 56/55, 128/125, 3773/3645


POTE generator: ~15/14 = 144.780
POTE generator: ~15/14 = 144.780
Line 388: Line 399:
EDOs: 24, 42e, 66ce, 108bce
EDOs: 24, 42e, 66ce, 108bce


Badness: 0.0807
Badness: 0.080738


== Niner ==
= 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.
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
Comma list: 128/125, 686/675


POTE generator: ~3/2 = 707.167
POTE generator: ~3/2 = 707.167
Line 403: Line 414:
EDOs: 9, 18, 27, 63c, 90c
EDOs: 9, 18, 27, 63c, 90c


Badness: 0.0672
Badness: 0.067157


=== 11-limit ===
== 11-limit ==
Commas: 56/55, 128/125, 540/539
Comma list: 56/55, 128/125, 540/539


POTE generator: ~3/2 = 706.726
POTE generator: ~3/2 = 706.726
Line 414: Line 425:
EDOs: 9, 27e, 36, 63ce
EDOs: 9, 27e, 36, 63ce


Badness: 0.0349
Badness: 0.034861


=== 13-limit ===
== 13-limit ==
Commas: 56/55, 78/77, 91/90, 128/125
Comma list: 56/55, 78/77, 91/90, 128/125


POTE generator: ~3/2 = 706.889
POTE generator: ~3/2 = 706.889
Line 425: Line 436:
EDOs: 9, 27e, 36, 63ce
EDOs: 9, 27e, 36, 63ce


Badness: 0.0241
Badness: 0.024143


== Music ==
= Music =
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]


Revision as of 02:18, 3 April 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.

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

Comma list: 36/35, 128/125

POTE generator: 696.011

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

Wedgie: <<3 0 6 -7 1 14||

EDOs: 9, 12, 45cd, 57cd, 69cd

Badness: 0.026459

11-limit

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

POTE generator: ~3/2 = 692.514

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

EDOs: 9, 12, 21, 33e, 45cde

Badness: 0.020191

13-limit

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

POTE generator: ~3/2 = 688.783

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

EDOs: 9, 12f, 21, 33ef, 54bcef

Badness: 0.018448

Augustus

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

POTE generator: ~3/2 = 685.356

Map: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]

EDOs: 9, 12, 21f

Badness: 0.022236

Augene

Comma list: 64/63, 126/125

POTE generator: 709.257

Map: [<3 0 7 18|, <0 1 0 -2|]

Wedgie: <<3 0 -6 -7 -18 -14||

EDOs: 12, 27, 39d, 66cd

Badness: 0.024816

11-limit

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

POTE generator: ~3/2 = 711.177

Map: [<3 0 7 18 20|, <0 1 0 -2 -2|]

EDOs: 12, 15, 27e

Badness: 0.019613

13-limit

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

POTE generator: ~3/2 = 712.013

Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]

EDOs: 12f, 15, 27ef

Badness: 0.020785

Ogene

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

POTE generator: ~3/2 = 712.609

Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]

EDOs: 12, 15, 27e, 69bcef

Badness: 0.022890

Agene

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

POTE generator: ~3/2 = 709.677

Map: [<3 0 7 18 20 35|, <0 1 0 -2 -2 -5|]

EDOs: 12f, 27e

Badness: 0.023113

Eugene

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

POTE generator: ~3/2 = 714.150

Map: [<3 0 7 18 -4|, <0 1 0 -2 3|]

EDOs: 15, 27, 42

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: 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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