Augmented family: Difference between revisions

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<span style="display: block; text-align: right;">
__FORCETOC__
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[[:de:übermässige_Temperaturen|Deutsch]]</span>
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 &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.


The 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo|monzo]] is |7 0 -3&gt;, and flipping that yields &lt;&lt;3 0 -7|| for the [[wedgie|wedgie]]. Hence the period is 1/3 octave, and this is what is used for a major third. The [[generator|generator]] can be taken as a fifth or a semitone, and [[12edo|12edo]], with its excellent fifth, is an obvious tuning for [[5-limit|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 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.


=Seven limit children=
== August ==
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit|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.
[[Comma]]s: 36/35, 128/125


=August=
[[POTE tuning|POTE generator]]: 696.011
[[Comma|Comma]]s: 36/35, 128/125


[[POTE_tuning|POTE generator]]: 696.011
[[Map]]: [&lt;3 0 7 -1|, &lt;0 1 0 2|]


[[Map|Map]]: [&lt;3 0 7 -1|, &lt;0 1 0 2|]
[[Wedgie]]: &lt;&lt;3 0 6 -7 1 14||
 
[[wedgie|Wedgie]]: &lt;&lt;3 0 6 -7 1 14||


[[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|Badness]]: 0.0265
[[Badness]]: 0.0265


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


Line 32: Line 29:
Badness: 0.0202
Badness: 0.0202


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


Line 43: Line 40:
Badness: 0.0184
Badness: 0.0184


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


Line 54: Line 51:
Badness: 0.0222
Badness: 0.0222


=Augene=
== Augene ==
Commas: 64/63, 126/125
Commas: 64/63, 126/125


Line 67: Line 64:
Badness: 0.0248
Badness: 0.0248


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


Line 78: Line 75:
Badness: 0.0196
Badness: 0.0196


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


Line 89: Line 86:
Badness: 0.0208
Badness: 0.0208


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


Line 100: Line 97:
Badness: 0.0229
Badness: 0.0229


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


Line 111: Line 108:
Badness: 0.0231
Badness: 0.0231


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


Line 122: Line 119:
Badness: 0.0356
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]
* [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 ==
Commas: 28/27, 128/125
Commas: 28/27, 128/125


Line 139: Line 136:
Badness: 0.0547
Badness: 0.0547


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


Line 150: Line 147:
Badness: 0.0312
Badness: 0.0312


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


Line 163: Line 160:
Badness: 0.0591
Badness: 0.0591


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


Line 176: Line 173:
Badness: 0.0577
Badness: 0.0577


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


Line 187: Line 184:
Badness: 0.0384
Badness: 0.0384


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


Line 198: Line 195:
Badness: 0.0359
Badness: 0.0359


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


Line 211: Line 208:
Badness: 0.0705
Badness: 0.0705


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


Line 222: Line 219:
Badness: 0.0382
Badness: 0.0382


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


Line 233: Line 230:
Badness: 0.0302
Badness: 0.0302


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


Line 248: Line 245:
[[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
Commas: 56/55, 77/75, 128/125


Line 259: Line 256:
Badness: 0.0262
Badness: 0.0262


===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
Commas: 49/48, 56/55, 66/65, 77/75


Line 275: Line 272:
[[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
Commas: 49/48, 56/55, 77/75, 507/500


Line 286: Line 283:
Badness: 0.0592
Badness: 0.0592


=== ===


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


Line 301: Line 297:
Badness: 0.1378
Badness: 0.1378


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


Line 312: Line 308:
Badness: 0.0681
Badness: 0.0681


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


Line 323: Line 319:
Badness: 0.0423
Badness: 0.0423


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


Line 334: Line 330:
Badness: 0.1383
Badness: 0.1383


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


Line 347: Line 343:
Badness: 0.1132
Badness: 0.1132


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


Line 358: Line 354:
Badness: 0.0592
Badness: 0.0592


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


Line 369: Line 365:
Badness: 0.0423
Badness: 0.0423


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


Line 382: Line 378:
Badness: 0.2110
Badness: 0.2110


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


Line 393: Line 389:
Badness: 0.0807
Badness: 0.0807


=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.


Line 408: Line 404:
Badness: 0.0672
Badness: 0.0672


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


Line 419: Line 415:
Badness: 0.0349
Badness: 0.0349


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


Line 430: Line 426:
Badness: 0.0241
Badness: 0.0241


==Music==
== Music ==
[https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]     [[Category:augmented]]
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
 
[[Category:augmented]]
[[Category:family]]
[[Category:family]]
[[Category:theory]]
[[Category:theory]]
<!-- interwiki links -->
[[de:Übermässige Temperaturen]]

Revision as of 17:55, 10 October 2018

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 hemis 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quarter-tones instead of semitones.

August

Commas: 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: 0.0265

11-limit

Commas: 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: 0.0202

13-limit

Commas: 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, 12, 21, 33ef, 54bcef

Badness: 0.0184

Augustus

Commas: 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: 0.0222

Augene

Commas: 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: 0.0248

11-limit

Commas: 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: 0.0196

13-limit

Commas: 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: 0.0208

Ogene

Commas: 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: 0.0229

Agene

Commas: 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: 0.0231

Eugene

Commas: 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: 0.0356

Inflated

Commas: 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: 15, 48bc, 63bc, 78bc

Badness: 0.0547

11-limit

Commas: 28/27, 55/54, 128/125

POTE generator: ~3/2 = 722.663

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

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

Badness: 0.0312

Deflated

Commas: 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, 18bd, 21d, 30bd

Badness: 0.0591

Hexe

Commas: 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, [[[42edo|42d]], 54cd

Badness: 0.0577

11-limit

Commas: 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: 0.0384

13-limit

Commas: 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: 12f

Badness: 0.0359

Hemiaug

Commas: 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: 0.0705

11-limit

Commas: 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: 0.0382

13-limit

Commas: 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: 0.0302

Triforce

Commas: 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: 0.0202

triforce9.jpg

11-limit

Commas: 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: 0.0262

Musical example

13-limit

Commas: 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: 0.0202

Triforce(9) with L:s = phi

Semitriforce

Commas: 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: 0.0592


Hemiug

Commas: 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: 0.1378

11-limit

Commas: 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: 0.0681

13-limit

Commas: 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: 0.0423

Trug

Commas: 128/125, 360/343

POTE generator: ~48/35 = 501.980

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

EDOs: 9bd, 12

Badness: 0.1383

Oodako

Commas: 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: 0.1132

11-limit

Commas: 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: 0.0592

13-limit

Commas: 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: 0.0423

Hemisemiaug

Commas: 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: 0.2110

11-limit

Commas: 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: 0.0807

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.

Commas: 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: 0.0672

11-limit

Commas: 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: 0.0349

13-limit

Commas: 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: 0.0241

Music

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