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 <<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 = | == Augmented == | ||
[[Comma]]: 128/125 | [[Comma]]: 128/125 | ||
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[[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 == | ||
[[Comma list]]: 36/35, 128/125 | [[Comma list]]: 36/35, 128/125 | ||
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[[Badness]]: 0.026459 | [[Badness]]: 0.026459 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 36/35, 45/44, 56/55 | Comma list: 36/35, 45/44, 56/55 | ||
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Badness: 0.020191 | Badness: 0.020191 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 27/26, 36/35, 45/44, 56/55 | Comma list: 27/26, 36/35, 45/44, 56/55 | ||
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Badness: 0.018448 | Badness: 0.018448 | ||
== Augustus == | === Augustus === | ||
Comma list: 26/25, 36/35, 45/44, 56/55 | Comma list: 26/25, 36/35, 45/44, 56/55 | ||
| Line 63: | Line 61: | ||
Badness: 0.022236 | Badness: 0.022236 | ||
= Augene = | == Augene == | ||
[[Comma list]]: 64/63, 126/125 | [[Comma list]]: 64/63, 126/125 | ||
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Badness: 0.024816 | Badness: 0.024816 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 64/63, 100/99 | Comma list: 56/55, 64/63, 100/99 | ||
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Badness: 0.019613 | Badness: 0.019613 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 40/39, 56/55, 64/63, 66/65 | Comma list: 40/39, 56/55, 64/63, 66/65 | ||
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Badness: 0.020785 | Badness: 0.020785 | ||
== Ogene == | === Ogene === | ||
Comma list: 56/55, 64/63, 91/90, 100/99 | Comma list: 56/55, 64/63, 91/90, 100/99 | ||
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Badness: 0.022890 | Badness: 0.022890 | ||
== Agene == | === Agene === | ||
Comma list: 56/55, 64/63, 78/77, 100/99 | Comma list: 56/55, 64/63, 78/77, 100/99 | ||
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Badness: 0.023113 | Badness: 0.023113 | ||
== Eugene == | === Eugene === | ||
Comma list: 55/54, 64/63, 77/75 | Comma list: 55/54, 64/63, 77/75 | ||
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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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Badness: 0.031171 | Badness: 0.031171 | ||
= Deflated = | == Deflated == | ||
Comma list: 21/20, 128/125 | Comma list: 21/20, 128/125 | ||
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Badness: 0.059079 | Badness: 0.059079 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 21/20, 33/32, 128/125 | Comma list: 21/20, 33/32, 128/125 | ||
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Badness: 0.037183 | Badness: 0.037183 | ||
= Hexe = | == Hexe == | ||
Comma list: 50/49, 128/125 | Comma list: 50/49, 128/125 | ||
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Badness: 0.057730 | Badness: 0.057730 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 50/49, 56/55, 125/121 | Comma list: 50/49, 56/55, 125/121 | ||
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Badness: 0.038412 | Badness: 0.038412 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 50/49, 56/55, 66/65, 105/104 | Comma list: 50/49, 56/55, 66/65, 105/104 | ||
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Badness: 0.035946 | Badness: 0.035946 | ||
= Hemiaug = | == Hemiaug == | ||
Comma list: 128/125, 245/243 | Comma list: 128/125, 245/243 | ||
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Badness: 0.070463 | Badness: 0.070463 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 245/243 | Comma list: 56/55, 128/125, 245/243 | ||
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Badness: 0.038232 | Badness: 0.038232 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 91/90, 128/125, 245/243 | Comma list: 56/55, 91/90, 128/125, 245/243 | ||
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Badness: 0.030159 | Badness: 0.030159 | ||
= Triforce = | == Triforce == | ||
Comma list: 49/48, 128/125 | Comma list: 49/48, 128/125 | ||
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[[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 | ||
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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 | ||
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[[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 | ||
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= Hemiug = | == Hemiug == | ||
Comma list: 128/125, 1323/1250 | Comma list: 128/125, 1323/1250 | ||
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Badness: 0.137764 | Badness: 0.137764 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 1323/1250 | Comma list: 56/55, 128/125, 1323/1250 | ||
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Badness: 0.068062 | Badness: 0.068062 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 66/65, 105/104, 507/500 | Comma list: 56/55, 66/65, 105/104, 507/500 | ||
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Badness: 0.042289 | Badness: 0.042289 | ||
= Trug = | == Trug == | ||
Comma list: 128/125, 360/343 | Comma list: 128/125, 360/343 | ||
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Badness: 0.138279 | Badness: 0.138279 | ||
= Oodako = | == Oodako == | ||
Comma list: 128/125, 2401/2400 | Comma list: 128/125, 2401/2400 | ||
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Badness: 0.113209 | Badness: 0.113209 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 3773/3750 | Comma list: 56/55, 128/125, 3773/3750 | ||
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Badness: 0.059192 | Badness: 0.059192 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 78/77, 128/125, 507/500 | Comma list: 56/55, 78/77, 128/125, 507/500 | ||
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Badness: 0.042323 | Badness: 0.042323 | ||
= Hemisemiaug = | == Hemisemiaug == | ||
Comma list: 128/125, 12005/11664 | Comma list: 128/125, 12005/11664 | ||
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Badness: 0.210984 | Badness: 0.210984 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 3773/3645 | Comma list: 56/55, 128/125, 3773/3645 | ||
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Badness: 0.080738 | 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. | ||
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Badness: 0.067157 | Badness: 0.067157 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 540/539 | Comma list: 56/55, 128/125, 540/539 | ||
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Badness: 0.034861 | Badness: 0.034861 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 78/77, 91/90, 128/125 | Comma list: 56/55, 78/77, 91/90, 128/125 | ||
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Badness: 0.024143 | 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] | ||