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Augene is a regular temperament of the [[Augmented_family#x-Augene|augmented family]], which means that 128/125 is tempered out and three 5/4s make a 2/1. Augene is distinguished from its relative [[august|august]] by tempering out 64/63, which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the MOS sequence of augene goes 12, 15, 27... in constrast to august's which goes 9, 12, 21...
{{Infobox regtemp
| Title = Augmented
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11
| Comma basis = [[128/125]] (5-limit); <br>[[64/63]], [[126/125]] (7-limit); <br>[[56/55]], [[64/63]], [[100/99]] (11-limit)
| Edo join 1 = 12 | Edo join 2 = 15
| Mapping = 3; 1 0 -2 -2
| Generators = 3/2
| Generators tuning = 711.6
| Optimization method = CWE
| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
| Pergen = (P8/3, P5)
| Odd limit 1 = 7 | Mistuning 1 = 13.7 | Complexity 1 = 12
| Odd limit 2 = 11 | Mistuning 2 = 22.4 | Complexity 2 = 15
}}
'''Augmented''' is a [[regular temperament|temperament]] that sets [[5/4]] to one third of an [[2/1|octave]], [[tempering out]] the diesis, [[128/125]], and has a [[generator]] of a [[3/2|perfect fifth]]. The fifth can be tuned to [[12edo]], but a sharper tuning is often perferred for it to blend with the sharp 5/4. This gives rise to the natural [[7-limit]] [[extension]] (known as '''augene''') that tempers out [[64/63]] and [[126/125]], where the whole tone stands in for [[8/7]][[~]][[9/8]].  


The first few augene MOSes are [[3L_3s|3L 3s]], [[3L_6s|3L 6s]], [[3L_9s|3L 9s]], [[12L_3s|12L 3s]]... and the first EDOs that reasonably support augene are [[12edo|12edo]], [[15edo|15edo]], and especially [[27edo|27edo]].
Further extension to the [[11-limit]] is available by noticing 128/125 = ([[56/55]])⋅([[176/175]]), and tempering out both commas means [[14/11]] is equated to the 1/3 octave as well. This does imply more [[damage]] in the 11-limit however, as the [[11/8]] is now conflated with [[7/5]].  


==Tuning spectrum==
The first few augmented [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably [[support]] it are [[12edo]], [[15edo]], and [[27edo]].


Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
Alternative 7-limit extensions of augmented include [[august]] (9 & 12), which tempers out [[36/35]] and favors a flat-of-just fifth.
Therefore the [[optimal ET sequence]] of august goes 9, 12, 21, … in constrast to augmented's which goes 12, 15, 27, ….


Gencom map: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]
See [[Augmented family #Augmented]] and [[Augmented family #Septimal augmented (augene)|#Septimal augmented (augene)]] for technical data. See [[Augmented extensions]] for a discussion on 13-limit extensions.


{| class="wikitable center-all"
== Interval chain ==
In the following table, odd harmonics 1–11 are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4 right-6"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
|-
! | ET<br>generator
| 0
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 0.0
! | subminor<br>third (¢)
| '''1/1'''
! | comments
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
|-
| |
| 1
| | 7/6
| 711.6
| | 266.871
| '''3/2'''
| |
| 1111.6
| 15/8, 40/21
| 311.6
| 6/5
|-
| 2
| 223.2
| '''8/7''', '''9/8'''
| 623.2
| 10/7, '''16/11'''
| 1023.2
| 9/5, 20/11
|-
| 3
| 934.8
| 12/7
| 134.8
| 12/11, 15/14
| 534.8
| 15/11
|-
| 4
| 446.4
| 9/7
| 846.4
| 18/11
| 46.4
| 45/44
|-
| 5
| 1158.0
| 27/14, 96/49
| 358.0
| 27/22
| 758.0
| 54/35
|}
<nowiki/>* In 11-limit CWE tuning, octave reduced
 
== Scales ==
=== Scala files ===
* [[12-27]] – 27edo tuning
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
| |
! rowspan="2" |  
| | 15/11
! colspan="3" | Euclidean
| | 268.475
| |
|-
|-
| |
! Constrained
| | 14/11
! Constrained & skewed
| | 269.585
! Destretched
| |
|-
|-
| |
! Tenney
| | 12/11
| CTE: ~3/2 = 701.9550{{c}}
| | 270.127
| CWE: ~3/2 = 705.0691{{c}}
| |
| POTE: ~3/2 = 706.6376{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
| |
! rowspan="2" |  
| | 15/14
! colspan="3" | Euclidean
| | 270.139
| |
|-
|-
| | 7\31
! Constrained
| |
! Constrained & skewed
| | 270.968
! Destretched
| |
|-
|-
| |
! Tenney
| | 11/9
| CTE: ~3/2 = 709.5949{{c}}
| | 271.049
| CWE: ~3/2 = 709.3249{{c}}
| |
| POTE: ~3/2 = 709.2568{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
|-
| |
! rowspan="2" |  
| | 8/7
! colspan="3" | Euclidean
| | 271.103
| |
|-
|-
| |
! Constrained
| | 7/5
! Constrained & skewed
| | 271.137
! Destretched
| |
|-
|-
| |
! Tenney
| | 5/4
| CTE: ~3/2 = 713.5701{{c}}
| | 271.229
| CWE: ~3/2 = 711.6031{{c}}
| |
| POTE: ~3/2 = 711.1766{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
| |
! Edo<br>generator
| | 1361367/1000000
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
| | 271.326
! Generator (¢)
| | 7 limit least squares
! Comments
|-
|-
| |
|  
| | 14/13
| 15/8
| | 271.418
| 688.269
| | 13 and 15 limit minimax
| -1/3 comma
|-
|-
| | 19\84
| '''7\12'''
| |
|  
| | 271.429
| '''700.000'''
| |
| '''Lower bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
| |
|  
| | {{vector|0 119 -46 20 -16}}
| 3/2
| | 271.445
| 701.955
| | 11 limit least squares
| Untempered
|-
|-
| |
| 30\51
| | <math>x^{10} + 2x^3 = 8</math>
|  
| | 271.508
| 705.882
| | equal beating
| 51cdeee val
|-
|-
| |
|  
| | 16/13
| 15/14
| | 271.551
| 706.481
| |
|  
|-
|-
| |
| 23\39
| | {{vector|0 90 -41 14}}
|  
| | 271.561
| 707.692
| | 9 limit least squares
| 39dee val
|-
|-
| |
|  
| | 6/5
| 7/5
| | 271.564
| 708.744
| | 5 limit minimax
|  
|-
|-
| |
|  
| | {{vector|0 -211 30 -47 -5 142}}
| 9/7
| | 271.567
| 708.771
| | 13 limit least squares
|  
|-
|-
| |
|  
| | {{vector|0 -236 5 -51 -3 165}}
| 9/5
| | 271.570
| 708.798
| | 15 limit least squares
| 1/6 comma, 9-odd-limit minimax
|-
|-
| |
| 39\66
| | 1220703125/1033121304
|  
| | 271.590
| 709.091
| | 5 limit least squares
| 66cdeee val
|-
|-
| |
|  
| | 13/12
| 7/6
| | 271.593
| 711.043
| |
|  
|-
|-
| |
| 16\27
| | 13/10
|  
| | 271.612
| 711.111
| |
| 27e val
|-
|-
| |
|  
| | 18/13
| 15/11
| | 271.618
| 712.317
| |
|  
|-
|-
| |
| 41\69
| | 10/9
|  
| | 271.623
| 713.043
| | 9 limit minimax
| 69bcee val
|-
|-
| |
|  
| | 15/13
| 11/9
| | 271.641
| 713.148
| |
| 11-odd-limit minimax
|-
|-
| | 12\53
| 25\42
| |
|  
| | 271.698
| 714.286
| |
| 42e val
|-
|-
| |
|  
| | 4/3
| 7/4
| | 271.708
| 715.587
| |
| 7-odd-limit minimax
|-
|-
| |
|  
| | 13/11
| 5/3
| | 271.942
| 715.641
| |
| 1/3 comma, 5-odd-limit minimax
|-
|-
| |
| 34\57
| | 16/15
|  
| | 272.067
| 715.789
| |
| 57bce val
|-
|-
| |
|  
| | 9/7
| 11/6
| | 272.514
| 716.879
| |
|  
|-
|-
| | 5\22
|  
| |
| 11/10
| | 272.727
| 717.498
| |
|  
|-
|-
| |
| '''9\15'''
| | 11/10
|  
| | 273.001
| '''720.000'''
| |
| '''Upper bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
| |
|  
| | 11/8
| 11/8
| | 275.659
| 724.341
| |
|  
|}
|}
<nowiki/>* Besides the octave
== Music ==
; [[Igliashon Jones]]
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves''] &ndash; in Augene[12] tuned to 27edo


=Music=
; [[Joel Grant Taylor]]
''[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] in Augene[12] tuned to 27edo
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] &ndash; in Augene[12] tuned to 27edo


''[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina]'' by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor] in Augene[12] tuned to 27edo
[[Category:Augmented| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Archytas clan]]
[[Category:Starling temperaments]]
Retrieved from "https://en.xen.wiki/w/Augmented_(temperament)"