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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 [[optimal ET 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]].  


== 11-limit interval chain ==
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]].
{| class="wikitable"
 
|+Generator based off of 11-limit minimax
Alternative 7-limit extensions of augmented include [[august]] (9 & 12), which tempers out [[36/35]] and favors a flat-of-just fifth.
!Gen.
Therefore the [[optimal ET sequence]] of august goes 9, 12, 21, … in constrast to augmented's which goes 12, 15, 27, ….
!Period 1
 
!Period 2
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.
!Period 3 (0)
 
== 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
|-
|-
|0
! Cents*
|400.000
! Approx. ratios
|800.000
! Cents*
|1200.000
! Approx. ratios
! Cents*
! Approx. ratios
|-
|-
|1
| 0
|313.148
| 0.0
|713.148
| '''1/1'''
|1113.148
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
|-
|2
| 1
|226.296
| 711.6
|626.296
| '''3/2'''
|1026.296
| 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
| 3
|139.444
| 934.8
|539.444
| 12/7
|939.444
| 134.8
| 12/11, 15/14
| 534.8
| 15/11
|-
|-
|4
| 4
|52.592
| 446.4
|452.592
| 9/7
|852.592
| 846.4
| 18/11
| 46.4
| 45/44
|-
|-
|5
| 5
|1165.740
| 1158.0
|365.740
| 27/14, 96/49
|765.740
| 358.0
| 27/22
| 758.0
| 54/35
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Scales ==
=== Scala files ===
* [[12-27]] – 27edo tuning


{| class="wikitable center-all"
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! | ET<br>generator
! rowspan="2" |  
! | [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! colspan="3" | Euclidean
! | fifth<br>(¢)
! | comments
|-
|-
| |
! Constrained
| | 14/13
! Constrained & skewed
| | 671.702
! Destretched
| |
|-
|-
| | 12\28
! Tenney
| |  
| CTE: ~3/2 = 701.9550{{c}}
| | 685.714
| 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" |  
| | 16/15
! colspan="3" | Euclidean
| | 688.269
| |
|-
|-
| |
! Constrained
| | 13/11
! Constrained & skewed
| | 689.210
! Destretched
| |
|-
|-
| | 7\12
! Tenney
| |  
| CTE: ~3/2 = 709.5949{{c}}
| | 700.000
| 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
|-
|-
| | 31\53
! rowspan="2" |  
| |
! colspan="3" | Euclidean
| | 701.887
| |
|-
|-
| |
! Constrained
| | 4/3
! Constrained & skewed
| | 701.955
! Destretched
| |
|-
|-
| | 30\51
! Tenney
| |
| CTE: ~3/2 = 713.5701{{c}}
| | 705.882
| CWE: ~3/2 = 711.6031{{c}}
| |
| POTE: ~3/2 = 711.1766{{c}}
|-
|}
| |
 
| | 15/14
=== Tuning spectrum ===
| | 706.481
{| class="wikitable center-all left-4"
| |
|-
|-
| | 23\39
! Edo<br>generator
| |
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
| | 707.692
! Generator (¢)
| |
! Comments
|-
|-
| |  
|  
| | 7/5
| 15/8
| | 708.744
| 688.269
| |
| -1/3 comma
|-
|-
| |
| '''7\12'''
| | 9/7
|  
| | 708.771
| '''700.000'''
| |
| '''Lower bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
| |  
|  
| | 10/9
| 3/2
| | 708.798
| 701.955
| | 9 limit minimax
| Untempered
|-
|-
| | 39\66
| 30\51
| |  
|  
| | 709.091
| 705.882
| |
| 51cdeee val
|-
|-
| |  
|  
| | 7/6
| 15/14
| | 711.043
| 706.481
| |  
|  
|-
|-
| | 16\27
| 23\39
| |  
|  
| | 711.111
| 707.692
| |
| 39dee val
|-
|-
| |  
|  
| | 15/11
| 7/5
| | 712.317
| 708.744
| |  
|  
|-
|-
| | 57\96
|  
| |
| 9/7
| | 712.5
| 708.771
| |  
|  
|-
|-
| |  
|  
| | 11/9
| 9/5
| | 713.148
| 708.798
| | 11 limit minimax
| 1/6 comma, 9-odd-limit minimax
|-
|-
| | 25\42
| 39\66
| |  
|  
| | 714.286
| 709.091
| |
| 66cdeee val
|-
|-
| |  
|  
| | 8/7
| 7/6
| | 715.587
| 711.043
| | 7, 13 and 15 limit minimax
|  
|-
|-
| |
| 16\27
| | 6/5
|  
| | 715.641
| 711.111
| | 5 limit minimax
| 27e val
|-
|-
| | 34\57
|  
| |
| 15/11
| | 715.789
| 712.317
| |  
|  
|-
|-
| | 25\72
| 41\69
| |  
|  
| | 716.667
| 713.043
| |
| 69bcee val
|-
|-
| |  
|  
| | 12/11
| 11/9
| | 716.879
| 713.148
| |
| 11-odd-limit minimax
|-
|-
| |
| 25\42
| | 11/10
|  
| | 717.498
| 714.286
| |
| 42e val
|-
|-
| | 9\15
|  
| |
| 7/4
| | 720.000
| 715.587
| |
| 7-odd-limit minimax
|-
|-
| |  
|  
| | 18/13
| 5/3
| | 721.127
| 715.641
| |
| 1/3 comma, 5-odd-limit minimax
|-
|-
| |
| 34\57
| | 15/13
|  
| | 723.871
| 715.789
| |
| 57bce val
|-
|-
| |  
|  
| | 11/8
| 11/6
| | 724.341
| 716.879
| |  
|  
|-
|-
| |  
|  
| | 13/12
| 11/10
| | 730.714
| 717.498
| |  
|  
|-
|-
| |
| '''9\15'''
| | 13/10
|  
| | 745.786
| '''720.000'''
| |
| '''Upper bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
| |  
|  
| | 16/13
| 11/8
| | 759.472
| 724.341
| |  
|  
|}
|}
<nowiki/>* Besides the octave


=Music=
== Music ==
''[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves]'' by [https://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones] in Augene[12] tuned to 27edo
; [[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


''[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina]'' by [https://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor] in Augene[12] tuned to 27edo
; [[Joel Grant Taylor]]
* [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


[[Category:Temperaments]]
[[Category:Augmented| ]] <!-- Main article -->
[[Category:Augene| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Archytas clan]]
[[Category:Starling temperaments]]
Retrieved from "https://en.xen.wiki/w/Augmented_(temperament)"