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Augene: Difference between revisions

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'''Augene''' is a [[regular temperament]] of the [[augmented family]], which means that [[128/125]] is [[tempering out|tempered out]] and three [[5/4]]'s make a [[2/1]]. Augene is distinguished from its relative [[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, ….
'''Augene''' is a [[regular temperament]] of the [[augmented family]], which means that [[128/125]] is [[tempering out|tempered out]] and three [[5/4]]'s make a [[2/1]]. Augene is distinguished from its relative [[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, ….
It is also the unique [[7-limit]] [[regular temperament]] that tempers out the [[essential tempering commas]] of the [[5-odd-limit|5-]] ([[128/125]]) and [[7-odd-limit]] ([[64/63]], [[126/125]]).


The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]].
The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]].


== 11-limit interval chain ==
See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions.
{| class="wikitable center-1 right-all"
 
|+Generator based off of 11-limit minimax
== Interval chain ==
! Gen.
In the following table, odd harmonics 1–11 are in '''bold'''.
! Period 0
 
! Period 1
{| class="wikitable center-1 right-2 right-4 right-6"
! Period 2
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
|-
| 0
| 0
| 0.000
| 0.0
| 400.000
| 1/1
| 800.000
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
|-
| 1
| 1
| 1113.148
| 1113.1
| 313.148
| 15/8, 40/21
| 713.148
| 313.1
| 6/5
| 713.1
| '''3/2'''
|-  
|-  
| 2
| 2
| 1026.296
| 1026.3
| 226.296
| 9/5, 20/11
| 626.296
| 226.3
| '''8/7''', '''9/8'''
| 626.3
| 10/7, '''16/11'''
|-
|-
| 3
| 3
| 939.444
| 939.4
| 139.444
| 12/7
| 539.444
| 139.4
| 12/11, 15/14
| 539.4
| 15/11
|-
|-
| 4
| 4
| 852.592
| 852.6
| 52.592
| 18/11
| 452.592
| 52.6
| 45/44
| 452.6
| 9/7
|-
|-
| 5
| 5
| 765.740
| 765.7
| 1165.740
| 54/35
| 365.740
| 1165.7
| 27/14, 96/49
| 365.7
| 27/22
|}
|}
<nowiki/>* In 11-odd-limit minimax tuning, octave reduced


== Tuning spectrum ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 709.5949{{c}}
| 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 prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 713.5701{{c}}
| CWE: ~3/2 = 711.6031{{c}}
| POTE: ~3/2 = 711.1766{{c}}
|}


=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval]])
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Fifth (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 14/13
| 15/8
| 671.702
|
|-
| 12\28
|
| 685.714
|
|-
|
| 16/15
| 688.269
| 688.269
|
| -1/3 comma
|-
|
| 13/11
| 689.210
|
|-
|-
| 7\12
| 7\12
Line 76: Line 124:
|  
|  
|-
|-
| 31\53
|  
|  
| 701.887
| 3/2
|
|-
|
| 4/3
| 701.955
| 701.955
|  
| Untempered
|-
|-
| 30\51
| 30\51
|  
|  
| 705.882
| 705.882
|  
| 51cdeee val
|-
|-
|  
|  
Line 99: Line 142:
|  
|  
| 707.692
| 707.692
|  
| 39dee val
|-
|-
|  
|  
Line 112: Line 155:
|-
|-
|  
|  
| 10/9
| 9/5
| 708.798
| 708.798
| 9-odd-limit minimax
| 1/6 comma, 9-odd-limit minimax
|-
|-
| 39\66
| 39\66
|  
|  
| 709.091
| 709.091
|  
| 66cdeee val
|-
|-
|  
|  
Line 129: Line 172:
|  
|  
| 711.111
| 711.111
|  
| 27e val
|-
|-
|  
|  
Line 136: Line 179:
|  
|  
|-
|-
| 57\96
| 41\69
|
| 712.500
|  
|  
| 713.043
| 69bcee val
|-
|-
|  
|  
Line 149: Line 192:
|  
|  
| 714.286
| 714.286
|  
| 42e val
|-
|-
|  
|  
| 8/7
| 7/4
| 715.587
| 715.587
| 7-, 13- and 15-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|  
|  
| 6/5
| 5/3
| 715.641
| 715.641
| 5-odd-limit minimax
| 1/3 comma, 5-odd-limit minimax
|-
|-
| 34\57
| 34\57
|  
|  
| 715.789
| 715.789
|  
| 53bce val
|-
|-
| 25\72
|  
|  
| 716.667
| 11/6
|
|-
|
| 12/11
| 716.879
| 716.879
|  
|  
Line 184: Line 222:
|  
|  
| 720.000
| 720.000
|
|-
|
| 18/13
| 721.127
|
|-
|
| 15/13
| 723.871
|  
|  
|-
|-
Line 199: Line 227:
| 11/8
| 11/8
| 724.341
| 724.341
|
|-
|
| 13/12
| 730.714
|
|-
|
| 13/10
| 745.786
|
|-
|
| 16/13
| 759.472
|  
|  
|}
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==
; [[Igliashon Jones]]
; [[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''] in Augene[12] tuned to 27edo
* [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


; [[Joel Grant Taylor]]
; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] – 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


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