Augmented (temperament): Difference between revisions

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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...
'''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, ….


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]].
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 ==
== 11-limit interval chain ==
{| class="wikitable"
{| class="wikitable"
|+Generator based off of 11-limit minimax
|+Generator based off of 11-limit minimax
!Gen.
! Gen.
!Period 1
! Period 1
!Period 2
! Period 2
!Period 3 (0)
! Period 3 (0)
|-
|-
|0
| 0
|400.000
| 400.000
|800.000
| 800.000
|1200.000
| 1200.000
|-
|-
|1
| 1
|313.148
| 313.148
|713.148
| 713.148
|1113.148
| 1113.148
|-
| 2
| 226.296
| 626.296
| 1026.296
|-
|-
|2
| 3
|226.296
| 139.444
|626.296
| 539.444
|1026.296
| 939.444
|-
|-
|3
| 4
|139.444
| 52.592
|539.444
| 452.592
|939.444
| 852.592
|-
|-
|4
| 5
|52.592
| 1165.740
|452.592
| 365.740
|852.592
| 765.740
|-
|5
|1165.740
|365.740
|765.740
|}
|}


Line 46: Line 46:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>Generator
! | [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval]])
! | fifth<br>(¢)
! Fifth (¢)
! | comments
! Comments
|-
|-
| |  
|  
| | 14/13
| 14/13
| | 671.702
| 671.702
| |  
|  
|-
|-
| | 12\28
| 12\28
| |  
|  
| | 685.714
| 685.714
| |  
|  
|-
|-
| |  
|  
| | 16/15
| 16/15
| | 688.269
| 688.269
| |  
|  
|-
|-
| |  
|  
| | 13/11
| 13/11
| | 689.210
| 689.210
| |  
|  
|-
|-
| | 7\12
| 7\12
| |  
|  
| | 700.000
| 700.000
| |  
|  
|-
|-
| | 31\53
| 31\53
| |  
|  
| | 701.887
| 701.887
| |  
|  
|-
|-
| |  
|  
| | 4/3
| 4/3
| | 701.955
| 701.955
| |  
|  
|-
|-
| | 30\51
| 30\51
| |  
|  
| | 705.882
| 705.882
| |  
|  
|-
|-
| |  
|  
| | 15/14
| 15/14
| | 706.481
| 706.481
| |  
|  
|-
|-
| | 23\39
| 23\39
| |  
|  
| | 707.692
| 707.692
| |  
|  
|-
|-
| |  
|  
| | 7/5
| 7/5
| | 708.744
| 708.744
| |  
|  
|-
|-
| |  
|  
| | 9/7
| 9/7
| | 708.771
| 708.771
| |  
|  
|-
|-
| |  
|  
| | 10/9
| 10/9
| | 708.798
| 708.798
| | 9 limit minimax
| 9 limit minimax
|-
|-
| | 39\66
| 39\66
| |  
|  
| | 709.091
| 709.091
| |  
|  
|-
|-
| |  
|  
| | 7/6
| 7/6
| | 711.043
| 711.043
| |  
|  
|-
|-
| | 16\27
| 16\27
| |  
|  
| | 711.111
| 711.111
| |  
|  
|-
|-
| |  
|  
| | 15/11
| 15/11
| | 712.317
| 712.317
| |  
|  
|-
|-
| | 57\96
| 57\96
| |  
|  
| | 712.5
| 712.5
| |  
|  
|-
|-
| |  
|  
| | 11/9
| 11/9
| | 713.148
| 713.148
| | 11 limit minimax
| 11 limit minimax
|-
|-
| | 25\42
| 25\42
| |  
|  
| | 714.286
| 714.286
| |  
|  
|-
|-
| |  
|  
| | 8/7
| 8/7
| | 715.587
| 715.587
| | 7, 13 and 15 limit minimax
| 7, 13 and 15 limit minimax
|-
|-
| |  
|  
| | 6/5
| 6/5
| | 715.641
| 715.641
| | 5 limit minimax
| 5 limit minimax
|-
|-
| | 34\57
| 34\57
| |  
|  
| | 715.789
| 715.789
| |  
|  
|-
|-
| | 25\72
| 25\72
| |  
|  
| | 716.667
| 716.667
| |  
|  
|-
|-
| |  
|  
| | 12/11
| 12/11
| | 716.879
| 716.879
| |  
|  
|-
|-
| |  
|  
| | 11/10
| 11/10
| | 717.498
| 717.498
| |  
|  
|-
|-
| | 9\15
| 9\15
| |  
|  
| | 720.000
| 720.000
| |  
|  
|-
|-
| |  
|  
| | 18/13
| 18/13
| | 721.127
| 721.127
| |  
|  
|-
|-
| |  
|  
| | 15/13
| 15/13
| | 723.871
| 723.871
| |  
|  
|-
|-
| |  
|  
| | 11/8
| 11/8
| | 724.341
| 724.341
| |  
|  
|-
|-
| |  
|  
| | 13/12
| 13/12
| | 730.714
| 730.714
| |  
|  
|-
|-
| |  
|  
| | 13/10
| 13/10
| | 745.786
| 745.786
| |  
|  
|-
|-
| |  
|  
| | 16/13
| 16/13
| | 759.472
| 759.472
| |  
|  
|}
|}


=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''] 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]]
* ''Galticeran Sonatina'' – [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 play] | [https://soundcloud.com/joelgranttaylor/galticeran_sonatina SoundCloud] – in Augene[12] tuned to 27edo


[[Category:Temperaments]]
[[Category:Temperaments]]
[[Category:Augene| ]] <!-- main article -->
[[Category:Augene| ]] <!-- main article -->

Revision as of 09:49, 13 May 2024

Augene is a regular temperament of the augmented family, which means that 128/125 is 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, ….

The first few augene mos scales 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

Generator based off of 11-limit minimax
Gen. Period 1 Period 2 Period 3 (0)
0 400.000 800.000 1200.000
1 313.148 713.148 1113.148
2 226.296 626.296 1026.296
3 139.444 539.444 939.444
4 52.592 452.592 852.592
5 1165.740 365.740 765.740

Tuning spectrum

Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Fifth (¢) Comments
14/13 671.702
12\28 685.714
16/15 688.269
13/11 689.210
7\12 700.000
31\53 701.887
4/3 701.955
30\51 705.882
15/14 706.481
23\39 707.692
7/5 708.744
9/7 708.771
10/9 708.798 9 limit minimax
39\66 709.091
7/6 711.043
16\27 711.111
15/11 712.317
57\96 712.5
11/9 713.148 11 limit minimax
25\42 714.286
8/7 715.587 7, 13 and 15 limit minimax
6/5 715.641 5 limit minimax
34\57 715.789
25\72 716.667
12/11 716.879
11/10 717.498
9\15 720.000
18/13 721.127
15/13 723.871
11/8 724.341
13/12 730.714
13/10 745.786
16/13 759.472

Music

Igliashon Jones
Joel Grant Taylor
  • Galticeran Sonatina – play | SoundCloud – in Augene[12] tuned to 27edo
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