Augmented (temperament): Difference between revisions

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→Spectrum of Augene Tunings by Eigenmonzos: improve and standardize tuning spectra tables
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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 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]].


== Tuning spectrum ==
==Tuning spectrum==
 
Gencom: [2 7/6; 99/98 121/120 176/175 275/273]
 
Gencom map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 9: Line 13:
! | ET<br>generator
! | ET<br>generator
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! | fifth<br>(¢)
! | subminor<br>third (¢)
! | comments
! | comments
|-
|-
| |  
| |
| | 14/13
| | 7/6
| | 671.702
| | 266.871
| |  
| |
|-
|-
| | 12\28
| |
| |  
| | 15/11
| | 685.714
| | 268.475
| |  
| |
|-
|-
| |  
| |
| | 16/15
| | 14/11
| | 688.269
| | 269.585
| |  
| |
|-
|-
| |  
| |
| | 13/11
| | 12/11
| | 689.210
| | 270.127
| |  
| |
|-
|-
| | 7\12
| |
| |  
| | 15/14
| | 700.000
| | 270.139
| |  
| |
|-
|-
| | 31\53
| | 7\31
| |  
| |
| | 701.887
| | 270.968
| |  
| |
|-
|-
| |  
| |
| | 4/3
| | 11/9
| | 701.955
| | 271.049
| |  
| |
|-
|-
| | 30\51
| |
| |  
| | 8/7
| | 705.882
| | 271.103
| |
| |
|-
| |
| | 15/14
| | 706.481
| |
|-
| | 23\39
| |
| | 707.692
| |  
|-
|-
| |  
| |
| | 7/5
| | 7/5
| | 708.744
| | 271.137
| |  
| |
|-
|-
| |  
| |
| | 9/7
| | 5/4
| | 708.771
| | 271.229
| |  
| |
|-
|-
| |  
| |
| | 10/9
| | 1361367/1000000
| | 708.798
| | 271.326
| | 9 limit minimax
| | 7 limit least squares
|-
|-
| | 39\66
| |
| |  
| | 14/13
| | 709.091
| | 271.418
| |  
| | 13 and 15 limit minimax
|-
|-
| |  
| | 19\84
| | 7/6
| |
| | 711.043
| | 271.429
| |  
| |
|-
|-
| | 16\27
| |
| |  
| | {{vector|0 119 -46 20 -16}}
| | 711.111
| | 271.445
| |  
| | 11 limit least squares
|-
|-
| |  
| |  
| | 15/11
| | <math>x^{10} + 2x^3 = 8</math>
| | 712.317
| | 271.508
| |  
| | equal beating
|-
|-
| | 57\96
| |
| |  
| | 16/13
| | 712.5
| | 271.551
| |
| |
|-
| |
| | 11/9
| | 713.148
| | 11 limit minimax
|-
| | 25\42
| |
| | 714.286
| |  
|-
|-
| |  
| |
| | 8/7
| | {{vector|0 90 -41 14}}
| | 715.587
| | 271.561
| | 7, 13 and 15 limit minimax
| | 9 limit least squares
|-
|-
| |  
| |
| | 6/5
| | 6/5
| | 715.641
| | 271.564
| | 5 limit minimax
| | 5 limit minimax
|-
|-
| | 34\57
| |
| |  
| | {{vector|0 -211 30 -47 -5 142}}
| | 715.789
| | 271.567
| |  
| | 13 limit least squares
|-
|-
| | 25\72
| |
| |  
| | {{vector|0 -236 5 -51 -3 165}}
| | 716.667
| | 271.570
| |  
| | 15 limit least squares
|-
|-
| |  
| |
| | 12/11
| | 1220703125/1033121304
| | 716.879
| | 271.590
| |  
| | 5 limit least squares
|-
|-
| |  
| |
| | 11/10
| | 13/12
| | 717.498
| | 271.593
| |  
| |
|-
|-
| | 9\15
| |
| |  
| | 13/10
| | 720.000
| | 271.612
| |  
| |
|-
|-
| |  
| |
| | 18/13
| | 18/13
| | 721.127
| | 271.618
| |  
| |
|-
| |
| | 10/9
| | 271.623
| | 9 limit minimax
|-
|-
| |  
| |
| | 15/13
| | 15/13
| | 723.871
| | 271.641
| |  
| |
|-
| | 12\53
| |
| | 271.698
| |
|-
| |
| | 4/3
| | 271.708
| |
|-
| |
| | 13/11
| | 271.942
| |
|-
| |
| | 16/15
| | 272.067
| |
|-
|-
| |  
| |
| | 11/8
| | 9/7
| | 724.341
| | 272.514
| |  
| |
|-
|-
| |  
| | 5\22
| | 13/12
| |
| | 730.714
| | 272.727
| |  
| |
|-
|-
| |  
| |
| | 13/10
| | 11/10
| | 745.786
| | 273.001
| |  
| |
|-
|-
| |  
| |
| | 16/13
| | 11/8
| | 759.472
| | 275.659
| |  
| |
|}
|}


Revision as of 18:22, 4 October 2021

Augene is a regular temperament of the 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 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...

The first few augene MOSes are 3L 3s, 3L 6s, 3L 9s, 12L 3s... and the first EDOs that reasonably support augene are 12edo, 15edo, and especially 27edo.

Tuning spectrum

Gencom: [2 7/6; 99/98 121/120 176/175 275/273]

Gencom map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]

ET
generator 		eigenmonzo
(unchanged interval) 		subminor
third (¢) 		comments
7/6 266.871
15/11 268.475
14/11 269.585
12/11 270.127
15/14 270.139
7\31 270.968
11/9 271.049
8/7 271.103
7/5 271.137
5/4 271.229
1361367/1000000 271.326 7 limit least squares
14/13 271.418 13 and 15 limit minimax
19\84 271.429
[0 119 -46 20 -16⟩ 271.445 11 limit least squares
[math]\displaystyle{ x^{10} + 2x^3 = 8 }[/math] 271.508 equal beating
16/13 271.551
[0 90 -41 14⟩ 271.561 9 limit least squares
6/5 271.564 5 limit minimax
[0 -211 30 -47 -5 142⟩ 271.567 13 limit least squares
[0 -236 5 -51 -3 165⟩ 271.570 15 limit least squares
1220703125/1033121304 271.590 5 limit least squares
13/12 271.593
13/10 271.612
18/13 271.618
10/9 271.623 9 limit minimax
15/13 271.641
12\53 271.698
4/3 271.708
13/11 271.942
16/15 272.067
9/7 272.514
5\22 272.727
11/10 273.001
11/8 275.659

Music

Sad Like Winter Leaves by Igliashon Jones in Augene[12] tuned to 27edo

Galticeran Sonatina by Joel Grant Taylor in Augene[12] tuned to 27edo

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