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]]).
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]].
See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions.


== Interval chain ==
== Interval chain ==
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<nowiki/>* In 11-odd-limit minimax tuning, octave reduced
<nowiki/>* In 11-odd-limit minimax tuning, octave reduced


== Tuning spectrum ==
== Tunings ==
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
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! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 13/7
| 671.702
|
|-
|-
|  
|  
| 15/8
| 15/8
| 688.269
| 688.269
|
| -1/3 comma
|-
|
| 13/11
| 689.210
|
|-
|-
| 7\12
| 7\12
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| 3/2
| 3/2
| 701.955
| 701.955
|  
| Untempered
|-
|-
| 30\51
| 30\51
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|  
|  
| 707.692
| 707.692
|  
| 39dee val
|-
|-
|  
|  
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| 9/5
| 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
|-
|-
|  
|  
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|  
|  
| 711.111
| 711.111
|  
| 27e val
|-
|-
|  
|  
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|  
|  
|-
|-
| 57\96
| 41\69
|
| 712.500
|  
|  
| 713.043
| 69bcee val
|-
|-
|  
|  
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|  
|  
| 714.286
| 714.286
|  
| 42e val
|-
|-
|  
|  
| 7/4
| 7/4
| 715.587
| 715.587
| 7-, 13- and 15-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|  
|  
| 5/3
| 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
|
|-
|-
|  
|  
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|  
|  
| 720.000
| 720.000
|
|-
|
| 13/9
| 721.127
|
|-
|
| 15/13
| 723.871
|  
|  
|-
|-
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| 11/8
| 11/8
| 724.341
| 724.341
|
|-
|
| 13/12
| 730.714
|
|-
|
| 13/10
| 745.786
|
|-
|
| 13/8
| 759.472
|  
|  
|}
|}

Revision as of 17:59, 21 August 2025

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, ….

It is also the unique 7-limit regular temperament that tempers out the essential tempering commas of the 5- (128/125) and 7-odd-limit (64/63, 126/125).

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.

See Augmented family #Augene for technical data. See Augene extensions for a discussion on 13-limit extensions.

Interval chain

In the following table, odd harmonics 1–11 are in bold.

# Period 0 Period 1 Period 2
Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios
0 0.0 1/1 400.0 5/4, 14/11 800.0 8/5, 11/7
1 1113.1 15/8, 40/21 313.1 6/5 713.1 3/2
2 1026.3 9/5, 20/11 226.3 8/7, 9/8 626.3 10/7, 16/11
3 939.4 12/7 139.4 12/11, 15/14 539.4 15/11
4 852.6 18/11 52.6 45/44 452.6 9/7
5 765.7 54/35 1165.7 27/14, 96/49 365.7 27/22

* In 11-odd-limit minimax tuning, octave reduced

Tunings

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
15/8 688.269 -1/3 comma
7\12 700.000
3/2 701.955 Untempered
30\51 705.882
15/14 706.481
23\39 707.692 39dee val
7/5 708.744
9/7 708.771
9/5 708.798 1/6 comma, 9-odd-limit minimax
39\66 709.091 66cdeee val
7/6 711.043
16\27 711.111 27e val
15/11 712.317
41\69 713.043 69bcee val
11/9 713.148 11-odd-limit minimax
25\42 714.286 42e val
7/4 715.587 7-odd-limit minimax
5/3 715.641 1/3 comma, 5-odd-limit minimax
34\57 715.789 53bce val
11/6 716.879
11/10 717.498
9\15 720.000
11/8 724.341

* Besides the octave

Music

Igliashon Jones
Joel Grant Taylor
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