Augene
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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.
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
Tuning spectrum
Edo Generator |
Eigenmonzo (Unchanged-interval)* |
Fifth (¢) | Comments |
---|---|---|---|
13/7 | 671.702 | ||
15/8 | 688.269 | ||
13/11 | 689.210 | ||
7\12 | 700.000 | ||
3/2 | 701.955 | ||
30\51 | 705.882 | ||
15/14 | 706.481 | ||
23\39 | 707.692 | ||
7/5 | 708.744 | ||
9/7 | 708.771 | ||
9/5 | 708.798 | 9-odd-limit minimax | |
39\66 | 709.091 | ||
7/6 | 711.043 | ||
16\27 | 711.111 | ||
15/11 | 712.317 | ||
57\96 | 712.500 | ||
11/9 | 713.148 | 11-odd-limit minimax | |
25\42 | 714.286 | ||
7/4 | 715.587 | 7-, 13- and 15-odd-limit minimax | |
5/3 | 715.641 | 5-odd-limit minimax | |
34\57 | 715.789 | ||
25\72 | 716.667 | ||
11/6 | 716.879 | ||
11/10 | 717.498 | ||
9\15 | 720.000 | ||
13/9 | 721.127 | ||
15/13 | 723.871 | ||
11/8 | 724.341 | ||
13/12 | 730.714 | ||
13/10 | 745.786 | ||
13/8 | 759.472 |
* besides the octave
Music
- Sad Like Winter Leaves in Augene[12] tuned to 27edo
- Galticeran Sonatina – in Augene[12] tuned to 27edo