Augene
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
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 709.5949 ¢ | CWE: ~3/2 = 709.3249 ¢ | POTE: ~3/2 = 709.2568 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 713.5701 ¢ | CWE: ~3/2 = 711.6031 ¢ | POTE: ~3/2 = 711.1766 ¢ |
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 | 51cdeee val | |
| 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
- Sad Like Winter Leaves – in Augene[12] tuned to 27edo
- Galticeran Sonatina – in Augene[12] tuned to 27edo