Augmented (temperament)
Augmented, 12 & 15, is a 5-limit temperament generated by a 1\3 period (representing 5/4) and a 3/2 generator; it has MOS scales triwood (3L 3s) and tcherepnin (3L 6s). It tempers out 128/125, the difference between a stack of three 5/4s and the octave.
Because 128/125 = 126/125 * 64/63, Augmented admits a structurally induced 7-limit extension, called Augene (2.3.5.7[12 & 15]), which adds Archy tempering to the fifth and provides sharp tunings for primes 3, 5, and 7. 15edo and 27edo are notable tunings of Augene. Augene is generally the preferred extension due to consistent sharpness on the third and fifth bringing the 4:5:6 chord closer to DR and making 6/5 better tuned (which is best supported by 15edo).
The other extension is 12 & 21, which tempers together 16/15 and 15/14 and makes it so 5/4 + 6/5 + 6/5 = 7/4, and therefore suggests a flat tuning of the fifth, as in 33edo.
List of patent vals
The following patent vals support Augmented. Vals contorted in the 5-limit are not included.
| Edo | 7 extension | 3/2 tuning | 7/4 tuning |
|---|---|---|---|
| 9 | 12 & 21 | 666.667 | 933.333 |
| 21 | 12 & 21 | 685.714 | 971.429 |
| 33 | 12 & 21 | 690.909 | 981.818 |
| 12 | Augene, 12 & 21 | 700.000 | 1000.000 |
| 39 | 707.692 | 953.846 | |
| 27 | Augene | 711.111 | 977.778 |
| 42 | Augene | 714.286 | 971.429 |
| 15 | Augene | 720.000 | 960.000 |
| 18 | 733.333 | 1000.000 | |
| 3 | Augene | 800.000 | 800.000 |
