Mabilic and trismegistus
| Mabilic; trismegistus |
1029/1024, 3125/3072 (2.3.5.7)
Mabilic is a temperament in the 2.5.7 subgroup where 5/2 is split into three generators, five of which octave-reduced reach 8/7. The generator is a sharpened fourth (or, conversely, a flattened fifth) in size, best tuned around 528 cents. Mabilic, as a result, tempers out the mabilisma 1071875/1048576. Mabilic is arrived at by removing the inaccurate 3/2 mapping from armodue temperament. As such, mabilic can be associated with MOS scales like antidiatonic and armotonic. Thus, while the generating interval is not 3/2, melodic chain-of-fifths notation makes some amount of sense to use here.
Mabilic can be extended into a full 7-limit temperament called trismegistus in which 3 is found at 15 steps. Since 5 is at 3 steps, that makes trismegistus a magic temperament. Similarly, since 8/7 is at 5 steps, trismegistus is a slendric temperament.Trismegistus is associated with the MOS scale 9L 7s, where 3/2 is found as the "augmented" version of the generator, and 25edo, 41edo, and 66edo make for good tunings.
There is an alternative extension, semabila (Mabila family#Semabila), which is a semaphore temperament (hence its name) and thus finds 4/3 at 10 generators. It is best tuned sharper than trismegistus.
Making the generator itself 4/3 leads to the exotemperament mavila, after which mabilic is named.
The tuning optimum of mabilic is 527.2 cents, which is almost exactly the golden antidiatonic generator.
For technical data, see No-threes subgroup temperaments#Mabilic and Magic family#Trismegistus.
Intervals
In the following tables, odd harmonics and subharmonics 1–15 are labeled in bold.
| Generators up | Generators down | |||
|---|---|---|---|---|
| # | Cents | Approximate ratios | Cents | Approximate ratios |
| 0 | 0 | 1/1 | 1200 | 2/1 |
| 1 | 527.236 | 672.764 | ||
| 2 | 1054.472 | 64/35 | 145.528 | 35/32 |
| 3 | 381.708 | 5/4 | 818.292 | 8/5 |
| 4 | 908.944 | 42/25 | 291.056 | 25/21 |
| 5 | 236.18 | 8/7 | 963.82 | 7/4 |
| 6 | 763.416 | 25/16 | 436.584 | 32/25 |
| 7 | 90.652 | 1109.348 | ||
| 8 | 617.888 | 10/7 | 582.112 | 7/5 |
| 9 | 1145.124 | 54.876 | ||
| 10 | 472.36 | 21/16 | 727.64 | 32/21 |
| 11 | 999.596 | 25/14 | 200.404 | 28/25 |
| 12 | 326.832 | 6/5 | 873.168 | 5/3 |
| 13 | 854.068 | 345.932 | ||
| 14 | 181.304 | 1018.696 | ||
| 15 | 708.54 | 3/2 | 491.46 | 4/3 |
| 16 | 35.776 | 1164.224 | ||