Bastille
Bastille is a rank-2 temperament associated with the [1426 0 -596 -15⟩ comma in the 2.5.7 subgroup, that is no-threes subgroup. It is extracted from a sequence of mos scales supported by 1407edo and 1789edo. It derives its name from the fact that the storming of Bastille happened on 14 July 1789, which in date format is written as 14/07/1789.
Theory
Since 1407edo and 1789edo have large relative errors on harmonics outside of the 2.5.7 subgroup, there initially was no single temperament named bastille, and the term used to refer to merely a sequence of mos scales to accommodate many possible vals that 1407edo and 1789edo support. However, the temperament can be firmly defined in the 2.5.7 subgroup.
The defining characteristic of Bastille is that the generator is a "meantone" flat fifth of around 694.5 cents. Fifteen stacked generators lead to 8/5. Given the precision, such a characteristic is notable for edos in the thousands to support.
Extensions
The 1407eff & 1789 temperament, named pure bastille, tempers out the jacobin comma, which brings two great French Revolution terms in one temperament. However, from a just intonation perspective, such temperament may be too complex, even for edos in the thousands. On the other hand, taking patent vals of 1407edo and 1789edo in the full 13-limit produces a temperament whose fifth is mapped to 112/75, a relatively simple interval on this scale. Comma basis for this version of bastille is {384912/384475, 41765625/41746432, 4501875/4499456, 540078539/540000000}. Since 112/75 is called marvelous fifth, the most straightforward name for this temperament is marvelous bastille.
MOS scales of Bastille
- 1L 1s
- 2L 1s
- 2L 3s
- 5L 2s - minisoft diatonic
- 7L 5s - minihard m-chromatic
- 7L 12s - ultrasoft m-enharmonic - can be used as a well-temperament for 19edo.