Circulating Temperaments
It is a sad fact that common practice Western music is based on triads, and the thirds of these triads are woefully out of tune. The major third in 12-equal is 13.69 cents sharp, and the minor third is 15.64 cents flat. Considering that anything beyond ten cents over or under a just intonation is entering the region of major cheeziness, we can see why 12-equal was not in general use during much of the common practice period, and that other systems held sway instead. One plan for dealing with the problem of the thirds is a circulating, or well-temperament. This makes the thirds which are most likely to be used a little bit better, at the expense of other thirds, which are made a little bit worse. The circulating temperments used on this website employ a new and bolder plan: the thirds are made a lot better, and instead of trying to make up the difference with out-of-tune versions of 5/4 or 6/5, we use instead in-tune versions of xenharmonic thirds: the supermajor third 9/7, the subminor third 7/6, along with 14/11 and 13/11. The resulting chords sound strange but not repulsive. Temperaments adopting this plan are grail, bifrost, and cauldron. Another approach leading to similar results is to well-tune a scale in just intonation, where a well-tuning is a regular tuning, meaning using fifths and thirds of the same size, whicb is chosen with an eye to tempering the scale to a circulating temperament. An example of this approach is duowell.
It is also possible to simply use a single meantone fifth for twelve notes, and bite the bullet of the resulting very sharp "wolf" fifth. One way to do this is to use a mild version of meantone, such as 1/6-comma. A way I think works better is to take a wolf fifth near or at 20/13. Systems adopting this approach are ratwolf and wilwolf.