Quartkeenlig

Quartkeenlig is a rank-2 temperament with a generator of about 53 cents, which stands for 33/32 and 36/35 tempered together.

For technical data see: Kleismic family#Quartkeenlig

Theory

EDOs which support quartkeenlig: 68, 91, 159. The simplest mos of quartkeenlig is 22L 1s.

The fifth in the standard sense constitutes 13 steps, and it is close to the 7edo fifth. In 91edo, it is exactly the 7edo fifth. However it should be noted that from a regular temperament theory perspective it is not mapped to 3/2. In order to reach just 3/2, one would need to stack 36 generators.

Relationship to 23edo and octave stretching

If 23 steps of pure TE tuning quartkeenlig in the 11-limit are considered without octave equivalence, the resulting scale would have an octave of 1215.62 cents, which is almost exactly the octave recommended for harmonic stretching of 23edo.

Other interval relationships also work. Quartkeenlig maps 5 steps to 7/6, and 6 steps to 6/5, which are the direct approximations stretched 23edo provides for these intervals. In addition, such a system would be fourthless like stretched 23edo, as 4/3 occurs nearly halfway between the 9th and 10th steps