Pentadacus
Pentadacus is a nonoctave regular temperament in the 5.7.11 subgroup which tempers out the comma 831875/823543. It is even more exotic than Bohlen-Pierce, lacking both 2/1 and 3/1, and typically it would be used with an equave of 5/1. It is generated by a meantone-esque small whole tone interval that represents 54/49. Stacking 3 of these tones gives 7/5 and 7 of them give 11/5. Pentadacus has both low complexity (especially by the standards of the 5/1-equivalent world, where scales have lots of notes) and low error if tuned correctly, providing an efficient traversal of the 5.7.11 subgroup. It was discovered by CompactStar in 2026.
| Pentadacus |
3.5.7 49-integer-limit: 2.80 ¢
3.5.7 49-integer-limit: 13 notes
14ed5 is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, although the whole tone is bigger than usual being around 9/8-sized, causing the approximations of 7/5 and 11/5 to be bad. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit 5/1. 14ed5 is also close to 6edo, the familiar whole-tone scale with octaves, and 6 generators in pentadacus can sound like a tempered octave but it’s usually bad enough to be dissonant.
Pentadacus is connected to the octave-repeating didacus temperament as both have a small whole tone generator for which 3 stack to 7/5, and 11-limit didacus is actually a weak extension of pentadacus to include octaves.