Pentadacus
| 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, also known as the pentave. 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. It was first discovered and named by CompactStar in 2026.
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 quite inaccurate and dissonant. Properly-tuned Pentadacus generates the 5/1-equivalent MOS scales 1L 1s<5/1>, 1L 2s<5/1>, etc. until ending the monolarge MOS chain at 1L 13s<5/1>, followed by 14L 1s<5/1>, 14L 15s<5/1>. After this it branches into.
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. In that respect it's vaguely similar to meantone in the 5-limit and Bohlen-Pierce-Stearns in the 3.5.7 subgroup, but it also differs because of the proximity of its generator to a step of 14ed5, while the generators of meantone and BPS are not similar to the step size of any non-trivial EDO or EDT.
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 undecimal didacus can actually be viewed not only as an extension of didacus to include the 11th harmonic, but also an extension of pentadacus to include octaves.