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.
| Pentadacus |
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.
14ed5 is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit 5/1. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as 15ed5 (although that one is barely a tuning, being even worse than 14ed5), 29ed5, 43ed5, and 57ed5. In this respect, it is similar to a cluster temperament, but does not seem to exactly meet the definition of a cluster temperament. 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>.
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. Pentadacus's connection to 14ed5 (which is effectively 6edo with a just 5/1) is a lot like didacus's connection to 6edo.