Pentadacus

Pentadacus

Subgroups	5.7.11
Comma basis	831875/823543
Reduced mapping	⟨1; 3 7]
ET join	43ed5 & 14ed5
Generators (CWE)	~55/49 = 194.820 ¢
MOS scales	1L 12s<5/1>, 1L 13s<5/1>, 14L 1s<5/1>, 14L 15s<5/1>
Ploidacot	tri7cleft
Minimax error	-integer-limit:  ¢
Target scale size	-integer-limit: notes

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. 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 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 first discovered and named by CompactStar in 2026.

Pentadacus can be thought of as a compressed 14ed5 until you hit 5/1, as is best exemplified by the 1L 13s<5/1> Pentadacus[14] MOS. Because of this, many good smaller pentadacus tunings are of the form (14n+1)ed5 such as 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, so Pentadacus is very alien (being a step above even tritave-equivalent temperaments) but can lapse into sounding like the familiar whole-tone scale at times. 6 generators in pentadacus can sound a bit like a compressed octave but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.

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.

Interval chain

Pentadacus

#	Cents*	Approximate ratios
		5.7.11 subgroup
0	0.0	1/1, 831875/823543