Pentadacus: Difference between revisions
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*CWE tuning (5.7.11) | <nowiki>*</nowiki>CWE tuning (5.7.11) | ||
== Tunings == | == Tunings == | ||
Revision as of 04:29, 7 August 2026
| 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 55/49. 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 is usually inaccurate unless you are 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.
Etymology
It is named as a play on didacus, with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).
Extensions
In terms of strong extensions to the 5.7.11.13 subgroup, there are the c57 & c71 extension which maps 13/5 to -20 generators (but this requires the usage of the 14L 15s MOS to access 13/5), and the c15 & c29 extension which maps 13/5 to -6 generators (but it has high error of about 20 cents on 11/5 and 13/5). These are roughly equal in Dirichlet badness.
The best extension to the 5.7.11.13 subgroup is actually the weak extension User:2^67-1 has named Pentapassion, that tempers out the Cuthbert comma 847/845. This halves the generator into a semitone of roughly 97 cents, and 17 of these new generators reach 13/5 (so essentially 8.5 of the original generator). It is so named because the generator is close to the generator of Passion temperament. Interestingly, the first few steps of Pentapassion can sound like 12edo (like how Pentadacus can sound like 6edo), although it quickly becomes out-of-tune after stacking a few generators. Pentapassion can be used with a 28L 1s<5/1> MOS.
Interval chain
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 194.9 | 55/49 |
| 2 | 389.8 | 343/275 |
| 3 | 584.7 | 7/5 |
| 4 | 779.6 | 11/7 |
| 5 | 974.5 | 605/343, 2401/1375 |
| 6 | 1169.4 | 49/25 |
| 7 | 1364.3 | 11/5 |
| 8 | 1559.2 | 121/49 |
| 9 | 1754.1 | 343/125 |
| 10 | 1949.0 | 77/25 |
| 11 | 2143.9 | 121/35 |
| 12 | 2338.8 | 1331/343, 2401/625 |
| 13 | 2533.7 | 539/125 |
| 14 | 2728.6 | 121/25 |
| 15 | 137.2 | 1331/1225 |
| 16 | 332.1 | 3773/3125 |
*CWE tuning (5.7.11)
Tunings
Tuning spectrum
| Ed5 generator |
Eigenmonzo (unchanged interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 1\15ed5 | 185.754 | Technically supported but below any reasonable tuning 11/5 has 65 cents of damage | |
| 2\29ed5 | 192.160 | ||
| 7\101ed5 | 193.111 | ||
| 5\72ed5 | 193.494 | ||
| 7/5 | 194.171 | ||
| 3\43ed5 | 194.394 | Probably the best smaller ed5 tuning | |
| 77/25 | 194.752 | ||
| 11/5 | 195.001 | ||
| 7\100ed5 | 195.042 | ||
| 4\57ed5 | 195.531 | ||
| 11/7 | 195.623 | ||
| 5\71ed5 | 196.219 | ||
| 6\85ed5 | 196.268 | ||
| 7\99ed5 | 197.012 | ||
| 1\14ed5 | 199.022 | ||
| 55/49 | 199.980 | Untempered tuning |