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| | Subgroups = 5.7.11 | | | Subgroups = 5.7.11 |
| | Comma basis = [[831875/823543]] | | | Comma basis = [[831875/823543]] |
| | Edo join 1 = 14ed5 | Edo join 2 = 15ed5 | | | Edo join 1 = 43ed5 | Edo join 2 = 14ed5 |
| | Mapping = 1; 3 7 | | | Mapping = 1; 3 7 |
| | Generators = 55/49 | | | Generators = 55/49 |
| | Generators tuning = 194.820 | | | Generators tuning = 194.820 |
| | Optimization method = CWE | | | Optimization method = CWE |
| | MOS scales = [[1L 1s (1/1-equivalent)|1L 1s<5/1>]], [[2L 1s (1/1-equivalent)|2L 1s<5/1>]],...,[[12L 1s (1/1-equivalent)|12L 1s<5/1>]], [[13L 1s (1/1-equivalent)|13L 1s<5/1>]], | | | MOS scales = [[1L 11s (1/1-equivalent)|1L 13s<5/1>]], [[1L 12s (5/1-equivalent)|1L 12s<5/1>]], [[1L 13s (1/1-equivalent)|1L 13s<5/1>]], [[14L 1s (1/1-equivalent)|14L 1s<5/1>]], [[1L 13s (1/1-equivalent)|1L 13s<5/1>]] |
| | Color name = | | | Color name = |
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| 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 [[efficiency|efficient]] traversal of the 5.7.11 subgroup. It was discovered by CompactStar in 2026.
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| [[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.
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| 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.
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| [[Category:Rank-2 temperaments]]
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| | Edo join 1 = [[13ed5|c13]] | Edo join 2 = b88
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| | Mapping = 1; -5 -4
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| | Generators = 7/5
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| | Generators tuning = 583.986
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| | Optimization method = CWE
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| | MOS scales = [[3L 1s (3/1-equivalent)|3L 1s <3/1>]], [[3L 4s (3/1-equivalent)|3L 4s <3/1>]], [[3L 7s (3/1-equivalent)|3L 7s <3/1>]]
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| | Odd limit 1 = 3.5.7 7 | Mistuning 1 = 1.40 | Complexity 1 = 7
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| | Odd limit 2 = 3.5.7 49 | Mistuning 2 = 2.80 | Complexity 2 = 13
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| }} | | }} |
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