Pentadacus: Difference between revisions

Line 2:

| Title = Pentadacus

| Subgroups = 5.7.11

| Comma basis =

| Comma basis = [[831875/823543]]

| Edo join 1 = 14ed5 | Edo join 2 = 15ed5

| Generators = 55/49

| Generators tuning = 194.820

| Optimization method = CWE

| 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>]]

| Color name =

}}

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.

[[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.

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.

[[Category:Rank-2 temperaments]]

| Edo join 1 = [[13ed5|c13]] | Edo join 2 = b88

| Mapping = 1; -5 -4

@@ Line 2: / Line 2: @@
 | Title = Pentadacus
 | Subgroups = 5.7.11
-| Comma basis =
+| Comma basis = [[831875/823543]]
+| Edo join 1 = 14ed5 | Edo join 2 = 15ed5
+| Generators = 55/49
+| Generators tuning = 194.820
+| Optimization method = CWE
+| 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>]]
+| Color name =
+}}
+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.
+[[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.
+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.
+[[Category:Rank-2 temperaments]]
 | Edo join 1 = [[13ed5|c13]] | Edo join 2 = b88
 | Mapping = 1; -5 -4