Pentadacus: Difference between revisions
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| Subgroups = 5.7.11 | | Subgroups = 5.7.11 | ||
| Comma basis = [[831875/823543]] | | Comma basis = [[831875/823543]] | ||
| Edo join 1 = | | Edo join 1 = c14 | Edo join 2 = c43 | ||
| Mapping = 1; 3 7 | | Mapping = 1; 3 7 | ||
| Generators = 55/49 | | Generators = 55/49 | ||
| Generators tuning = 194. | | Generators tuning = 194.8 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| MOS scales = [[1L 12s (5/1-equivalent)|1L 12s<5/1>]], [[1L 13s (5/1-equivalent)|1L 13s<5/1>]], [[14L 1s (5/1-equivalent)|14L 1s<5/1>]], [[14L 15s (5/1-equivalent)|14L 15s<5/1>]] | | MOS scales = [[1L 12s (5/1-equivalent)|1L 12s<5/1>]], [[1L 13s (5/1-equivalent)|1L 13s<5/1>]], [[14L 1s (5/1-equivalent)|14L 1s<5/1>]], [[14L 15s (5/1-equivalent)|14L 15s<5/1>]] | ||
| | | Odd limit 1 = ? | Mistuning 1 = ? | Complexity 1 = ? | ||
}} | }} | ||
'''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]]. | '''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-equivalent)|1L 1s<5/1>]], [[1L 2s (5/1-equivalent)|1L 2s<5/1>]], etc. until ending the monolarge mos chain at [[1L 13s (5/1-equivalent)|1L 13s<5/1>]], followed by [[14L 1s (5/1-equivalent)|14L 1s<5/1>]], [[14L 15s (5/1-equivalent)|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 [[efficiency|efficient]] traversal of the 5.7.11 subgroup. It was first discovered and named by [[User:CompactStar|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-equivalent)|1L 13s<5/1>]] Pentadacus[14] mos. Because of this, many good smaller pentadacus tunings are of the form (14''n'' + 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 [[stretched and compressed tuning|compressed octave]] but it is usually inaccurate unless you are using a very sharp tuning like 14ed5. | |||
Pentadacus | 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 [[didacus|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 == | |||
{{Todo|inline=1|complete table}} | |||
{| class="wikitable center-1 right-2" | |||
! # !! Cents* !! Approximate ratios | |||
|- | |||
| 0 || 0.0 || '''1/1''' | |||
|} | |||
[[Category:Pentadacus| ]] <!-- main article --> | |||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category: | [[Category:Subgroup temperaments]] | ||
Latest revision as of 08:49, 8 March 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 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 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.
Interval chain
|
Todo: complete table |
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |