Pentadacus: Difference between revisions

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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]], 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]]. It was first discovered and named by [[User:CompactStar|CompactStar]] in 2026.

'''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>]]. It was first discovered and named by [[User:CompactStar|CompactStar]] in 2026.

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.

[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5~~, at least~~ until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 [[Octave stretching|compressed octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.

[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5 until you hit [[5/1]], as is best exmplified by the [[1L 13s (5/1-equivalent)|1L 13s<5/1>]] Pentadacus[14] MOS. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 [[Octave stretching|compressed octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.

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 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]].

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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]], 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]].  It was first discovered and named by [[User:CompactStar|CompactStar]] in 2026.
+'''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>]]. It was first discovered and named by [[User:CompactStar|CompactStar]] in 2026.
 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.
-[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 [[Octave stretching|compressed octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.
+[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5 until you hit [[5/1]], as is best exmplified by the [[1L 13s (5/1-equivalent)|1L 13s<5/1>]] Pentadacus[14] MOS. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 [[Octave stretching|compressed octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.
-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 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]].