Pentadacus: Difference between revisions

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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, 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, and 6 generators in pentadacus can sound like a tempered octave but it’s usually quite inaccurate and dissonant. 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 [[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]].

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]].

[[Category:Rank-2 temperaments]]

[[Category:Didacus]]

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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, 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, and 6 generators in pentadacus can sound like a tempered octave but it’s usually quite inaccurate and dissonant. 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 [[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]].
+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]].
 [[Category:Rank-2 temperaments]]
 [[Category:Didacus]]