Pentadacus: Difference between revisions

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| Title = Pentadacus

| Subgroups = 5.7.11

| Comma basis =

| Comma basis = [[831875/823543]]

| Edo join 1 = ~~c13~~ | Edo join 2 = ~~b88~~

| Edo join 1 = c14 | Edo join 2 = c43

| Mapping = 1; ~~-5 -4~~

| Mapping = 1; 3 7

| Generators = 7/5

| Generators = 55/49

| Generators tuning = ~~583~~.~~986~~

| Generators tuning = 194.8

| 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 =~~

| Odd limit 1 = ? | Mistuning 1 = ? | Complexity 1 = ?

| Odd limit 1 = ~~3.5.7 7~~ | Mistuning 1 = ~~1.40~~ | Complexity 1 = 7

~~| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 2.80 | Complexity 2 = 13~~

}}

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.

'''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.

[[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 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 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.

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 ==

{| class="wikitable center-1 right-2"

! # !! Cents* !! Approximate ratios

|-

| 0 || 0.0 || '''1/1'''

|}

[[Category:Pentadacus| ]]

[[Category:Rank-2 temperaments]]

[[Category:Subgroup temperaments]]

@@ Line 2: / Line 2: @@
 | Title = Pentadacus
 | Subgroups = 5.7.11
-| Comma basis =
+| Comma basis = [[831875/823543]]
-| Edo join 1 = c13 | Edo join 2 = b88
+| Edo join 1 = c14 | Edo join 2 = c43
-| Mapping = 1; -5 -4
+| Mapping = 1; 3 7
-| Generators = 7/5
+| Generators = 55/49
-| Generators tuning = 583.986
+| Generators tuning = 194.8
 | 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>]]
+| 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>]]
-| Color name =
+| Odd limit 1 = ? | Mistuning 1 = ? | Complexity 1 = ?
-| Odd limit 1 = 3.5.7 7 | Mistuning 1 = 1.40 | Complexity 1 = 7
-| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 2.80 | Complexity 2 = 13
 }}
-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.
+'''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.
-[[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 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 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.
+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:Subgroup temperaments]]