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 [[55/49]]. 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.

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.

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== Extensions ==

In terms of strong extensions to the 5.7.11.13 subgroup, there are the c57 & c71 extension which maps [[13/5]] to -20 generators (but this requires the usage of the 14L 15s MOS to access 13/5), and the c15 & c29 extension which maps [[13/5]] to -6 generators (but it has high [[error]]). These are roughly equal in Dirichlet [[badness]].

In terms of strong extensions to the 5.7.11.13 subgroup, there are the c57 & c71 extension which maps [[13/5]] to -20 generators (but this requires the usage of the 14L 15s MOS to access 13/5), and the c15 & c29 extension which maps [[13/5]] to -6 generators (but it has high [[error]] of about 20 cents on 11/5 and 13/5). These are roughly equal in Dirichlet [[badness]].

The best extension to the 5.7.11.13 subgroup is actually the weak extension [[User:2^67-1]] has named ''Pentapassion'', that tempers out the Cuthbert comma [[847/845]]. This halves the generator into a semitone of roughly 97 cents, and 17 of these new generators reach 13/5 (so essentially 8.5 of the original generator). It is so named because the generator is close to the generator of [[Passion]] temperament (and as in septimal passion, 7/5 is mapped to 6 generators up). Interestingly, the first few steps of Pentapassion can sound like [[12edo]] (like how Pentadacus can sound like 6edo), although it quickly becomes out-of-tune after stacking a few generators. Pentapassion can be used with a 28L 1s<5/1> MOS.

The best extension to the 5.7.11.13 subgroup is actually the weak extension [[User:2^67-1]] has named ''Pentapassion'', that tempers out the Cuthbert comma [[847/845]]. This halves the generator into a semitone of roughly 97 cents, and 17 of these new generators reach 13/5 (so essentially 8.5 of the original generator). It is so named because the generator is close to the generator of [[Passion]] temperament. Interestingly, the first few steps of Pentapassion can sound like [[12edo]] (like how Pentadacus can sound like 6edo), although it quickly becomes out-of-tune after stacking a few generators.

== Interval chain ==

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

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|16||332.1||[[3773/3125]]

|}

*CWE tuning (5.7.11)

<nowiki>*</nowiki>CWE tuning (5.7.11)

== Tunings ==

=== Tuning spectrum ===

~~{{todo|inline=1|complete table}}~~

{| class="wikitable center-all left-4"

|-

@@ Line 14: / Line 14: @@
 '''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 [[55/49]]. 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.
+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.
@@ Line 24: / Line 24: @@
 == Extensions ==
-In terms of strong extensions to the 5.7.11.13 subgroup, there are the c57 & c71 extension which maps [[13/5]] to -20 generators (but this requires the usage of the 14L 15s MOS to access 13/5), and the c15 & c29 extension which maps [[13/5]] to -6 generators (but it has high [[error]]). These are roughly equal in Dirichlet [[badness]].
+In terms of strong extensions to the 5.7.11.13 subgroup, there are the c57 & c71 extension which maps [[13/5]] to -20 generators (but this requires the usage of the 14L 15s MOS to access 13/5), and the c15 & c29 extension which maps [[13/5]] to -6 generators (but it has high [[error]] of about 20 cents on 11/5 and 13/5). These are roughly equal in Dirichlet [[badness]].
+The best extension to the 5.7.11.13 subgroup is actually the weak extension [[User:2^67-1]] has named ''Pentapassion'', that tempers out the Cuthbert comma [[847/845]]. This halves the generator into a semitone of roughly 97 cents, and 17 of these new generators reach 13/5 (so essentially 8.5 of the original generator). It is so named because the generator is close to the generator of [[Passion]] temperament (and as in septimal passion, 7/5 is mapped to 6 generators up). Interestingly, the first few steps of Pentapassion can sound like [[12edo]] (like how Pentadacus can sound like 6edo), although it quickly becomes out-of-tune after stacking a few generators. Pentapassion can be used with a 28L 1s<5/1> MOS.
-The best extension to the 5.7.11.13 subgroup is actually the weak extension [[User:2^67-1]] has named ''Pentapassion'', that tempers out the Cuthbert comma [[847/845]]. This halves the generator into a semitone of roughly 97 cents, and 17 of these new generators reach 13/5 (so essentially 8.5 of the original generator). It is so named because the generator is close to the generator of [[Passion]] temperament. Interestingly, the first few steps of Pentapassion can sound like [[12edo]] (like how Pentadacus can sound like 6edo), although it quickly becomes out-of-tune after stacking a few generators.
 == Interval chain ==
 {| class="wikitable center-1 right-2"
@@ Line 65: / Line 66: @@
 |16||332.1||[[3773/3125]]
 |}
-*CWE tuning (5.7.11)
+<nowiki>*</nowiki>CWE tuning (5.7.11)
 == Tunings ==
 === Tuning spectrum ===
-{{todo|inline=1|complete table}}
 {| class="wikitable center-all left-4"
 |-