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| Subgroups = 5.7.11
| Subgroups = 5.7.11
| Comma basis = [[831875/823543]]
| Comma basis = [[831875/823543]]
| Edo join 1 = 43ed5 | Edo join 2 = 14ed5
| Edo join 1 = c14 | Edo join 2 = c43
| Mapping = 1; 3 7
| Mapping = 1; 3 7
| Generators = 55/49
| Generators = 55/49
| Generators tuning = 194.820
| 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>]]
| Color name =  
| 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]]. 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''' 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]].


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 (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 [[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 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 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]].
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 [[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]].
 
== Etymology ==
It is named as a play on [[didacus]], with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).
 
== 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]] 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.


== Interval chain ==
== Interval chain ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Pentadacus
! # !! Cents* !! Approximate ratios
|-
| -14||57.7||[[125/121]]
|-
| -13||252.6||[[625/539]]
|-
| -12||447.5||[[1715/1331]],[[3125/2401]]
|-
| -11||642.4||[[175/121]]
|-
| -10||837.3||[[125/77]]
|-
| -9||1032.2||[[625/343]]
|-
| -8||1227.1||[[245/121]]
|-
| -7||1422.0||[[25/11]]
|-
| -6||1616.9||[[125/49]]
|-
| -5|| 1811.8 ||[[343/121]], [[6875/2401]]
|-
| -4|| 2006.7 ||[[35/11]]
|-
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
| -3|| 2201.6||'''[[25/7]]'''
|-
|-
! 5.7.11 subgroup !!
| -2|| 2396.5||[[1375/343]]
|-
|-
| 0 || 0.0 || '''1/1''', 831875/823543  ||
| -1|| 2591.4||[[49/11]]
|-
| 0 || 0.0 || '''1/1'''
|-
|1||194.9||[[55/49]]
|-
|2||389.8||[[343/275]]
|-
|3||584.7||'''[[7/5]]'''
|-
|4||779.6||[[11/7]]
|-
|5||974.5||[[605/343]], [[2401/1375]]
|-
|6||1169.4||[[49/25]]
|-
|7||1364.3|| '''[[11/5]]'''
|-
|8||1559.2||[[121/49]]
|-
|9||1754.1||[[343/125]]
|-
|10||1949.0||[[77/25]]
|-
|11||2143.9||[[121/35]]
|-
|12||2338.8||[[1331/343]], [[2401/625]]
|-
|13||2533.7||[[539/125]]
|-
|14||2728.6||[[121/25]]
|-
|15||137.2||[[1331/1225]]
|-
|16||332.1||[[3773/3125]]
|}
<nowiki>*</nowiki>CWE tuning (5.7.11)


== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Ed5<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
! Generator (¢)
! Comments
|-
| 1\[[15ed5]]
|
| 185.754
| Technically supported but below any reasonable tuning<br>11/5 has 65 cents of damage
|-
| 2\[[29ed5]]
|
| 192.160
|
|-
| 7\[[101ed5]]
|
| 193.111
|
|-
| 5\[[72ed5]]
|
| 193.494
|
|-
|
| [[7/5]]
| 194.171
|
|-
| 3\[[43ed5]]
|
| 194.394
| Probably the best smaller ed5 tuning
|-
|
| [[77/25]]
| 194.752
|
|-
|
| [[11/5]]
| 195.001
|
|-
| 7\[[100ed5]]
|
| 195.042
|
|-
| 4\[[57ed5]]
|
| 195.531
|
|-
|
| [[11/7]]
| 195.623
|
|-
| 5\[[71ed5]]
|
| 196.219
|
|-
| 6\[[85ed5]]
|
| 196.268
|
|-
| 7\[[99ed5]]
|
| 197.012
|
|-
| 1\[[14ed5]]
|
| 199.022
|
|-
|
| [[55/49]]
| 199.980
| Untempered tuning
|}
|}


[[Category:Pentadacus| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Didacus]]
[[Category:Subgroup temperaments]]
[[Category:Non-octave temperaments]]
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