Pentadacus: Difference between revisions

(31 intermediate revisions by 2 users not shown)

Line 12:

}}

'''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.

'''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~~ (~~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~~.

It is suitable for a harmonic system based exclusively on [https://oeis.org/wiki/5-rough_numbers 5-rough numbers] (numbers not divisible by 2 or 3), similar to how [[Bohlen-Pierce]]/[[BPS]] is suitable for a harmonic system based on odd numbers. 5:7:11(:25) could be used as the fundamental consonant chord in such a system, in the place of 3:5:7(:9) in BPS or of 4:5:6(:8) in meantone.

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]].

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. It should also be noted that -2 generators pentave-reduced can sound very close to [[4/1]].

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 a [[weak extension]] of pentadacus to include octaves, with 2/1 being mapped to half a pentave minus one generator. Pentadacus's connection to [[14ed5]] (which is effectively 6edo with a just 5/1) is a lot like Didacus's connection to [[6edo]].

For technical data, see [[No-2s subgroup temperaments#Pentadacus]].

== 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 ==

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

! # !! 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]]

|-

| -3|| 2201.6||'''[[25/7]]'''

|-

| -2|| 2396.5||[[1375/343]]

|-

| -1|| 2591.4||[[49/11]]

|-

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

Line 32:

Line 73:

|4||779.6||[[11/7]]

|-

|5||974.5||[[605/343]]

|5||974.5||[[605/343]], [[2401/1375]]

|-

|6||1169.4||[[49/25]]

Line 46:

Line 87:

|11||2143.9||[[121/35]]

|-

|12||2338.8||[[1331/343]],[[2401/625]]

|12||2338.8||[[1331/343]], [[2401/625]]

|-

|13||2533.7||[[539/125]]

Line 56:

Line 97:

|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 67:

Line 107:

! Generator (¢)

! Comments

|-

|

| '''0.000'''

| '''Lower bound of 5.7.11 tonality diamond monotone'''

|-

| 1\[[15ed5]]

|

| 185.754

| ~~Below~~ any reasonable tuning, 11/5 has 65 cents of damage

| Technically supported but below any reasonable tuning<br>11/5 has 65 cents of damage

|-

| 2\[[29ed5]]

|

| 192.~~1596~~

| 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

|

|-

Line 86:

Line 151:

| [[11/5]]

| 195.001

|

|-

| 7\[[100ed5]]

|

| 195.042

|

|-

| 4\[[57ed5]]

|

| 195.531

|

|-

Line 93:

Line 168:

|

|-

| 1\[[~~14ed5~~]]

| 5\[[71ed5]]

|

| 196.219

|

| ~~199~~.~~022~~

|-

| 6\[[85ed5]]

|

| 196.268

|

|-

| 7\[[99ed5]]

|

| 197.012

|

|-

| '''1\[[14ed5]]'''

|

| '''199.022'''

| '''Upper bound of 5.7.11 tonality diamond monotone'''

|-

|

Line 107:

Line 197:

[[Category:Rank-2 temperaments]]

[[Category:Subgroup temperaments]]

[[Category:Non-octave temperaments]]

@@ Line 12: / Line 12: @@
 }}
-'''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.
+'''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 (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.
+It is suitable for a harmonic system based exclusively on [https://oeis.org/wiki/5-rough_numbers 5-rough numbers] (numbers not divisible by 2 or 3), similar to how [[Bohlen-Pierce]]/[[BPS]] is suitable for a harmonic system based on odd numbers. 5:7:11(:25) could be used as the fundamental consonant chord in such a system, in the place of 3:5:7(:9) in BPS or of 4:5:6(:8) in meantone.
-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]].
+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. It should also be noted that -2 generators pentave-reduced can sound very close to [[4/1]].
+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 a [[weak extension]] of pentadacus to include octaves, with 2/1 being mapped to half a pentave minus one generator. Pentadacus's connection to [[14ed5]] (which is effectively 6edo with a just 5/1) is a lot like Didacus's connection to [[6edo]].
+For technical data, see [[No-2s subgroup temperaments#Pentadacus]].
+== 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 ==
 {| class="wikitable center-1 right-2"
 ! # !! 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]]
+|-
+| -3|| 2201.6||'''[[25/7]]'''
+|-
+| -2|| 2396.5||[[1375/343]]
+|-
+| -1|| 2591.4||[[49/11]]
 |-
 | 0 || 0.0 || '''1/1'''
@@ Line 32: / Line 73: @@
 |4||779.6||[[11/7]]
 |-
-|5||974.5||[[605/343]]
+|5||974.5||[[605/343]], [[2401/1375]]
 |-
 |6||1169.4||[[49/25]]
@@ Line 46: / Line 87: @@
 |11||2143.9||[[121/35]]
 |-
-|12||2338.8||[[1331/343]],[[2401/625]]
+|12||2338.8||[[1331/343]], [[2401/625]]
 |-
 |13||2533.7||[[539/125]]
@@ Line 56: / Line 97: @@
 |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 67: / Line 107: @@
 ! Generator (¢)
 ! Comments
+|-
+|
+|
+| '''0.000'''
+| '''Lower bound of 5.7.11 tonality diamond monotone'''
 |-
 | 1\[[15ed5]]
 |
 | 185.754
-| Below any reasonable tuning, 11/5 has 65 cents of damage
+| Technically supported but below any reasonable tuning<br>11/5 has 65 cents of damage
 |-
 | 2\[[29ed5]]
 |
-| 192.1596
+| 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
 |
 |-
@@ Line 86: / Line 151: @@
 | [[11/5]]
 | 195.001
+|
+|-
+| 7\[[100ed5]]
+|
+| 195.042
+|
+|-
+| 4\[[57ed5]]
+|
+| 195.531
 |
 |-
@@ Line 93: / Line 168: @@
 |
 |-
-| 1\[[14ed5]]
+| 5\[[71ed5]]
+|
+| 196.219
 |
-| 199.022
+|-
+| 6\[[85ed5]]
+|
+| 196.268
+|
+|-
+| 7\[[99ed5]]
+|
+| 197.012
+|
+|-
+| '''1\[[14ed5]]'''
 |
+| '''199.022'''
+| '''Upper bound of 5.7.11 tonality diamond monotone'''
 |-
 |
@@ Line 107: / Line 197: @@
 [[Category:Rank-2 temperaments]]
 [[Category:Subgroup temperaments]]
+[[Category:Non-octave temperaments]]