Pentadacus - Revision history

Overthink: info on how mapping relates to didacus

2026-08-27T04:07:12Z

info on how mapping relates to didacus

← Older revision		Revision as of 04:07, 27 August 2026
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	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 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]].		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]].

	== Etymology ==		== Etymology ==

CompactStar: /* Interval chain */

2026-08-18T07:30:57Z

Interval chain

← Older revision		Revision as of 07:30, 18 August 2026
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	{\| class="wikitable center-1 right-2"		{\| class="wikitable center-1 right-2"
	! # !! Cents* !! Approximate ratios		! # !! 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]]		\| -9\|\|1032.2\|\|[[625/343]]
Line 36:		Line 46:
	\| -8\|\|1227.1\|\|[[245/121]]		\| -8\|\|1227.1\|\|[[245/121]]
	\|-		\|-
	\| -7\|\|1422.0\|\|25/11		\| -7\|\|1422.0\|\|[[25/11]]
	\|-		\|-
	\| -6\|\|1616.9\|\|[[125/49]]		\| -6\|\|1616.9\|\|[[125/49]]

CompactStar: /* Interval chain */

2026-08-18T07:28:03Z

Interval chain

← Older revision		Revision as of 07:28, 18 August 2026
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	! # !! Cents* !! Approximate ratios		! # !! Cents* !! Approximate ratios
	\|-		\|-
	\|-9\|\|1032.2\|\|[[625/343]]		\| -9\|\|1032.2\|\|[[625/343]]
	\|-		\|-
	\|-8\|\|1227.1\|\|[[245/121]]		\| -8\|\|1227.1\|\|[[245/121]]
	\|-		\|-
	\|-7\|\|1422.0\|\|25/11		\| -7\|\|1422.0\|\|25/11
	\|-		\|-
	\|-6\|\|1616.9\|\|[[125/49]]		\| -6\|\|1616.9\|\|[[125/49]]
	\|-		\|-
	\|-5\|\| 1811.8 \|\|[[343/121]], [[6875/2401]]		\| -5\|\| 1811.8 \|\|[[343/121]], [[6875/2401]]
	\|-		\|-
	\|-4\|\| 2006.7 \|\|[[35/11]]		\| -4\|\| 2006.7 \|\|[[35/11]]
	\|-		\|-
	\|-3\|\| 2201.6\|\|'''[[25/7]]'''		\| -3\|\| 2201.6\|\|'''[[25/7]]'''
	\|-		\|-
	\|-2\|\| 2396.5\|\|[[1375/343]]		\| -2\|\| 2396.5\|\|[[1375/343]]
	\|-		\|-
	\|-1\|\| 2591.4\|\|[[49/11]]		\| -1\|\| 2591.4\|\|[[49/11]]
	\|-		\|-
	\| 0 \|\| 0.0 \|\| '''1/1'''		\| 0 \|\| 0.0 \|\| '''1/1'''

CompactStar at 07:27, 18 August 2026

2026-08-18T07:27:33Z

@@ Line 31: / Line 31: @@
 {| class="wikitable center-1 right-2"
 ! # !! Cents* !! Approximate ratios
 |-
 | 0 || 0.0 || '''1/1'''

CompactStar: /* Extensions */

2026-08-08T00:00:27Z

Extensions

← Older revision		Revision as of 00:00, 8 August 2026
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	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]].		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 (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 ==

CompactStar at 00:00, 8 August 2026

2026-08-08T00:00:15Z

← Older revision		Revision as of 00:00, 8 August 2026
Line 26:		Line 26:
	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]].		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. 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, 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 ==

CompactStar at 04:31, 7 August 2026

2026-08-07T04:31:54Z

← Older revision		Revision as of 04:31, 7 August 2026
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]].		'''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.		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.

CompactStar: /* Interval chain */

2026-08-07T04:29:34Z

Interval chain

← Older revision		Revision as of 04:29, 7 August 2026
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	\|16\|\|332.1\|\|[[3773/3125]]		\|16\|\|332.1\|\|[[3773/3125]]
	\|}		\|}
	*CWE tuning (5.7.11)		<nowiki>*</nowiki>CWE tuning (5.7.11)

	== Tunings ==		== Tunings ==

CompactStar: /* Tuning spectrum */

2026-08-07T04:29:22Z

Tuning spectrum

← Older revision		Revision as of 04:29, 7 August 2026
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	== Tunings ==		== Tunings ==
	=== Tuning spectrum ===		=== Tuning spectrum ===
	~~{{todo\|inline=1\|complete table}}~~
	{\| class="wikitable center-all left-4"		{\| class="wikitable center-all left-4"
	\|-		\|-

CompactStar: /* Extensions */

2026-08-07T04:22:22Z

Extensions

← Older revision		Revision as of 04:22, 7 August 2026
Line 24:		Line 24:

	== Extensions ==		== 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. 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. Pentapassion can be used with a 28L 1s<5/1> MOS.