Pentadacus - Revision history

FloraC: Fix infobox. Cleanup. Style. Recategorize.

2026-03-08T08:49:34Z

Fix infobox. Cleanup. Style. Recategorize.

← Older revision		Revision as of 08:49, 8 March 2026
Line 3:		Line 3:
	\| 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 [[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 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.		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 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 ==		== Interval chain ==
			{{Todo\|inline=1\|complete table}}

	{\| class="wikitable center-1 right-2"		{\| class="wikitable center-1 right-2"
	~~\|+ style="font-size: 105%;" \| Pentadacus~~		! # !! Cents* !! Approximate ratios
	\|-		\|-
	~~! rowspan="2" \| # !! rowspan="2" \| Cents* !! colspan="2" \| Approximate ratios~~		\| 0 \|\| 0.0 \|\| '''1/1'''
	\|-
	~~! 5.7.11 subgroup !!~~
	\|-
	\| 0 \|\| 0.0 \|\| '''1/1'''~~, 831875/823543 \|\|~~

	\|}		\|}

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

CompactStar at 01:08, 8 March 2026

2026-03-08T01:08:35Z

@@ Line 17: / Line 17: @@
 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]].
 [[Category:Rank-2 temperaments]]
 [[Category:Didacus]]

CompactStar at 01:07, 8 March 2026

2026-03-08T01:07:24Z

← Older revision		Revision as of 01:07, 8 March 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 [[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 [[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 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 pentadacus tunings are of the form (14n+1)ed5 such as ~~[[15ed5]] (although that one is barely a tuning, being even worse than 14ed5),~~ [[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.		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.

	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 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]].

CompactStar at 01:06, 8 March 2026

2026-03-08T01:06:59Z

← Older revision		Revision as of 01:06, 8 March 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 [[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 [[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, but it is a much larger than usual~~ Pentadacus ~~generator which results in poor approximations. Basically, pentadacus~~ can be thought of as a compressed 14ed5 until you hit [[5/1]], as is best ~~exmplified~~ by the [[1L 13s (5/1-equivalent)\|1L 13s<5/1>]] Pentadacus[14] MOS. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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.		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 pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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.

	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 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]].

CompactStar at 01:02, 8 March 2026

2026-03-08T01:02:06Z

← Older revision		Revision as of 01:02, 8 March 2026
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 [[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>]]. ~~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 [[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 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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5 until you hit [[5/1]], as is best exmplified by the [[1L 13s (5/1-equivalent)\|1L 13s<5/1>]] Pentadacus[14] MOS. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5 until you hit [[5/1]], as is best exmplified by the [[1L 13s (5/1-equivalent)\|1L 13s<5/1>]] Pentadacus[14] MOS. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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.


	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 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]].

CompactStar at 01:01, 8 March 2026

2026-03-08T01:01:36Z

← Older revision		Revision as of 01:01, 8 March 2026
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 [[54/49]]. Stacking 3 of these tones gives [[7/5]] and 7 of them give [[11/5]]. 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 [[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>]]. It was first discovered and named by [[User:CompactStar\|CompactStar]] in 2026.

	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.		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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5~~, at least~~ until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5 until you hit [[5/1]], as is best exmplified by the [[1L 13s (5/1-equivalent)\|1L 13s<5/1>]] Pentadacus[14] MOS. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 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 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]].

CompactStar at 01:00, 8 March 2026

2026-03-08T01:00:18Z

← Older revision		Revision as of 01:00, 8 March 2026
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	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.		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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 ~~can often be~~ very alien (being a step above even tritave-equivalent temperaments) but ~~also~~ 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.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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>]].		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>]].

CompactStar at 00:59, 8 March 2026

2026-03-08T00:59:58Z

← Older revision		Revision as of 00:59, 8 March 2026
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	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.		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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 can often be very alien (being a step above even tritave-equivalent temperaments) but also lapse into sounding like the familiar whole-tone scale at times. 6 generators in pentadacus can sound a bit like a [[~~tempered~~ octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 can often be very alien (being a step above even tritave-equivalent temperaments) but also 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>]].		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>]].

CompactStar at 00:59, 8 March 2026

2026-03-08T00:59:48Z

← Older revision		Revision as of 00:59, 8 March 2026
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	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.		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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 can often be very alien (being a step above even tritave-equivalent temperaments) but also lapse into sounding like the familiar whole-tone scale at times. 6 generators in pentadacus can sound a bit like a [[~~compressed~~ octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 can often be very alien (being a step above even tritave-equivalent temperaments) but also lapse into sounding like the familiar whole-tone scale at times. 6 generators in pentadacus can sound a bit like a [[tempered 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>]].		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>]].

CompactStar at 00:59, 8 March 2026

2026-03-08T00:59:20Z

← Older revision		Revision as of 00:59, 8 March 2026
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	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.		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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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~~. Because of this~~, Pentadacus can often be very alien (being a step above even tritave-equivalent temperaments) but also lapse into sounding like the familiar whole-tone scale at times. 6 generators in pentadacus can sound a bit like a [[compressed octave]] but it’s usually inaccurate unless you're using a very sharp tuning like 14ed5.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 can often be very alien (being a step above even tritave-equivalent temperaments) but also lapse into sounding like the familiar whole-tone scale at times. 6 generators in pentadacus can sound a bit like a [[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>]].		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>]].

← Older revision		Revision as of 01:00, 8 March 2026
Line 16:		Line 16:
	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.		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.

	[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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 ~~can often be~~ very alien (being a step above even tritave-equivalent temperaments) but ~~also~~ 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.		[[14ed5]] is an inaccurate but important tuning of Pentadacus, because in 14ed5, the whole tone generator corresponds to a single step of 14ed5, but it is a much larger than usual Pentadacus generator which results in poor approximations. Basically, pentadacus can be thought of as a compressed 14ed5, at least until you hit [[5/1]]. Because of this, many pentadacus tunings are of the form (14n+1)ed5 such as [[15ed5]] (although that one is barely a tuning, being even worse than 14ed5), [[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>]].		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>]].