5L 3s - Revision history

Fredg999: Interwiki

2026-07-17T04:24:01Z

Interwiki

← Older revision		Revision as of 04:24, 17 July 2026
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	\| de =		\| de =
	\| es =		\| es =
	\| ja =		\| ja = 5L 3s
	\| ko = 5L3s (Korean)		\| ko = 5L3s (Korean)
	}}		}}

Overthink: /* Ultrahard tunings */ use words for these

2026-05-31T19:36:33Z

Ultrahard tunings: use words for these

← Older revision		Revision as of 19:36, 31 May 2026
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	[[Buzzard]] is a rank-2 temperament in the [[Step ratio\|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.		[[Buzzard]] is a rank-2 temperament in the [[Step ratio\|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.

	In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago\|archipelago]] harmonies, as by dividing the ~~5th~~ in 4 it obviously also divides it in two as well.		In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] and [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago\|archipelago]] harmonies, as by dividing the perfect fifth in four it obviously also divides it in two as well.

	Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.		Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.

Overthink: /* Parasoft and ultrasoft tunings */ tridec

2026-05-31T19:30:18Z

Parasoft and ultrasoft tunings: tridec

← Older revision		Revision as of 19:30, 31 May 2026
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	=== Parasoft and ultrasoft tunings ===		=== Parasoft and ultrasoft tunings ===
	The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.		The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo. The generator is close to the optimal range for [[tridec]] temperament, which approximates the 5:7:11:13 chord.

	{{MOS tunings		{{MOS tunings

ArrowHead294: substitute deprecated template

2025-05-05T13:59:08Z

substitute deprecated template

← Older revision		Revision as of 13:59, 5 May 2025
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	== Scale properties ==		== Scale properties ==
	{{MOS ~~data~~}}
			=== Intervals ===
			{{MOS intervals}}

			=== Generator chain ===
			{{MOS genchain}}

			=== Modes ===
			{{MOS mode degrees}}

	==== Proposed mode names ====		==== Proposed mode names ====

ArrowHead294: whoops

2025-05-05T13:57:06Z

whoops

← Older revision		Revision as of 13:57, 5 May 2025
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	\| Neutral = 2L 6s		\| Neutral = 2L 6s
	}}		}}
	: ''For the tritave-equivalent MOS structure with the same step pattern, see [[~~5L5L~~ ~~3s3s~~ (3/1-equivalent)]].''		: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
	{{MOS intro}}		{{MOS intro}}
	5L 3s can be seen as a [[Warped diatonic\|warped diatonic scale]], because it has one extra small step compared to diatonic ([[~~5L5L~~ ~~3s2s~~]]).		5L 3s can be seen as a [[Warped diatonic\|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).

	== Name ==		== Name ==

ArrowHead294 at 13:56, 5 May 2025

2025-05-05T13:56:38Z

← Older revision		Revision as of 13:56, 5 May 2025
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	\| Neutral = 2L 6s		\| Neutral = 2L 6s
	}}		}}
	: ''For the tritave-equivalent MOS structure with the same step pattern, see [[~~5L 3s~~ (3/1-equivalent)]].''		: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L5L 3s3s (3/1-equivalent)]].''
	{{MOS intro}}		{{MOS intro}}
	5L 3s can be seen as a [[Warped diatonic\|warped diatonic scale]], because it has one extra small step compared to diatonic ([[~~5L 2s~~]]).		5L 3s can be seen as a [[Warped diatonic\|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L5L 3s2s]]).

	== Name ==		== Name ==
	{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.		{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.

	'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].		'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].

	== Scale properties ==		== Scale properties ==
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	== Tunings==		== Tunings==
	=== Simple tunings ===		=== Simple tunings ===
	The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.		The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.

	{{MOS tunings\|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}		{{MOS tunings\|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}
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	EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team\|A-Team]] temperament.		EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team\|A-Team]] temperament.
	* 13edo has characteristically small 1-mossteps of about ~~185¢~~. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.		* 13edo has characteristically small 1-mossteps of about 185{{c}}. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.
	* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3¢, a perfect 5-mosstep) and falling fifths (666.7¢, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.		* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
	* 31edo can be used to make the major 2-mosstep a near-just 5/4.		* 31edo can be used to make the major 2-mosstep a near-just 5/4.
	* [[44edo]] (generator 17\44 = 463.~~64¢~~), [[57edo]] (generator 22\57 = 463.~~16¢~~), and [[70edo]] (generator 27\70 = 462.~~857¢~~) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.		* [[44edo]] (generator {{nowrap\|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap\|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

	{{MOS tunings\|Step Ratios=Hypohard\|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1\|Tolerance=15}}		{{MOS tunings\|Step Ratios=Hypohard\|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1\|Tolerance=15}}
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	=== Hyposoft tunings ===		=== Hyposoft tunings ===
	[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,		[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,
	* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about ~~92¢~~ to ~~114¢~~.		* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{c}} to 114{{c}}.
	* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (~~342¢~~) to 4\13 (~~369¢~~).		* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).

	* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.~~71¢~~) and Baroque diatonic semitones (114.~~29¢~~, close to quarter-comma meantone's 117.~~11¢~~).		* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
	* [[34edo]]'s 9:10:11:13 is even better.		* [[34edo]]'s 9:10:11:13 is even better.

	This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri\|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)		This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri\|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)

	{{MOS tunings\|Step Ratios=Hyposoft\|JI Ratios=1/1;		{{MOS tunings
			\| Step Ratios = Hyposoft
			\| JI Ratios =
			1/1;
	16/15;		16/15;
	10/9; 11/10;		10/9; 11/10;
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	9/5;		9/5;
	15/8;		15/8;
	2/1}}		2/1
			}}

	=== Parasoft and ultrasoft tunings ===		=== Parasoft and ultrasoft tunings ===
	The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.		The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.

	{{MOS tunings\|Step Ratios=6/5; 3/2; 4/3\|JI Ratios=1/1;		{{MOS tunings
			\| Step Ratios = 6/5; 3/2; 4/3
			\| JI Ratios =
			1/1;
	14/13;		14/13;
	11/10;		11/10;
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	20/11;		20/11;
	13/7;		13/7;
	2/1}}		2/1
			}}

	=== Parahard tunings ===		=== Parahard tunings ===
	23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).		23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).

	{{MOS tunings\|JI Ratios=1/1;		{{MOS tunings
			\| JI Ratios =
			1/1;
	21/17;		21/17;
	17/16;		17/16;
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	32/17;		32/17;
	34/21;		34/21;
	2/1\|Step Ratios=4/1}}		2/1
			\| Step Ratios = 4/1
			}}

	=== Ultrahard tunings ===		=== Ultrahard tunings ===
	{{Main\|5L 3s/Temperaments#Buzzard}}		{{Main\|5L 3s/Temperaments#Buzzard}}

	[[Buzzard]] is a rank-2 temperament in the [[Step ratio\|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.		[[Buzzard]] is a rank-2 temperament in the [[Step ratio\|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
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	Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.		Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.

	{{MOS tunings\|JI Ratios=1/1;		{{MOS tunings
			\| JI Ratios =
			1/1;
	8/7;		8/7;
	13/10;		13/10;
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	26/15;		26/15;
	49/25, 160/81;		49/25, 160/81;
	2/1\|Step Ratios=7/1; 10/1; 12/1\|Tolerance=30}}		2/1
			\| Step Ratios = 7/1; 10/1; 12/1
			\| Tolerance = 30
			}}

	== Approaches ==		== Approaches ==
	* [[5L 3s/Temperaments]]		* [[5L 3s/Temperaments]]

	== Samples ==		== Samples ==

ArrowHead294 at 18:50, 28 February 2025

2025-02-28T18:50:54Z

← Older revision		Revision as of 18:50, 28 February 2025
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	\| Neutral = 2L 6s		\| Neutral = 2L 6s
	}}		}}
	:''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''		: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
	{{MOS intro}}		{{MOS intro}}
	5L 3s can be seen as a [[Warped diatonic\|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).		5L 3s can be seen as a [[Warped diatonic\|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).
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	==== Proposed mode names ====		==== Proposed mode names ====
	The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.		The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
	{{MOS modes\|Mode Names=		{{MOS modes
	Dylathian;		\| Mode Names=
	Ilarnekian;		Dylathian $
	Celephaïsian;		Ilarnekian $
	Ultharian;		Celephaïsian $
	Mnarian;		Ultharian $
	Kadathian;		Mnarian $
	Hlanithian;		Kadathian $
	Sarnathian;		Hlanithian $
	\|Collapsed=1}}		Sarnathian $
			\| Collapsed=1
			}}

	== Tunings==		== Tunings==
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	2/1}}		2/1}}

	===Parahard tunings===		=== Parahard tunings ===
	23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).		23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).

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	== Scale tree ==		== Scale tree ==
	{{~~Scale tree~~\|~~Comments=~~		{{MOS tuning spectrum
	13/8: Golden oneirotonic (458.3592 ¢);		\| 13/8 = Golden oneirotonic (458.3592{{c}})
	13/5: Golden A-Team (465.0841 ¢);		\| 13/5 = Golden A-Team (465.0841{{c}})
	}}		}}

	[[Category:Oneirotonic\| ]] <!-- sort order in category: this page shows above A -->		[[Category:Oneirotonic\| ]] <!-- sort order in category: this page shows above A -->
	[[Category:Pages with internal sound examples]]		[[Category:Pages with internal sound examples]]

FloraC: -redundant category

2025-02-11T04:30:51Z

-redundant category

← Older revision		Revision as of 04:30, 11 February 2025
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	[[Category:Oneirotonic\| ]] <!-- sort order in category: this page shows above A -->		[[Category:Oneirotonic\| ]] <!-- sort order in category: this page shows above A -->
	[[Category:Pages with internal sound examples]]		[[Category:Pages with internal sound examples]]
	~~[[Category:8-tone scales]]~~

FloraC: /* Scale tree */ restore contents

2025-02-11T04:30:11Z

Scale tree: restore contents

← Older revision		Revision as of 04:30, 11 February 2025
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	== Scale tree ==		== Scale tree ==
	{{Scale tree}}		{{Scale tree\|Comments=
			13/8: Golden oneirotonic (458.3592 ¢);
			13/5: Golden A-Team (465.0841 ¢);
			}}

	[[Category:Oneirotonic\| ]] <!-- sort order in category: this page shows above A -->		[[Category:Oneirotonic\| ]] <!-- sort order in category: this page shows above A -->
	[[Category:Pages with internal sound examples]]		[[Category:Pages with internal sound examples]]
	[[Category:8-tone scales]]		[[Category:8-tone scales]]

FloraC: Move modes names to the "modes" section, following similar pages

2025-02-11T04:23:11Z

Move modes names to the "modes" section, following similar pages

← Older revision		Revision as of 04:23, 11 February 2025
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	'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].		'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].

	=== Proposed mode names ===		== Scale properties ==
			{{MOS data}}

			==== Proposed mode names ====
	The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.		The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
	{{MOS modes\|Mode Names=Dylathian;		{{MOS modes\|Mode Names=
			Dylathian;
	Ilarnekian;		Ilarnekian;
	Celephaïsian;		Celephaïsian;
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	Kadathian;		Kadathian;
	Hlanithian;		Hlanithian;
	Sarnathian;\|Collapsed=1}}		Sarnathian;
			\|Collapsed=1}}
	~~== Scale properties ==~~
	~~{{MOS data~~}}

	== Tunings==		== Tunings==
	===Simple tunings ===		=== Simple tunings ===
	The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.		The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.