5L 3s: Difference between revisions
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| Neutral = 2L 6s | | Neutral = 2L 6s | ||
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: ''For the tritave-equivalent MOS structure with the same step pattern, see [[ | : ''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 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 | * 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. | * 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. | * [[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 | * 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 ( | * 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. | * [[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 == | ||