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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = 5L 3s
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-05-23 15:16:06 UTC</tt>.<br>
| de =
: The original revision id was <tt>551971290</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| ko = 5L3s (Korean)
<h4>Original Wikitext content:</h4>
}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this:
The only notable harmonic entropy minimum is Vulture/[[Hemifamity temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>5L 3s</title></head><body>5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of <a class="wiki_link" href="/5edo">5edo</a> = 480¢) to 3\8 (three degrees of <a class="wiki_link" href="/8edo">8edo</a> = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this:<br />
== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.
<table class="wiki_table">
'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]].
The only notable harmonic entropy minimum is Vulture/<a class="wiki_link" href="/Hemifamity%20temperaments">Buzzard</a>, in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.</body></html></pre></div>
== Scale properties ==
=== Intervals ===
{{MOS intervals}}
=== Generator chain ===
{{MOS genchain}}
=== Modes ===
{{MOS mode degrees}}
==== Proposed mode names ====
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 $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}
== 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.
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", around the range of [[10/9]] to [[9/8]].
* The major 2-mosstep is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].
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{{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{{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.
* [[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}}
=== Hyposoft 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{{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{{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{{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.
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;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}
=== 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.
{{MOS tunings
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =
1/1;
14/13;
11/10;
9/8;
15/13;
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}
=== 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).
{{MOS tunings
| JI Ratios =
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}
=== Ultrahard tunings ===
{{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.
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.
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;
8/7;
13/10;
21/16;
3/2;
12/7, 22/13;
26/15;
49/25, 160/81;
2/1
| Step Ratios = 7/1; 10/1; 12/1
| Tolerance = 30
}}
== Approaches ==
* [[5L 3s/Temperaments]]
== Samples ==
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])
[[File:13edo Prelude in J Oneirominor.mp3]]
[[WT13C]] [[:File:13edo Prelude in J Oneirominor.mp3|Prelude II (J Oneirominor)]] ([[:File:13edo Prelude in J Oneirominor Score.pdf|score]]) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.
[[File:13edo_1MC.mp3]]
(13edo, first 30 seconds is in J Celephaïsian)
[[File:A Moment of Respite.mp3]]
(13edo, L Ilarnekian)
[[File:Lunar Approach.mp3]]
(by [[Igliashon Jones]], 13edo, J Celephaïsian)
=== 13edo Oneirotonic Modal Studies ===
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden oneirotonic (458.3592{{c}})
| 13/5 = Golden A-Team (465.0841{{c}})
}}
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).
5L 3s, named oneirotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 450 ¢ to 480 ¢, or from 720 ¢ to 750 ¢.
5L 3s can be seen as a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s).
TAMNAMS suggests the temperament-agnostic name oneirotonic as the name of 5L 3s. The name was originally used as a name for the 5L 3s scale in 13edo. '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.
Scale properties
Intervals
Intervals of 5L 3s
Intervals
Steps subtended
Range in cents
Generic
Specific
Abbrev.
0-oneirostep
Perfect 0-oneirostep
P0oneis
0
0.0 ¢
1-oneirostep
Minor 1-oneirostep
m1oneis
s
0.0 ¢ to 150.0 ¢
Major 1-oneirostep
M1oneis
L
150.0 ¢ to 240.0 ¢
2-oneirostep
Minor 2-oneirostep
m2oneis
L + s
240.0 ¢ to 300.0 ¢
Major 2-oneirostep
M2oneis
2L
300.0 ¢ to 480.0 ¢
3-oneirostep
Diminished 3-oneirostep
d3oneis
L + 2s
240.0 ¢ to 450.0 ¢
Perfect 3-oneirostep
P3oneis
2L + s
450.0 ¢ to 480.0 ¢
4-oneirostep
Minor 4-oneirostep
m4oneis
2L + 2s
480.0 ¢ to 600.0 ¢
Major 4-oneirostep
M4oneis
3L + s
600.0 ¢ to 720.0 ¢
5-oneirostep
Perfect 5-oneirostep
P5oneis
3L + 2s
720.0 ¢ to 750.0 ¢
Augmented 5-oneirostep
A5oneis
4L + s
750.0 ¢ to 960.0 ¢
6-oneirostep
Minor 6-oneirostep
m6oneis
3L + 3s
720.0 ¢ to 900.0 ¢
Major 6-oneirostep
M6oneis
4L + 2s
900.0 ¢ to 960.0 ¢
7-oneirostep
Minor 7-oneirostep
m7oneis
4L + 3s
960.0 ¢ to 1050.0 ¢
Major 7-oneirostep
M7oneis
5L + 2s
1050.0 ¢ to 1200.0 ¢
8-oneirostep
Perfect 8-oneirostep
P8oneis
5L + 3s
1200.0 ¢
Generator chain
Generator chain of 5L 3s
Bright gens
Scale degree
Abbrev.
12
Augmented 4-oneirodegree
A4oneid
11
Augmented 1-oneirodegree
A1oneid
10
Augmented 6-oneirodegree
A6oneid
9
Augmented 3-oneirodegree
A3oneid
8
Augmented 0-oneirodegree
A0oneid
7
Augmented 5-oneirodegree
A5oneid
6
Major 2-oneirodegree
M2oneid
5
Major 7-oneirodegree
M7oneid
4
Major 4-oneirodegree
M4oneid
3
Major 1-oneirodegree
M1oneid
2
Major 6-oneirodegree
M6oneid
1
Perfect 3-oneirodegree
P3oneid
0
Perfect 0-oneirodegree Perfect 8-oneirodegree
P0oneid P8oneid
−1
Perfect 5-oneirodegree
P5oneid
−2
Minor 2-oneirodegree
m2oneid
−3
Minor 7-oneirodegree
m7oneid
−4
Minor 4-oneirodegree
m4oneid
−5
Minor 1-oneirodegree
m1oneid
−6
Minor 6-oneirodegree
m6oneid
−7
Diminished 3-oneirodegree
d3oneid
−8
Diminished 8-oneirodegree
d8oneid
−9
Diminished 5-oneirodegree
d5oneid
−10
Diminished 2-oneirodegree
d2oneid
−11
Diminished 7-oneirodegree
d7oneid
−12
Diminished 4-oneirodegree
d4oneid
Modes
Scale degrees of the modes of 5L 3s
UDP
Cyclic order
Step pattern
Scale degree (oneirodegree)
0
1
2
3
4
5
6
7
8
7|0
1
LLsLLsLs
Perf.
Maj.
Maj.
Perf.
Maj.
Aug.
Maj.
Maj.
Perf.
6|1
4
LLsLsLLs
Perf.
Maj.
Maj.
Perf.
Maj.
Perf.
Maj.
Maj.
Perf.
5|2
7
LsLLsLLs
Perf.
Maj.
Min.
Perf.
Maj.
Perf.
Maj.
Maj.
Perf.
4|3
2
LsLLsLsL
Perf.
Maj.
Min.
Perf.
Maj.
Perf.
Maj.
Min.
Perf.
3|4
5
LsLsLLsL
Perf.
Maj.
Min.
Perf.
Min.
Perf.
Maj.
Min.
Perf.
2|5
8
sLLsLLsL
Perf.
Min.
Min.
Perf.
Min.
Perf.
Maj.
Min.
Perf.
1|6
3
sLLsLsLL
Perf.
Min.
Min.
Perf.
Min.
Perf.
Min.
Min.
Perf.
0|7
6
sLsLLsLL
Perf.
Min.
Min.
Dim.
Min.
Perf.
Min.
Min.
Perf.
Proposed mode names
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
Hypohard oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with meantone diatonic tunings:
The large step is a "meantone", around the range of 10/9 to 9/8.
The major 2-mosstep is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to 7/6.
EDOs that are in the hypohard range include 13edo, 18edo, and 31edo, and are associated with 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.
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.
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.
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 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 ¢).
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 ¢).
This set of JI identifications is associated with 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.)
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.
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 22edosuperpythdiatonic and a warped 24edosemaphoresemiquartal (and both nearby scales are superhard MOSes).
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 archipelago harmonies, as by dividing the 5th in 4 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.