5L 3s

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Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 3\8 to 2\5 (450.0¢ to 480.0¢)
Dark 3\5 to 5\8 (720.0¢ to 750.0¢)
Related MOS scales
Parent 3L 2s
Sister 3L 5s
Daughters 8L 5s, 5L 8s
Neutralized 2L 6s
2-Flought 13L 3s, 5L 11s
Equal tunings
Equalized (L:s = 1:1) 3\8 (450.0¢)
Supersoft (L:s = 4:3) 11\29 (455.2¢)
Soft (L:s = 3:2) 8\21 (457.1¢)
Semisoft (L:s = 5:3) 13\34 (458.8¢)
Basic (L:s = 2:1) 5\13 (461.5¢)
Semihard (L:s = 5:2) 12\31 (464.5¢)
Hard (L:s = 3:1) 7\18 (466.7¢)
Superhard (L:s = 4:1) 9\23 (469.6¢)
Collapsed (L:s = 1:0) 2\5 (480.0¢)
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).

5L 3s 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).

Name

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.

Proposed mode names

The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.

Modes of 5L 3s
UDP Cyclic
order		 Step
pattern		 Mode names
7|0 1 LLsLLsLs Dylathian
6|1 4 LLsLsLLs Ilarnekian
5|2 7 LsLLsLLs Celephaïsian
4|3 2 LsLLsLsL Ultharian
3|4 5 LsLsLLsL Mnarian
2|5 8 sLLsLLsL Kadathian
1|6 3 sLLsLsLL Hlanithian
0|7 6 sLsLLsLL Sarnathian

Scale properties

Intervals

The intervals of 5L 3s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-mosstep and perfect 8-mosstep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.

Intervals of 5L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0¢
1-mosstep Minor 1-mosstep m1ms s 0.0¢ to 150.0¢
Major 1-mosstep M1ms L 150.0¢ to 240.0¢
2-mosstep Minor 2-mosstep m2ms L + s 240.0¢ to 300.0¢
Major 2-mosstep M2ms 2L 300.0¢ to 480.0¢
3-mosstep Diminished 3-mosstep d3ms L + 2s 240.0¢ to 450.0¢
Perfect 3-mosstep P3ms 2L + s 450.0¢ to 480.0¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 480.0¢ to 600.0¢
Major 4-mosstep M4ms 3L + s 600.0¢ to 720.0¢
5-mosstep Perfect 5-mosstep P5ms 3L + 2s 720.0¢ to 750.0¢
Augmented 5-mosstep A5ms 4L + s 750.0¢ to 960.0¢
6-mosstep Minor 6-mosstep m6ms 3L + 3s 720.0¢ to 900.0¢
Major 6-mosstep M6ms 4L + 2s 900.0¢ to 960.0¢
7-mosstep Minor 7-mosstep m7ms 4L + 3s 960.0¢ to 1050.0¢
Major 7-mosstep M7ms 5L + 2s 1050.0¢ to 1200.0¢
8-mosstep Perfect 8-mosstep P8ms 5L + 3s 1200.0¢

Generator chain

A chain of bright generators, each a perfect 3-mosstep, produces the following scale degrees. A chain of 8 bright generators contains the scale degrees of one of the modes of 5L 3s. Expanding the chain to 13 scale degrees produces the modes of either 8L 5s (for soft-of-basic tunings) or 5L 8s (for hard-of-basic tunings).

Generator chain of 5L 3s
Bright gens Scale Degree - 12 Augmented 4-mosdegree A4md
11 Augmented 1-mosdegree A1md
10 Augmented 6-mosdegree A6md
9 Augmented 3-mosdegree A3md
8 Augmented 0-mosdegree A0md
7 Augmented 5-mosdegree A5md
6 Major 2-mosdegree M2md
5 Major 7-mosdegree M7md
4 Major 4-mosdegree M4md
3 Major 1-mosdegree M1md
2 Major 6-mosdegree M6md
1 Perfect 3-mosdegree P3md
0 Perfect 0-mosdegree
Perfect 8-mosdegree		 P0md
P8md
-1 Perfect 5-mosdegree P5md
-2 Minor 2-mosdegree m2md
-3 Minor 7-mosdegree m7md
-4 Minor 4-mosdegree m4md
-5 Minor 1-mosdegree m1md
-6 Minor 6-mosdegree m6md
-7 Diminished 3-mosdegree d3md
-8 Diminished 8-mosdegree d8md
-9 Diminished 5-mosdegree d5md
-10 Diminished 2-mosdegree d2md
-11 Diminished 7-mosdegree d7md
-12 Diminished 4-mosdegree d4md

Modes

Scale degrees of the modes of 5L 3s 
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
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.

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.


Simple Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Hard (3:1)
18edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\13 0.0 0\18 0.0 0\21 0.0
Minor 1-mosdegree m1md 1\13 92.3 1\18 66.7 2\21 114.3
Major 1-mosdegree M1md 2\13 184.6 3\18 200.0 3\21 171.4
Minor 2-mosdegree m2md 3\13 276.9 4\18 266.7 5\21 285.7
Major 2-mosdegree M2md 4\13 369.2 6\18 400.0 6\21 342.9
Diminished 3-mosdegree d3md 4\13 369.2 5\18 333.3 7\21 400.0
Perfect 3-mosdegree P3md 5\13 461.5 7\18 466.7 8\21 457.1
Minor 4-mosdegree m4md 6\13 553.8 8\18 533.3 10\21 571.4
Major 4-mosdegree M4md 7\13 646.2 10\18 666.7 11\21 628.6
Perfect 5-mosdegree P5md 8\13 738.5 11\18 733.3 13\21 742.9
Augmented 5-mosdegree A5md 9\13 830.8 13\18 866.7 14\21 800.0
Minor 6-mosdegree m6md 9\13 830.8 12\18 800.0 15\21 857.1
Major 6-mosdegree M6md 10\13 923.1 14\18 933.3 16\21 914.3
Minor 7-mosdegree m7md 11\13 1015.4 15\18 1000.0 18\21 1028.6
Major 7-mosdegree M7md 12\13 1107.7 17\18 1133.3 19\21 1085.7
Perfect 8-mosdegree P8md 13\13 1200.0 18\18 1200.0 21\21 1200.0


Hypohard tunings

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.


Hypohard Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Semihard (5:2)
31edo 		Hard (3:1)
18edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\13 0.0 0\31 0.0 0\18 0.0
Minor 1-mosdegree m1md 1\13 92.3 2\31 77.4 1\18 66.7
Major 1-mosdegree M1md 2\13 184.6 5\31 193.5 3\18 200.0
Minor 2-mosdegree m2md 3\13 276.9 7\31 271.0 4\18 266.7
Major 2-mosdegree M2md 4\13 369.2 10\31 387.1 6\18 400.0
Diminished 3-mosdegree d3md 4\13 369.2 9\31 348.4 5\18 333.3
Perfect 3-mosdegree P3md 5\13 461.5 12\31 464.5 7\18 466.7
Minor 4-mosdegree m4md 6\13 553.8 14\31 541.9 8\18 533.3
Major 4-mosdegree M4md 7\13 646.2 17\31 658.1 10\18 666.7
Perfect 5-mosdegree P5md 8\13 738.5 19\31 735.5 11\18 733.3
Augmented 5-mosdegree A5md 9\13 830.8 22\31 851.6 13\18 866.7
Minor 6-mosdegree m6md 9\13 830.8 21\31 812.9 12\18 800.0
Major 6-mosdegree M6md 10\13 923.1 24\31 929.0 14\18 933.3
Minor 7-mosdegree m7md 11\13 1015.4 26\31 1006.5 15\18 1000.0
Major 7-mosdegree M7md 12\13 1107.7 29\31 1122.6 17\18 1133.3
Perfect 8-mosdegree P8md 13\13 1200.0 31\31 1200.0 18\18 1200.0


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¢ 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¢).
  • 34edo's 9:10:11:13 is even better.

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.)


Hyposoft Tunings of 5L 3s
Scale degree Abbrev. Soft (3:2)
21edo 		Semisoft (5:3)
34edo 		Basic (2:1)
13edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\21 0.0 0\34 0.0 0\13 0.0
Minor 1-mosdegree m1md 2\21 114.3 3\34 105.9 1\13 92.3
Major 1-mosdegree M1md 3\21 171.4 5\34 176.5 2\13 184.6
Minor 2-mosdegree m2md 5\21 285.7 8\34 282.4 3\13 276.9
Major 2-mosdegree M2md 6\21 342.9 10\34 352.9 4\13 369.2
Diminished 3-mosdegree d3md 7\21 400.0 11\34 388.2 4\13 369.2
Perfect 3-mosdegree P3md 8\21 457.1 13\34 458.8 5\13 461.5
Minor 4-mosdegree m4md 10\21 571.4 16\34 564.7 6\13 553.8
Major 4-mosdegree M4md 11\21 628.6 18\34 635.3 7\13 646.2
Perfect 5-mosdegree P5md 13\21 742.9 21\34 741.2 8\13 738.5
Augmented 5-mosdegree A5md 14\21 800.0 23\34 811.8 9\13 830.8
Minor 6-mosdegree m6md 15\21 857.1 24\34 847.1 9\13 830.8
Major 6-mosdegree M6md 16\21 914.3 26\34 917.6 10\13 923.1
Minor 7-mosdegree m7md 18\21 1028.6 29\34 1023.5 11\13 1015.4
Major 7-mosdegree M7md 19\21 1085.7 31\34 1094.1 12\13 1107.7
Perfect 8-mosdegree P8md 21\21 1200.0 34\34 1200.0 13\13 1200.0


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.


Soft Tunings of 5L 3s
Scale degree Abbrev. 6:5
45edo 		Supersoft (4:3)
29edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\45 0.0 0\29 0.0 0\21 0.0
Minor 1-mosdegree m1md 5\45 133.3 3\29 124.1 2\21 114.3
Major 1-mosdegree M1md 6\45 160.0 4\29 165.5 3\21 171.4
Minor 2-mosdegree m2md 11\45 293.3 7\29 289.7 5\21 285.7
Major 2-mosdegree M2md 12\45 320.0 8\29 331.0 6\21 342.9
Diminished 3-mosdegree d3md 16\45 426.7 10\29 413.8 7\21 400.0
Perfect 3-mosdegree P3md 17\45 453.3 11\29 455.2 8\21 457.1
Minor 4-mosdegree m4md 22\45 586.7 14\29 579.3 10\21 571.4
Major 4-mosdegree M4md 23\45 613.3 15\29 620.7 11\21 628.6
Perfect 5-mosdegree P5md 28\45 746.7 18\29 744.8 13\21 742.9
Augmented 5-mosdegree A5md 29\45 773.3 19\29 786.2 14\21 800.0
Minor 6-mosdegree m6md 33\45 880.0 21\29 869.0 15\21 857.1
Major 6-mosdegree M6md 34\45 906.7 22\29 910.3 16\21 914.3
Minor 7-mosdegree m7md 39\45 1040.0 25\29 1034.5 18\21 1028.6
Major 7-mosdegree M7md 40\45 1066.7 26\29 1075.9 19\21 1085.7
Perfect 8-mosdegree P8md 45\45 1200.0 29\29 1200.0 21\21 1200.0


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).


Superhard Tuning of 5L 3s
Scale degree Abbrev. Superhard (4:1)
23edo
Steps ¢
Perfect 0-mosdegree P0md 0\23 0.0
Minor 1-mosdegree m1md 1\23 52.2
Major 1-mosdegree M1md 4\23 208.7
Minor 2-mosdegree m2md 5\23 260.9
Major 2-mosdegree M2md 8\23 417.4
Diminished 3-mosdegree d3md 6\23 313.0
Perfect 3-mosdegree P3md 9\23 469.6
Minor 4-mosdegree m4md 10\23 521.7
Major 4-mosdegree M4md 13\23 678.3
Perfect 5-mosdegree P5md 14\23 730.4
Augmented 5-mosdegree A5md 17\23 887.0
Minor 6-mosdegree m6md 15\23 782.6
Major 6-mosdegree M6md 18\23 939.1
Minor 7-mosdegree m7md 19\23 991.3
Major 7-mosdegree M7md 22\23 1147.8
Perfect 8-mosdegree P8md 23\23 1200.0


Ultrahard tunings

Main article: 5L 3s/Temperaments § Buzzard

Buzzard is a rank-2 temperament in the 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 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.


Ultrahard Tunings of 5L 3s
Scale degree Abbrev. 7:1
38edo 		10:1
53edo 		12:1
63edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\38 0.0 0\53 0.0 0\63 0.0
Minor 1-mosdegree m1md 1\38 31.6 1\53 22.6 1\63 19.0
Major 1-mosdegree M1md 7\38 221.1 10\53 226.4 12\63 228.6
Minor 2-mosdegree m2md 8\38 252.6 11\53 249.1 13\63 247.6
Major 2-mosdegree M2md 14\38 442.1 20\53 452.8 24\63 457.1
Diminished 3-mosdegree d3md 9\38 284.2 12\53 271.7 14\63 266.7
Perfect 3-mosdegree P3md 15\38 473.7 21\53 475.5 25\63 476.2
Minor 4-mosdegree m4md 16\38 505.3 22\53 498.1 26\63 495.2
Major 4-mosdegree M4md 22\38 694.7 31\53 701.9 37\63 704.8
Perfect 5-mosdegree P5md 23\38 726.3 32\53 724.5 38\63 723.8
Augmented 5-mosdegree A5md 29\38 915.8 41\53 928.3 49\63 933.3
Minor 6-mosdegree m6md 24\38 757.9 33\53 747.2 39\63 742.9
Major 6-mosdegree M6md 30\38 947.4 42\53 950.9 50\63 952.4
Minor 7-mosdegree m7md 31\38 978.9 43\53 973.6 51\63 971.4
Major 7-mosdegree M7md 37\38 1168.4 52\53 1177.4 62\63 1181.0
Perfect 8-mosdegree P8md 38\38 1200.0 53\53 1200.0 63\63 1200.0


Approaches

Samples

The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo oneirotonic (score)

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

(13edo, first 30 seconds is in J Celephaïsian)

(13edo, L Ilarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

13edo Oneirotonic Modal Studies

Scale tree

Scale Tree and Tuning Spectrum of 5L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
3\8 450.000 750.000 1:1 1.000 Equalized 5L 3s
17\45 453.333 746.667 6:5 1.200
14\37 454.054 745.946 5:4 1.250
25\66 454.545 745.455 9:7 1.286
11\29 455.172 744.828 4:3 1.333 Supersoft 5L 3s
30\79 455.696 744.304 11:8 1.375
19\50 456.000 744.000 7:5 1.400
27\71 456.338 743.662 10:7 1.429
8\21 457.143 742.857 3:2 1.500 Soft 5L 3s
29\76 457.895 742.105 11:7 1.571
21\55 458.182 741.818 8:5 1.600
34\89 458.427 741.573 13:8 1.625
13\34 458.824 741.176 5:3 1.667 Semisoft 5L 3s
31\81 459.259 740.741 12:7 1.714
18\47 459.574 740.426 7:4 1.750
23\60 460.000 740.000 9:5 1.800
5\13 461.538 738.462 2:1 2.000 Basic 5L 3s
Scales with tunings softer than this are proper
22\57 463.158 736.842 9:4 2.250
17\44 463.636 736.364 7:3 2.333
29\75 464.000 736.000 12:5 2.400
12\31 464.516 735.484 5:2 2.500 Semihard 5L 3s
31\80 465.000 735.000 13:5 2.600
19\49 465.306 734.694 8:3 2.667
26\67 465.672 734.328 11:4 2.750
7\18 466.667 733.333 3:1 3.000 Hard 5L 3s
23\59 467.797 732.203 10:3 3.333
16\41 468.293 731.707 7:2 3.500
25\64 468.750 731.250 11:3 3.667
9\23 469.565 730.435 4:1 4.000 Superhard 5L 3s
20\51 470.588 729.412 9:2 4.500
11\28 471.429 728.571 5:1 5.000
13\33 472.727 727.273 6:1 6.000
2\5 480.000 720.000 1:0 → ∞ Collapsed 5L 3s
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