5L 3s

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For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (tritave-equivalent).
5L 3s
Pattern LLsLLsLs
Period 2/1
Generator range 3\8 (450.0¢) to 2\5 (480.0¢)
Parent MOS 3L 2s
Daughter MOSes 8L 5s, 5L 8s
Sister MOS 3L 5s
Neutralized MOS 2L 6s
TAMNAMS name oneirotonic
Equal tunings
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¢)

5L 3s refers to the octave-equivalent MOS structure with generator 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).

5L 3s is a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s): for example, the Ionian diatonic mode LLsLLLs can be distorted to the Dylathian mode LLsLLsLs.

Any edo with an interval between 450¢ and 480¢ has a 5L 3s scale. 13edo is the smallest edo with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.

5L 3s has a pentatonic MOS subset 3L 2s (SLSLL). (Note: 3L 5s scales also have 3L 2s subsets.)

Standing assumptions

The TAMNAMS system is used in this article to name 5L 3s intervals and step size ratios and step ratio ranges.

The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)

The chain of perfect 3-mossteps becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...

Thus the 13edo gamut is as follows:

J/Q& J&/K@ K/L@ L/K& L&/M@ M M&/N@ N/O@ O/N& O&/P@ P Q Q&/J@ J

The 18edo gamut is notated as follows:

J Q&/K@ J&/L@ K L K&/M@ L& M N@ M&/O@ N O P@ O& P Q P&/J@ Q@ J

The 21edo gamut:

J J& K@ K K&/L@ L L& M@ M M& N@ N N&/O@ O O& P@ P P&/Q@ Q Q& J@ J

Names

The TAMNAMS system suggests the name oneirotonic (/oʊnaɪrəˈtɒnɪk/ oh-ny-rə-TON-ik or /ənaɪrə-/ ə-ny-rə-) or 'oneiro' for short. The name oneirotonic (from Greek oneiros 'dream') is coined after the Dreamlands in H.P. Lovecraft's Dream Cycle mythos.

'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. A more 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.

Intervals

The table of oneirotonic intervals below takes the flat fourth as the generator. Given the size of the subfourth generator g, any oneirotonic interval can easily be found by noting what multiple of g it is, and multiplying the size by the number k of generators it takes to reach the interval and reducing mod 1200 if necessary (so you can use "k*g % 1200" for search engines, for plugged-in values of k and g). For example, since the major 2-step is reached by six subfourth generators, 18edo's major 2-step is 6*466.67 mod 1200 = 2800 mod 1200 = 400¢, same as the 12edo major third.

Note: In TAMNAMS, a k-step interval class in oneirotonic may be called a "k-step", "k-mosstep", or "k-oneirostep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

Notation (1/1 = J) TAMNAMS name In L's and s's # generators up Notation of 2/1 inverse TAMNAMS name In L's and s's
The 8-note MOS has the following intervals (from some root):
0 J perfect unison 0L + 0s 0 J octave 5L + 3s
1 M perfect 3-step 2L + 1s -1 O perfect 5-step 3L + 2s
2 P major 6-step 4L + 2s -2 L minor 2-step 1L + 1s
3 K major (1-)step 1L + 0s -3 Q minor 7-step 4L + 3s
4 N major 4-step 3L + 1s -4 N@ minor 4-step 2L + 2s
5 Q& major 7-step 5L + 2s -5 K@ minor (1-)step 0L + 1s
6 L& major 2-step 2L + 0s -6 P@ minor 6-step 3L + 3s
7 O& augmented 5-step 4L + 1s -7 M@ diminished 3-step 1L + 2s
The chromatic 13-note MOS (either 5L 8s, 8L 5s, or 13edo) also has the following intervals (from some root):
8 J& augmented 0-step (aka moschroma) 1L - 1s -8 J@ diminished 8-step (aka diminished mosoctave) 4L + 4s
9 M& augmented 3-step 3L + 0s -9 O@ diminished 5-step 2L + 3s
10 P& augmented 6-step 5L + 1s -10 L@ diminished 2-step 0L + 2s
11 K& augmented 1-step 2L - 1s -11 Q@ diminished 7-step 3L + 4s
12 N& augmented 4-step 4L + 0s -12 N@@ diminished 4-step 1L + 3s

Tuning ranges

Simple tunings

Table of intervals in the simplest oneirotonic tunings:

Degree Size in 13edo (basic) Size in 18edo (hard) Size in 21edo (soft) Note name on J #Gens up
unison 0\13, 0.00 0\18, 0.00 0\21, 0.00 J 0
minor step 1\13, 92.31 1\18, 66.67 2\21, 114.29 K@ -5
major step 2\13, 184.62 3\18, 200.00 3\21, 171.43 K +3
minor 2-step 3\13, 276.92 4\18, 266.67 5\21, 285.71 L -2
major 2-step 4\13, 369.23 6\18, 400.00 6\21, 342.86 L& +6
dim. 3-step 4\13, 369.23 5\18, 333.33 7\21, 400.00 M@ -7
perf. 3-step 5\13, 461.54 7\18, 466.67 8\21, 457.14 M +1
minor 4-step 6\13, 553.85 8\18, 533.33 10\21, 571.43 N@ -4
major 4-step 7\13, 646.15 10\18, 666.66 11\31, 628.57 N +4
perf. 5-step 8\13, 738.46 11\18, 733.33 13\21, 742.86 O -1
aug. 5-step 9\13, 830.77 13\18, 866.66 14\21, 800.00 O& +7
minor 6-step 9\13, 830.77 12\18, 800.00 15\21, 857.14 P@ -6
major 6-step 10\13, 923.08 14\18, 933.33 16\21, 914.29 P +2
minor 7-step 11\13, 1015.39 15\18, 1000.00 18\21, 1028.57 Q -3
major 7-step 12\13, 1107.69 17\18, 1133.33 19\21, 1085.71 Q& +5

Hypohard

Hypohard oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

Hypohard oneirotonic can be considered "meantone oneirotonic". This is because these tunings share the following features with meantone diatonic tunings:

  • The large step is a "meantone", somewhere between near-10/9 (as in 13edo) and near-9/8 (as in 18edo).
  • The major 2-mosstep (made of two large steps) is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.

Also, in 18edo and 31edo, the minor 2-mosstep is close to 7/6.

The set of identifications above is associated with A-Team temperament.

EDOs that are in the hypohard range include 13edo, 18edo, and 31edo.

  • 13edo has characteristically small 1-mossteps of about 185c. 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 (733.3c, a perfect 5-mosstep) and falling fifths (666.7c, 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.

The sizes of the generator, large step and small step of oneirotonic are as follows in various hypohard oneiro tunings.

13edo (basic) 18edo (hard) 31edo (semihard)
generator (g) 5\13, 461.54 7\18, 466.67 12\31, 464.52
L (3g - octave) 2\13, 184.62 3\18, 200.00 5\31, 193.55
s (-5g + 2 octaves) 1\13, 92.31 1\18, 66.67 2\31, 77.42

Intervals

Sortable table of major and minor intervals in hypohard oneiro tunings:

Degree Size in 13edo (basic) Size in 18edo (hard) Size in 31edo (semihard) Note name on J Approximate ratios[1] #Gens up
unison 0\13, 0.00 0\18, 0.00 0\31, 0.00 J 1/1 0
minor step 1\13, 92.31 1\18, 66.67 2\31, 77.42 K@ 21/20, 22/21 -5
major step 2\13, 184.62 3\18, 200.00 5\31, 193.55 K 9/8, 10/9 +3
minor 2-step 3\13, 276.92 4\18, 266.67 7\31, 270.97 L 7/6 -2
major 2-step 4\13, 369.23 6\18, 400.00 10\31, 387.10 L& 5/4 +6
dim. 3-step 4\13, 369.23 5\18, 333.33 9\31, 348.39 M@ 16/13, 11/9 -7
perf. 3-step 5\13, 461.54 7\18, 466.67 12\31, 464.52 M 21/16, 13/10, 17/13 +1
minor 4-step 6\13, 553.85 8\18, 533.33 14\31, 541.94 N@ 11/8 -4
major 4-step 7\13, 646.15 10\18, 666.66 17\31, 658.06 N 13/9, 16/11 +4
perf. 5-step 8\13, 738.46 11\18, 733.33 19\31, 735.48 O 26/17 -1
aug. 5-step 9\13, 830.77 13\18, 866.66 22\31, 851.61 O& 13/8, 18/11 +7
minor 6-step 9\13, 830.77 12\18, 800.00 21\31, 812.90 P@ 8/5 -6
major 6-step 10\13, 923.08 14\18, 933.33 24\31, 929.03 P 12/7 +2
minor 7-step 11\13, 1015.39 15\18, 1000.00 26\31, 1006.45 Q 9/5, 16/9 -3
major 7-step 12\13, 1107.69 17\18, 1133.33 29\31, 1122.58 Q& +5
  1. The ratio interpretations that are not valid for 18edo are italicized.

Hyposoft

Hyposoft oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of oneirotonic tunings 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.)

The sizes of the generator, large step and small step of oneirotonic are as follows in various hyposoft oneiro tunings (13edo not shown).

21edo (soft) 34edo (semisoft)
generator (g) 8\21, 457.14 13\34, 458.82
L (3g - octave) 3\21, 171.43 5\34, 176.47
s (-5g + 2 octaves) 2\21, 114.29 3\34, 105.88

Intervals

Sortable table of major and minor intervals in hyposoft tunings (13edo not shown):

Degree Size in 21edo (soft) Size in 34edo (semisoft) Note name on J Approximate ratios #Gens up
unison 0\21, 0.00 0\34, 0.00 J 1/1 0
minor step 2\21, 114.29 3\34, 105.88 K@ 16/15 -5
major step 3\21, 171.43 5\34, 176.47 K 10/9, 11/10 +3
minor 2-step 5\21, 285.71 8\34, 282.35 L 13/11, 20/17 -2
major 2-step 6\21, 342.86 10\34, 352.94 L& 11/9 +6
dim. 3-step 7\21, 400.00 11\34, 388.24 M@ 5/4 -7
perf. 3-step 8\21, 457.14 12\31, 458.82 M 13/10 +1
minor 4-step 10\21, 571.43 16\34, 564.72 N@ 18/13, 32/23 -4
major 4-step 11\21, 628.57 18\34, 635.29 N 13/9, 23/16 +4
perf. 5-step 13\21, 742.86 21\34, 741.18 O 20/13 -1
aug. 5-step 14\21, 800.00 23\34, 811.77 O& 8/5 +7
minor 6-step 15\21, 857.14 24\34, 847.06 P@ 18/11 -6
major 6-step 16\21, 914.29 26\34, 917.65 P 22/13, 17/10 +2
minor 7-step 18\21, 1028.57 29\34, 1023.53 Q 9/5 -3
major 7-step 19\21, 1085.71 31\34, 1094.12 Q& 15/8 +5

Parasoft to ultrasoft tunings

The range of oneirotonic tunings of step ratio between 6/5 and 3/2 (thus in the parasoft to ultrasoft range) may be of interest because it 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. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] The chord 10:11:13 is very well approximated in 29edo.

The sizes of the generator, large step and small step of oneirotonic are as follows in various tunings in this range.

29edo (supersoft) 37edo
generator (g) 11\29, 455.17 14\37, 454.05
L (3g - octave) 4\29, 165.52 5\37, 162.16
s (-5g + 2 octaves) 3\29, 124.14 4\37, 129.73

Intervals

The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft oneirotonic tunings.

Degree Size in 29edo (supersoft) Note name on J Approximate ratios (29edo) #Gens up
unison 0\29, 0.00 J 1/1 0
oneirochroma 1\29, 41.4 J& +8
dim. step 2\29, 82.8 K@@ -13
minor step 3\29, 124.1 K@ 14/13 -5
major step 4\29, 165.5 K 11/10 +3
aug. step 5\29, 206.9 K& 9/8 +11
dim. 2-step 6\29, 248.3 L@ 15/13 -10
minor 2-step 7\29, 289.7 L 13/11 -2
major 2-step 8\29, 331.0 L& +6
aug. 2-step 9\29, 372.4 L&& +14
doubly dim. 3-step 9\29, 372.4 M@@ -15
dim. 3-step 10\29, 413.8 M@ 14/11 -7
perf. 3-step 11\29, 455.2 M 13/10 +1
aug. 3-step 12\29, 496.6 M& 4/3 +9
dim. 4-step 13\29, 537.9 N@@ 15/11 -12
minor 4-step 14\29, 579.3 N@ 7/5 -4
major 4-step 15\29 620.7 N 10/7 +4
aug. 4-step 16\29 662.1 N& 22/15 +12
dim. 5-step 17\29, 703.4 O@ 3/2 -9
perf. 5-step 18\29, 755.2 O 20/13 -1
aug. 5-step 19\29, 786.2 O& 11/7 +7
doubly aug. 5-step 20\29 827.6 O&& +15
dim. 6-step 20\29 827.6 P@@ -14
minor 6-step 21\29 869.0 P@ -6
major 6-step 22\29, 910.3 P 22/13 +2
aug. 6-step 23\29, 951.7 P& 26/15 +10
dim. 7-step 24\29, 993.1 Q@ 16/9 -11
minor 7-step 25\29, 1034.5 Q 20/11 -3
major 7-step 26\29, 1075.9 Q& 13/7 +5
aug. 7-step 27\29, 1117.2 Q&& +13
dim. 8-step 28\29, 1158.6 J@ -8

Parahard

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).

Intervals

The intervals of the extended generator chain (-12 to +12 generators) are as follows in various oneirotonic tunings close to 23edo.

Degree Size in 23edo (superhard) Note name on J Approximate ratios (23edo) #Gens up
unison 0\23, 0.0 J 1/1 0
oneirochroma 3\23, 156.5 J& +8
minor step 1\23, 52.2 K@ -5
major step 4\23, 208.7 K +3
aug. step 7\23, 365.2 K& 21/17, inverse φ +11
dim. 2-step 2\23, 104.3 L@ 17/16 -10
minor 2-step 5\23, 260.9 L -2
major 2-step 8\23, 417.4 L& 14/11 +6
dim. 3-step 6\23, 313.0 M@ 6/5 -7
perf. 3-step 9\23, 469.6 M 21/16 +1
aug. 3-step 12\23, 626.1 M& +9
dim. 4-step 7\23, 365.2 N@@ 21/17, inverse φ -12
minor 4-step 10\23, 521.7 N@ -4
major 4-step 13\23, 678.3 N +4
aug. 4-step 16\23, 834.8 N& 34/21, φ +12
dim. 5-step 11\23, 573.9 O@ -9
perf. 5-step 14\23, 730.4 O 32/21 -1
aug. 5-step 17\23, 887.0 O& 5/3 +7
minor 6-step 15\23 782.6 P@ 11/7 -6
major 6-step 18\23, 939.1 P +2
aug. 6-step 21\23, 1095.7 P& 32/17 +10
dim. 7-step 16\23, 834.8 Q@ 34/21, φ -11
minor 7-step 19\23, 991.3 Q -3
major 7-step 22\23, 1147.8 Q& +5
dim. 8-step 20\23, 1043.5 J@ -8

Ultrahard

Buzzard is an oneirotonic rank-2 temperament in the pseudopaucitonic 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.

The sizes of the generator, large step and small step of oneirotonic are as follows in various buzzard tunings.

38edo 53edo 63edo Optimal (POTE) Buzzard tuning JI intervals represented (2.3.5.7.13 subgroup)
generator (g) 15\38, 473.68 21\53, 475.47 25\63, 476.19 475.69 4/3 21/16
L (3g - octave) 7/38, 221.04 10/53, 226.41 12/63, 228.57 227.07 8/7
s (-5g + 2 octaves) 1/38, 31.57 1/53 22.64 1/63 19.05 21.55 50/49 81/80 91/90

Intervals

Sortable table of intervals in the Dylathian mode and their Buzzard interpretations:

Degree Size in 38edo Size in 53edo Size in 63edo Size in POTE tuning Note name on Q Approximate ratios #Gens up
1 0\38, 0.00 0\53, 0.00 0\63, 0.00 0.00 Q 1/1 0
2 7\38, 221.05 10\53, 226.42 12\63, 228.57 227.07 J 8/7 +3
3 14\38, 442.10 20\53, 452.83 24\63, 457.14 453.81 K 13/10, 9/7 +6
4 15\38, 473.68 21\53, 475.47 25\63, 476.19 475.63 L 21/16 +1
5 22\38, 694.73 31\53, 701.89 37\63, 704.76 702.54 M 3/2 +4
6 29\38, 915.78 41\53, 928.30 49\63, 933.33 929.45 N 12/7, 22/13 +7
7 30\38, 947.36 42\53, 950.94 50\63, 952.38 951.27 O 26/15 +2
8 37\38, 1168.42 52\53, 1177.36 62\63, 1180.95 1178.18 P 98/50, 160/81 +5

Modes

Oneirotonic modes are named after cities in the Dreamlands.

Mode UDP Name
LLsLLsLs 7|0 Dylathian (də-LA(H)TH-iən)
LLsLsLLs 6|1 Illarnekian (ill-ar-NEK-iən)
LsLLsLLs 5|2 Celephaïsian (kel-ə-FAY-zhən)
LsLLsLsL 4|3 Ultharian (ul-THA(I)R-iən)
LsLsLLsL 3|4 Mnarian (mə-NA(I)R-iən)
sLLsLLsL 2|5 Kadathian (kə-DA(H)TH-iən)
sLLsLsLL 1|6 Hlanithian (lə-NITH-iən)
sLsLLsLL 0|7 Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

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 Illarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

See also

Scale tree

Generator Cents L s L/s Comments
3\8 450.000 1 1 1.000
17\45 453.333 6 5 1.200
14\37 454.054 5 4 1.250
34\59 454.545 9 7 1.286
11\29 455.172 4 3 1.333
30\79 455.696 11 8 1.375
19\50 456.000 7 5 1.400
27\71 456.338 10 7 1.429
8\21 457.143 3 2 1.500 L/s = 3/2
29\76 457.895 11 7 1.571
21\55 458.182 8 5 1.600
34\89 458.427 13 8 1.625 Golden oneirotonic
13\34 458.824 5 3 1.667
31\81 459.259 12 7 1.714
18\47 459.574 7 4 1.750
23\60 460.000 9 5 1.800
5\13 461.538 2 1 2.000 Basic oneirotonic
(generators smaller than this are proper)
22\57 463.158 9 4 2.250
17\44 463.636 7 3 2.333
29\75 464.000 12 5 2.400
12\31 464.516 5 2 2.500
31\80 465.000 13 5 2.600
19\49 465.306 8 3 2.667
26\67 465.672 11 4 2.750
7\18 466.667 3 1 3.000 L/s = 3/1
23\59 467.797 10 3 3.333
16\41 468.293 7 2 3.500
25\64 468.750 11 3 3.667
9\23 469.565 4 1 4.000
20\51 470.588 9 2 4.500
11\28 471.429 5 1 5.000
13\33 472.727 6 1 6.000
2\5 480.000 1 0 → inf