5L 3s

↖ 4L 2s	↑ 5L 2s	6L 2s ↗
← 4L 3s	5L 3s	6L 3s →
↙ 4L 4s	↓ 5L 4s	6L 4s ↘

┌╥╥┬╥╥┬╥┬┐ │║║│║║│║││ ││││││││││ └┴┴┴┴┴┴┴┴┘
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 ¢)
TAMNAMS information
Name	oneirotonic
Prefix	oneiro-
Abbrev.	onei
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, 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 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 warped to the Dylathian mode LLsLLsLs.

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

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.

'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

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

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 ¢

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 names

Oneirotonic modes are named after cities in the Dreamlands.

Mode	UDP	Name
LLsLLsLs	7|0	Dylathian
LLsLsLLs	6|1	Ilarnekian
LsLLsLLs	5|2	Celephaïsian
LsLLsLsL	4|3	Ultharian
LsLsLLsL	3|4	Mnarian
sLLsLLsL	2|5	Kadathian
sLLsLsLL	1|6	Hlanithian
sLsLLsLL	0|7	Sarnathian

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)	#Gens up
unison	0\13, 0.00	0\18, 0.00	0\21, 0.00	0
minor step	1\13, 92.31	1\18, 66.67	2\21, 114.29	-5
major step	2\13, 184.62	3\18, 200.00	3\21, 171.43	+3
minor 2-step	3\13, 276.92	4\18, 266.67	5\21, 285.71	-2
major 2-step	4\13, 369.23	6\18, 400.00	6\21, 342.86	+6
dim. 3-step	4\13, 369.23	5\18, 333.33	7\21, 400.00	-7
perf. 3-step	5\13, 461.54	7\18, 466.67	8\21, 457.14	+1
minor 4-step	6\13, 553.85	8\18, 533.33	10\21, 571.43	-4
major 4-step	7\13, 646.15	10\18, 666.66	11\31, 628.57	+4
perf. 5-step	8\13, 738.46	11\18, 733.33	13\21, 742.86	-1
aug. 5-step	9\13, 830.77	13\18, 866.66	14\21, 800.00	+7
minor 6-step	9\13, 830.77	12\18, 800.00	15\21, 857.14	-6
major 6-step	10\13, 923.08	14\18, 933.33	16\21, 914.29	+2
minor 7-step	11\13, 1015.39	15\18, 1000.00	18\21, 1028.57	-3
major 7-step	12\13, 1107.69	17\18, 1133.33	19\21, 1085.71	+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 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.

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)	Approximate ratios	#Gens up
unison	0\13, 0.00	0\18, 0.00	0\31, 0.00	1/1	0
minor step	1\13, 92.31	1\18, 66.67	2\31, 77.42	21/20, 22/21	-5
major step	2\13, 184.62	3\18, 200.00	5\31, 193.55	9/8, 10/9	+3
minor 2-step	3\13, 276.92	4\18, 266.67	7\31, 270.97	7/6	-2
major 2-step	4\13, 369.23	6\18, 400.00	10\31, 387.10	5/4	+6
dim. 3-step	4\13, 369.23	5\18, 333.33	9\31, 348.39	16/13, 11/9	-7
perf. 3-step	5\13, 461.54	7\18, 466.67	12\31, 464.52	21/16, 13/10, 17/13	+1
minor 4-step	6\13, 553.85	8\18, 533.33	14\31, 541.94	11/8	-4
major 4-step	7\13, 646.15	10\18, 666.66	17\31, 658.06	13/9, 16/11	+4
perf. 5-step	8\13, 738.46	11\18, 733.33	19\31, 735.48	26/17	-1
aug. 5-step	9\13, 830.77	13\18, 866.66	22\31, 851.61	13/8, 18/11	+7
minor 6-step	9\13, 830.77	12\18, 800.00	21\31, 812.90	8/5	-6
major 6-step	10\13, 923.08	14\18, 933.33	24\31, 929.03	12/7	+2
minor 7-step	11\13, 1015.39	15\18, 1000.00	26\31, 1006.45	9/5, 16/9	-3
major 7-step	12\13, 1107.69	17\18, 1133.33	29\31, 1122.58		+5

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)	Approximate ratios	#Gens up
unison	0\21, 0.00	0\34, 0.00	1/1	0
minor step	2\21, 114.29	3\34, 105.88	16/15	-5
major step	3\21, 171.43	5\34, 176.47	10/9, 11/10	+3
minor 2-step	5\21, 285.71	8\34, 282.35	13/11, 20/17	-2
major 2-step	6\21, 342.86	10\34, 352.94	11/9	+6
dim. 3-step	7\21, 400.00	11\34, 388.24	5/4	-7
perf. 3-step	8\21, 457.14	12\31, 458.82	13/10	+1
minor 4-step	10\21, 571.43	16\34, 564.72	18/13, 32/23	-4
major 4-step	11\21, 628.57	18\34, 635.29	13/9, 23/16	+4
perf. 5-step	13\21, 742.86	21\34, 741.18	20/13	-1
aug. 5-step	14\21, 800.00	23\34, 811.77	8/5	+7
minor 6-step	15\21, 857.14	24\34, 847.06	18/11	-6
major 6-step	16\21, 914.29	26\34, 917.65	22/13, 17/10	+2
minor 7-step	18\21, 1028.57	29\34, 1023.53	9/5	-3
major 7-step	19\21, 1085.71	31\34, 1094.12	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)	Approximate ratios (29edo)	#Gens up
unison	0\29, 0.00	1/1	0
oneirochroma	1\29, 41.4		+8
dim. step	2\29, 82.8		-13
minor step	3\29, 124.1	14/13	-5
major step	4\29, 165.5	11/10	+3
aug. step	5\29, 206.9	9/8	+11
dim. 2-step	6\29, 248.3	15/13	-10
minor 2-step	7\29, 289.7	13/11	-2
major 2-step	8\29, 331.0		+6
aug. 2-step	9\29, 372.4		+14
doubly dim. 3-step	9\29, 372.4		-15
dim. 3-step	10\29, 413.8	14/11	-7
perf. 3-step	11\29, 455.2	13/10	+1
aug. 3-step	12\29, 496.6	4/3	+9
dim. 4-step	13\29, 537.9	15/11	-12
minor 4-step	14\29, 579.3	7/5	-4
major 4-step	15\29 620.7	10/7	+4
aug. 4-step	16\29 662.1	22/15	+12
dim. 5-step	17\29, 703.4	3/2	-9
perf. 5-step	18\29, 755.2	20/13	-1
aug. 5-step	19\29, 786.2	11/7	+7
doubly aug. 5-step	20\29 827.6		+15
dim. 6-step	20\29 827.6		-14
minor 6-step	21\29 869.0		-6
major 6-step	22\29, 910.3	22/13	+2
aug. 6-step	23\29, 951.7	26/15	+10
dim. 7-step	24\29, 993.1	16/9	-11
minor 7-step	25\29, 1034.5	20/11	-3
major 7-step	26\29, 1075.9	13/7	+5
aug. 7-step	27\29, 1117.2		+13
dim. 8-step	28\29, 1158.6		-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)	Approximate ratios (23edo)	#Gens up
unison	0\23, 0.0	1/1	0
oneirochroma	3\23, 156.5		+8
minor step	1\23, 52.2		-5
major step	4\23, 208.7		+3
aug. step	7\23, 365.2	21/17, inverse φ	+11
dim. 2-step	2\23, 104.3	17/16	-10
minor 2-step	5\23, 260.9		-2
major 2-step	8\23, 417.4	14/11	+6
dim. 3-step	6\23, 313.0	6/5	-7
perf. 3-step	9\23, 469.6	21/16	+1
aug. 3-step	12\23, 626.1		+9
dim. 4-step	7\23, 365.2	21/17, inverse φ	-12
minor 4-step	10\23, 521.7		-4
major 4-step	13\23, 678.3		+4
aug. 4-step	16\23, 834.8	34/21, φ	+12
dim. 5-step	11\23, 573.9		-9
perf. 5-step	14\23, 730.4	32/21	-1
aug. 5-step	17\23, 887.0	5/3	+7
minor 6-step	15\23 782.6	11/7	-6
major 6-step	18\23, 939.1		+2
aug. 6-step	21\23, 1095.7	32/17	+10
dim. 7-step	16\23, 834.8	34/21, φ	-11
minor 7-step	19\23, 991.3		-3
major 7-step	22\23, 1147.8		+5
dim. 8-step	20\23, 1043.5		-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	Approximate ratios	#Gens up
1	0\38, 0.00	0\53, 0.00	0\63, 0.00	0.00	1/1	0
2	7\38, 221.05	10\53, 226.42	12\63, 228.57	227.07	8/7	+3
3	14\38, 442.10	20\53, 452.83	24\63, 457.14	453.81	13/10	+6
4	15\38, 473.68	21\53, 475.47	25\63, 476.19	475.63	21/16	+1
5	22\38, 694.73	31\53, 701.89	37\63, 704.76	702.54	3/2	+4
6	29\38, 915.78	41\53, 928.30	49\63, 933.33	929.45	12/7, 22/13	+7
7	30\38, 947.36	42\53, 950.94	50\63, 952.38	951.27	26/15	+2
8	37\38, 1168.42	52\53, 1177.36	62\63, 1180.95	1178.18	49/25, 160/81	+5

Approaches

• 5L 3s/Temperaments

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

• : Tonal Study in Dylathian
• : Tonal Study in Ultharian
• : Tonal Study in Hlanithian
• : Tonal Study in Ilarnekian
• : Tonal Study in Mnarian
• : Tonal Study in Sarnathian
• : Tonal Study in Celephaïsian
• : Tonal Study in Kadathian

Scale tree

Generator ranges:

• Bright generator: 450 cents (3\8) to 480 cents (2\5)
• Dark generator: 720 cents (3\5) to 750 cents (5\8)