3L 2s (3/2-equivalent)

↖ 2L 1s⟨3/2⟩	↑ 3L 1s⟨3/2⟩	4L 1s⟨3/2⟩ ↗
← 2L 2s⟨3/2⟩	3L 2s (3/2-equivalent)	4L 2s⟨3/2⟩ →
↙ 2L 3s⟨3/2⟩	↓ 3L 3s⟨3/2⟩	4L 3s⟨3/2⟩ ↘

┌╥╥┬╥┬┐ │║║│║││ │││││││ └┴┴┴┴┴┘
Scale structure
Step pattern	LLsLs sLsLL
Equave	3/2 (702.0 ¢)
Period	3/2 (702.0 ¢)
Generator size(edf)
Bright	3\5 to 2\3 (421.2 ¢ to 468.0 ¢)
Dark	1\3 to 2\5 (234.0 ¢ to 280.8 ¢)
Related MOS scales
Parent	2L 1s⟨3/2⟩
Sister	2L 3s⟨3/2⟩
Daughters	5L 3s⟨3/2⟩, 3L 5s⟨3/2⟩
Neutralized	1L 4s⟨3/2⟩
2-Flought	8L 2s⟨3/2⟩, 3L 7s⟨3/2⟩
Equal tunings(edf)
Equalized (L:s = 1:1)	3\5 (421.2 ¢)
Supersoft (L:s = 4:3)	11\18 (429.0 ¢)
Soft (L:s = 3:2)	8\13 (432.0 ¢)
Semisoft (L:s = 5:3)	13\21 (434.5 ¢)
Basic (L:s = 2:1)	5\8 (438.7 ¢)
Semihard (L:s = 5:2)	12\19 (443.3 ¢)
Hard (L:s = 3:1)	7\11 (446.7 ¢)
Superhard (L:s = 4:1)	9\14 (451.3 ¢)
Collapsed (L:s = 1:0)	2\3 (468.0 ¢)

3L 2s<3/2> (sometimes called uranian), is a fifth-repeating MOS scale. The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating 3L 2s. The name of the period interval is called the sesquitave (by analogy to the tritave). It is a warped diatonic scale because it has one extra small step compared to diatonic (3L 1s (fifth-equivalent)): for example, the Ionian diatonic fifth LLsL can be distorted to the Oberonan mode LsLLs.

The generator range is 234 to 280.8 cents, placing it in between the diatonic major second and the diatonic minor third, usually representing a subminor third of some type (like 7/6). The bright (chroma-positive) generator is, however, its fifth complement (468 to 421.2 cents).

Because uranian is a fifth-repeating scale, each tone has a 3/2 perfect fifth above it. The scale has three major chords and two minor chords, all voiced so that the third of the triad is an octave higher, a tenth. Uranian also has two harmonic 7th chords.

Basic uranian is in 8edf, which is a very good fifth-based equal tuning similar to 88cET.

Notation

There are 2 main ways to notate the uranian scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 5 naturals (A-E). Given that 1-7/4-5/2 is fifth-equivalent to a tone cluster of 1-10/9-7/6, it may be more convenient to notate uranian scales as repeating at the double sesquitave (major ninth), however it does make navigating the genchain harder. This way, 7/4 is its own pitch class, distinct from 7/6. Notating this way produces a major ninth which is the Aeolian mode of Annapolis[6L 4s]. Since there are exactly 10 naturals in double sesquitave notation, Greek numerals 1-10 may be used.

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Uranian	Annapolis	18edf	13edf	21edf	8edf	19edf	11edf	14edf
A#	Α#	1\18 38.9975	1\13 53.9965	2\21 66.8529	1\8 87.7444	3\19 110.835	2\11 [[1]]	3\14 [[2]]
Bb	Βb	3\18 [[3]]	2\13 [[4]]	3\21 [[5]]		2\19 73.89	1\11 63.814	1\14 50.1396
B	Β	4\18 155.99	3\13 [[6]]	5\21 [[7]]	2\8 175.48875	5\19 184.725	3\11 [[8]]	4\14 [[9]]
B#	Β#	5\18 [[10]]	4\13 [[11]]	7\21 233.985	3\8 [[12]]	8\19 295.56	5\11 319.07045	7\14 [[13]]
Cb	Γb	6\18 233.985		6\21 [[14]]	2\8 175.48875	4\19 147.78	2\11 [[15]]	2\14 [[16]]
C	Γ	7\18 [[17]]	5\13 [[18]]	8\21 [[19]]	3\8 [[20]]	7\19 258.615	4\11 [[21]]	5\14 [[22]]
C#	Γ#	8\18 311.98	6\13 [[23]]	10\21 [[24]]	4\8 [[25]]	9\19 332.505	6\11 382.88455	8\14 [[26]]
Db	Δb	10\18 389.975	7\13 [[27]]	11\21 [[28]]		10\19 369.45	5\11 319.07045	6\14 [[29]]
D	Δ	11\18 [[30]]	8\13 [[31]]	13\21 [[32]]	5\8 [[33]]	12\19 470.285	7\11 [[34]]	9\14 [[35]]
D#	Δ#	12\18 467.97	9\13 [[36]]	15\21 [[37]]	6\8 526.46625	15\19 554.175	9\11 [[38]]	12\14 [[39]]
Eb	Εb	14\18 545.965	10\13 [[40]]	16\21 [[41]]		14\19 516.23	8\11 [[42]]	10\14 [[43]]
E	Ε	15\18 [[44]]	11\13 [[45]]	18\21 [[46]]	7\8 [[47]]	17\19 628.065	10\11 [[48]]	13\14 [[49]]
E#	Ε#	16\18 622.96	12\13 [[50]]	20\21 [[51]]	8\8 701.955	20\19 738.9	12\11 765.769	16\14 [[52]]
Ab	Ϛb/Ϝb	17\18 [[53]]		19\21 [[54]]	7\8 [[55]]	16\19 591.12	9\11 [[56]]	11\14 551.636
A	Ϛ/Ϝ	701.955
A#	Ϛ#/Ϝ#	19\18 [[57]]	14\13 [[58]]	23\21 [[59]]	9\8 [[60]]	22\19 812.79	13\11 [[61]]	17\14 [[62]]
Bb	Ζb	21\18 [[63]]	15\13 [[64]]	24\21 [[65]]		21\19 775.845	12\11 765.769	15\14 [[66]]
B	Ζ	22\18 857.945	16\13 [[67]]	26\21 [[68]]	10\8 877.44375	24\19 886.68	14\11 [[69]]	18\14 [[70]]
B#	Ζ#	23\18 [[71]]	17\13 [[72]]	28\21 [[73]]	11\8 [[74]]	27\19 997.515	16\11 1021.02545	21\14 1052.9235
Cb	Ηb	24\18 935.94		27\21 [[75]]	10\8 877.44375	23\19 849.753	13\11 [[76]]	16\14 [[77]]
C	Η	25\18 [[78]]	18\13 [[79]]	29\21 [[80]]	11\8 [[81]]	26\19 960.57	15\11 [[82]]	19\14 [[83]]
C#	Η#	26\18 1012.935	19\13 1025.9342	31\21 1036.2193	12\8 1052.9235	29\19 1071.405	17\11 1084.83955	22\14 1103.0721
Db	Θb	28\18 1091.93	20\13 1079.9308	32\21 1069.9157		28\19 1034.46	16\11 1021.02545	20\14 1002.7929
D	Θ	29\18 1130.9275	21\13 1133.9273	34\21 1136.4986	13\8 1140.7769	31\19 1145.295	18\11 1148.6536	23\14 1153.2118
D#	Θ#	30\18 1169.925	22\13 1187.9238	36\21 1203.3514	14\8 1228.42125	34\19 1256.13	20\11 1276.2818	26\14 1303.6307
Eb	Ιb	32\18 1247.92	23\13 1241.9203	37\21 1236.7779		33\19 1218.285	19\11 1212.5678	24\14 1203.3514
E	Ι	33\18 1286.9175	24\13 1295.9169	39\21 1303.6307	15\8 1316.1656	36\19 1330.02	21\11 1340.0959	27\14 1353.8704
E#	Ι#	34\18 1323.915	25\13 1348.9135	41\21 1370.4836	16\8 1403.91	39\19 1440.855	23\11 1468.724	30\14 1504.1892
Ab	Αb	35\18 1364.9125		40\21 1337.0571	15\8 1316.1656	35\19 1293.075	20\11 1276.2818	25\14 1253.591
A	Α	1403.91

Intervals

Generators	Sesquitave notation	Interval category name	Generators	Notation of 3/2 inverse	Interval category name
The 5-note MOS has the following intervals (from some root):
0	A	perfect unison	0	A	sesquitave (just fifth)
1	C	perfect mosthird (min third)	-1	D	perfect mosfourth (maj third)
2	Eb	minor mosfifth	-2	B	major mossecond
3	Bb	minor mossecond	-3	E	major mosfifth
4	Db	diminished mosfourth	-4	C#	augmented mosthird
The chromatic 8-note MOS also has the following intervals (from some root):
5	Ab	diminished sesquitave	-5	A#	augmented unison (chroma)
6	Cb	diminished mosthird	-6	D#	augmented mosfourth
7	Ebb	diminished mosfifth	-7	B#	augmented mossecond

Genchain

The generator chain for this scale is as follows:

Bbb

Ebb

Cb

Ab

Db

Bb

Eb

C

A

D

B

E

C#

A#

D#

B#

E#

d2

d5

d3

d6

d4

m2

m5

P3

P1

P4

M2

M5

A3

A1

A4

A2

A5

Modes

The mode names are based on the major satellites of Uranus, in order of size:

Mode	Scale	UDP	Interval type (mos-)
name	pattern	notation	2nd	3rd	4th	5th
Titanian	LLsLs	4|0	M	A	P	M
Oberonan	LsLLs	3|1	M	P	P	M
Umbrielan	LsLsL	2|2	M	P	P	m
Arielan	sLLsL	1|3	m	P	P	m
Mirandan	sLsLL	0|4	m	P	d	m

Temperaments

The most basic rank-2 temperament interpretation of uranian is semiwolf, which has 4:7:10 chords spelled root-(p+1g)-(3p-2g) (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two 7/6 generators approximating a 27/20 wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine where 2 major mos2nds (LL) equal 11/9, and hemilycan where 1 major and 2 minor mos2nds (sLs) equal 11/9. Basic 8edf fits both extensions.

Semiwolf

Subgroup: 3/2.7/4.5/2

Comma list: 245/243

POL2 generator: ~7/6 = [[84]]

Mapping: [⟨1 1 3], ⟨0 1 -2]]

Vals: Template:Val list

Semilupine

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 100/99

POL2 generator: ~7/6 = [[85]]

Mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]

Vals: Template:Val list

Hemilycan

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 441/440

POL2 generator: ~7/6 = [[86]]

Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]

Vals: Template:Val list

Scale tree

The spectrum looks like this:

Generator (bright)						Cents		L	s	L/s	Comments
						Chroma-positive	Chroma-negative
3\5						421.173	280.782	1	1	1.000	Equalised
11\18						428.973	272.983	4	3	1.333
	30\49					429.768	272.187	11	8	1.375
	19\31					[[87]]	[[88]]	7	5	1.400
8\13						431.972	269.983	3	2	1.500	Semiwolf and Semilupine start here
		37\60				432.872	269.083	14	9	1.556
	29\47					433.121	268.834	11	7	1.571
	21\34					433.56	268.395	8	5	1.600
		34\55				433.935	268.02	13	8	1.625
	13\21					435.084	266.871	5	3	1.667
	18\29					435.696	266.259	7	4	1.750
	23\37					436.35	265.605	9	5	1.800
	28\45					436.772	265.183	11	6	1.833
		33\53				437.066	264.889	13	7	1.857
5\8						438.722	263.233	2	1	2.000	Semilupine ends, Hemilycan begins
					47\75	439.892	262.063	19	9	2.111
				42\67		440.031	261.924	17	8	2.125
			37\59			440.209	261.746	15	7	2.143
		32\51				440.442	261.513	13	6	2.167
	27\43					440.762	261.193	11	5	2.200
	22\35					441.229	260.726	9	4	2.250
	17\27					441.972	259.973	7	3	2.333
		29\46				442.537	259.418	12	5	2.400
	12\19					443.34	258.615	5	2	2.500
	19\30					[[89]]	[[90]]	8	3	2.667
	26\41					445.142	256.813	11	4	2.750
7\11						446.699	255.256	3	1	3.000	Semiwolf and Hemilycan end here
		37\58				447.799	254.156	16	5	3.200
	30\47					448,056	253.899	13	4	3.250
	23\36					448.471	253.484	10	3	3.333
	16\25					449.251	252.704	7	2	3.500
	25\39					449.971	251.984	11	3	3.667
	34\53					450.311	251.644	15	4	3.750
9\14						451.257	250.698	4	1	4.000	Near 24edo
2\3						467.97	233.985	1	0	→ inf	Paucitonic