3L 2s (3/2-equivalent)

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.

↖ 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 ¢)
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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 [[12]]	3\8 [[13]]	8\19 295.56	5\11 319.07045	7\14 [[14]]
Cb	Γb	6\18 233.985		6\21 [[15]]	2\8 175.48875	4\19 147.78	2\11 [[16]]	2\14 [[17]]
C	Γ	7\18 [[18]]	5\13 [[19]]	8\21 [[20]]	3\8 [[21]]	7\19 258.615	4\11 [[22]]	5\14 [[23]]
C#	Γ#	8\18 311.98	6\13 [[24]]	10\21 [[25]]	4\8 [[26]]	9\19 332.505	6\11 382.88455	8\14 [[27]]
Db	Δb	10\18 389.975	7\13 [[28]]	11\21 [[29]]		10\19 369.45	5\11 319.07045	6\14 [[30]]
D	Δ	11\18 [[31]]	8\13 [[32]]	13\21 [[33]]	5\8 [[34]]	12\19 470.285	7\11 [[35]]	9\14 [[36]]
D#	Δ#	12\18 467.97	9\13 [[37]]	15\21 [[38]]	6\8 526.46625	15\19 554.175	9\11 [[39]]	12\14 [[40]]
Eb	Εb	14\18 545.965	10\13 [[41]]	16\21 [[42]]		14\19 516.23	8\11 [[43]]	10\14 [[44]]
E	Ε	15\18 [[45]]	11\13 [[46]]	18\21 [[47]]	7\8 [[48]]	17\19 628.065	10\11 [[49]]	13\14 [[50]]
E#	Ε#	16\18 622.96	12\13 [[51]]	20\21 [[52]]	8\8 701.955	20\19 738.9	12\11 765.769	16\14 [[53]]
Ab	Ϛb/Ϝb	17\18 [[54]]		19\21 [[55]]	7\8 [[56]]	16\19 591.12	9\11 [[57]]	11\14 551.636
A	Ϛ/Ϝ	701.955
A#	Ϛ#/Ϝ#	19\18 [[58]]	14\13 [[59]]	23\21 [[60]]	9\8 [[61]]	22\19 812.79	13\11 [[62]]	17\14 [[63]]
Bb	Ζb	21\18 [[64]]	15\13 [[65]]	24\21 [[66]]		21\19 775.845	12\11 765.769	15\14 [[67]]
B	Ζ	22\18 857.945	16\13 [[68]]	26\21 [[69]]	10\8 877.44375	24\19 886.68	14\11 [[70]]	18\14 [[71]]
B#	Ζ#	23\18 [[72]]	17\13 [[73]]	28\21 [[74]]	11\8 [[75]]	27\19 997.515	16\11 1021.02545	21\14 1052.9235
Cb	Ηb	24\18 935.94		27\21 [[76]]	10\8 877.44375	23\19 849.753	13\11 [[77]]	16\14 [[78]]
C	Η	25\18 [[79]]	18\13 [[80]]	29\21 [[81]]	11\8 [[82]]	26\19 960.57	15\11 [[83]]	19\14 [[84]]
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 = [[85]]

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 = [[86]]

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 = [[87]]

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
		8\13		431.972	269.983	3	2	1.500	Semiwolf and Semilupine start here
			13\21	435.084	266.871	5	3	1.667
	5\8			438.722	263.233	2	1	2.000	Semilupine ends, Hemilycan begins
			12\19	443.34	258.615	5	2	2.500
		7\11		446.699	255.256	3	1	3.000	Semiwolf and Hemilycan end here
			9\14	451.257	250.698	4	1	4.000	Near 24edo
2\3				467.97	233.985	1	0	→ inf	Paucitonic