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

800edf

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Uranian	Annapolis	18edf	13edf	21edf	8edf	19edf	11edf	14edf
A#	Α#	1\18 44.4444	1\13 61.5385	2\21 76.1905	1\8 100	3\19 126.3158	2\11 145.45455	3\14 171.4286
Bb	Βb	3\18 133.3333	2\13 123.0679	3\21 114.2857		2\19 84.2105	1\11 72.7273	1\14 57.1429
B	Β	4\18 177.7778	3\13 184.6154	5\21 190.4762	2\8 200	5\19 210.5263	3\11 218.1818	4\14 228.5714
B#	Β#	5\18 222.2222	4\13 246.15385	7\21 266.6667	3\8 300	8\19 336.8947	5\11 363.6364	7\14 400
Cb	Γb	6\18 266.6667		6\21 228.5714	2\8 200	4\19 168.42105	2\11 145.45455	2\14 114.8571
C	Γ	7\18 311.1111	5\13 307.6923	8\21 304.7619	3\8 300	7\19 294.7368	4\11 290.9091	5\14 285.7143
C#	Γ#	8\18 355.5556	6\13 369.2307	10\21 380.9523	4\8 400	10\19 421.0526	6\11 436.3636	8\14 457.1429
Db	Δb	10\18 444.4444	7\13 430.7692	11\21 419.0477		9\19 378.9474	5\11 363.6364	6\14 342.8571
D	Δ	11\18 488.8889	8\13 492.3077	13\21 495.2381	5\8 500	12\19 505.2632	7\11 509.0909	9\14 514.2857
D#	Δ#	12\18 533.3333	9\13 553.84615	15\21 571.4286	6\8 600	15\19 631.57895	9\11 654.54545	12\14 685.7143
Eb	Εb	14\18 622.2222	10\13 615.3846	16\21 610.5238		14\19 589.4737	8\11 581.8182	10\14 571.4286
E	Ε	15\18 666.6667	11\13 676.9321	18\21 685.7143	7\8 700	17\19 715.7895	10\11 727.2727	13\14 742.8571
E#	Ε#	16\18 711.1111	12\13 738.4615	20\21 761.9048	8\8 800	20\19 842.1053	12\11 872.7273	16\14 914.8571
Ab	Ϛb/Ϝb	17\18 755.5556		19\21 723.8195	7\8 700	16\19 673.6842	9\11 654.54545	11\14 628.5714
A	Ϛ/Ϝ	800
A#	Ϛ#/Ϝ#	19\18 844.4444	14\13 861.5385	23\21 876.1905	9\8 900	22\19 926.3158	13\11 945.45455	17\14 971.4286
Bb	Ζb	21\18 933.3333	15\13 923.0769	24\21 914.2857		21\19 884.2105	12\11 872.7273	15\14 857.1429
B	Ζ	22\18 977.7778	16\13 984.6154	26\21 990.4762	10\8 1000	24\19 1010.5263	14\11 1018.1818	18\14 1028.5714
B#	Ζ#	23\18 1022.2222	17\13 1046.15385	28\21 1066.6667	11\8 1100	27\19 1136.8947	16\11 1163.6364	21\14 1200
Cb	Ηb	24\18 1066.6667		27\21 1028.5714	10\8 1000	23\19 968.42105	13\11 945.45455	16\14 914.2857
C	Η	25\18 1111.1111	18\13 1107.6923	29\21 1104.7619	11\8 1100	26\19 1094.7368	15\11 1090.9091	19\14 1085.7143
C#	Η#	26\18 1155.5556	19\13 1169.2307	31\21 1180.9523	12\8 1200	29\19 1221.0526	17\11 1236.3636	22\14 1257.1429
Db	Θb	28\18 1244.4444	20\13 1230.7692	32\21 1219.0477		28\19 1178.9474	16\11 1163.6364	20\14 1142.8571
D	Θ	29\18 1288.8889	21\13 1292.3077	34\21 1295.2381	13\8 1300	31\19 1305.2632	18\11 1309.0909	23\14 1314.2857
D#	Θ#	30\18 1333.3333	22\13 1353.84615	36\21 1371.4286	14\8 1400	34\19 1431.57895	20\11 1454.54545	26\14 1485.7143
Eb	Ιb	32\18 1422.2222	23\13 1415.3845	37\21 1410.5238		33\19 1389.4737	19\11 1381.8182	24\14 1371.4286
E	Ι	33\18 1466.6667	24\13 1476.9231	39\21 1485.7143	15\8 1500	36\19 1515.7895	21\11 1527.2727	27\14 1542.8571
E#	Ι#	34\18 1511.1111	25\13 1538.4615	41\21 1561.9048	16\8 1600	39\19 1642.1053	23\11 1672.7273	30\14 1714.2857
Ab	Αb	35\18 1555.556		40\21 1523.8195	15\8 1500	35\19 1473.6842	20\11 1454.54545	25\14 1428.5714
A	Α	1600

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		800edf		L	s	L/s	Comments
			Chroma-positive	Chroma-negative	Chroma-positive	Chroma-negative
3\5			421.173	280.782	480	320	1	1	1.000	Equalised
11\18			428.973	272.983	488.889	311.111	4	3	1.333
	30\49		429.768	272.187	489.796	310.204	11	8	1.375
	19\31		[[87]]	[[88]]	490.323	309.677	7	5	1.400
	27\44		430.745	271.21	490.909	309.091	10	7	1.429
	35\57		431.025	270.93	491.228	308.772	13	9	1.444
8\13			431.972	269.983	492.308	307.692	3	2	1.500	Semiwolf and Semilupine start here
	37\60		432.872	269.083	493.333	306.667	14	9	1.556
	29\47		433.121	268.834	493.617	306.383	11	7	1.571
	21\34		433.56	268.395	494.118	305.882	8	5	1.600
		34\55	433.935	268.02	494.5455	305.4545	13	8	1.625
		47\76	434.104	267.851	494.737	305.263	18	11	1.636
	13\21		435.084	266.871	495.238	304.762	5	3	1.667
	18\29		435.696	266.259	496.552	303.441	7	4	1.750
	23\37		436.35	265.605	497.297	302.703	9	5	1.800
	28\45		436.772	265.183	497.778	302.222	11	6	1.833
	33\53		437.066	264.889	498.113	301.887	13	7	1.857
	38\61		437.283	264.672	498.361	301.639	15	8	1.875
	43\69		437.45	264.555	498.551	301.449	17	9	1.889
	48\77		437.582	264.373	498.701	301.299	19	10	1.900
5\8			438.722	263.233	500	300	2	1	2.000	Semilupine ends, Hemilycan begins
	47\75		439.892	262.063	501.333	298.667	19	9	2.111
	42\67		440.031	261.924	501.4925	298.5075	17	8	2.125
	37\59		440.209	261.746	501.695	298.305	15	7	2.143
	32\51		440.442	261.513	501.961	298.039	13	6	2.167
	27\43		440.762	261.193	502.326	297.624	11	5	2.200
	22\35		441.229	260.726	502.857	297.143	9	4	2.250
	17\27		441.972	259.973	503.704	296.296	7	3	2.333
		29\46	442.537	259.418	504.348	295.652	12	5	2.400
	12\19		443.34	258.615	505.263	294.737	5	2	2.500
	19\30		[[89]]	[[90]]	506.667	293.333	8	3	2.667
	26\41		445.142	256.813	507.317	292.683	11	4	2.750
	33\52		445.471	256.484	507.692	292.308	14	5	2.800
	40\63		445.686	256.269	507.9365	292.0635	17	6	2.833
	47\74		445.836	256.119	508.108	291.892	20	7	2.857
7\11			446.699	255.256	509.091	290.909	3	1	3.000	Semiwolf and Hemilycan end here
	37\58		447.799	254.156	510.345	289.655	16	5	3.200
	30\47		448,056	253.899	510.638	289.362	13	4	3.250
	23\36		448.471	253.484	511.111	288.889	10	3	3.333
	16\25		449.251	252.704	512	288	7	2	3.500
	25\39		449.971	251.984	512.8205	287.1795	11	3	3.667
	34\53		450.311	251.644	513.2075	286.7925	15	4	3.750
9\14			451.257	250.698	514.286	285.714	4	1	4.000	Near 24edo
2\3			467.97	233.985	533.333	266.667	1	0	→ inf	Paucitonic