3L 2s (8/5-equivalent)

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3L 2s<8/5> (sometimes called diatonic), is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating 3L 2s. The name of the period interval is called the sextave (by analogy to the tritave).

↖ 2L 1s⟨8/5⟩	↑ 3L 1s⟨8/5⟩	4L 1s⟨8/5⟩ ↗
← 2L 2s⟨8/5⟩	3L 2s (8/5-equivalent)	4L 2s⟨8/5⟩ →
↙ 2L 3s⟨8/5⟩	↓ 3L 3s⟨8/5⟩	4L 3s⟨8/5⟩ ↘

Scale structure
Step pattern	LLsLs sLsLL
Equave	8/5 (813.7 ¢)
Period	8/5 (813.7 ¢)
Generator size(ed8/5)
Bright	3\5 to 2\3 (488.2 ¢ to 542.5 ¢)
Dark	1\3 to 2\5 (271.2 ¢ to 325.5 ¢)
Related MOS scales
Parent	2L 1s⟨8/5⟩
Sister	2L 3s⟨8/5⟩
Daughters	5L 3s⟨8/5⟩, 3L 5s⟨8/5⟩
Neutralized	1L 4s⟨8/5⟩
2-Flought	8L 2s⟨8/5⟩, 3L 7s⟨8/5⟩
Equal tunings(ed8/5)
Equalized (L:s = 1:1)	3\5 (488.2 ¢)
Supersoft (L:s = 4:3)	11\18 (497.3 ¢)
Soft (L:s = 3:2)	8\13 (500.7 ¢)
Semisoft (L:s = 5:3)	13\21 (503.7 ¢)
Basic (L:s = 2:1)	5\8 (508.6 ¢)
Semihard (L:s = 5:2)	12\19 (513.9 ¢)
Hard (L:s = 3:1)	7\11 (517.8 ¢)
Superhard (L:s = 4:1)	9\14 (523.1 ¢)
Collapsed (L:s = 1:0)	2\3 (542.5 ¢)
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The generator range is 240 to 342.9 cents, placing it on the diatonic minor third, usually representing a minor third of some type (like 6/5). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).

Because this diatonic is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.

Basic diatonic is in 8ed8/5, which is a very good minor sixth-based equal tuning similar to 12edo.

Notation

There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sesquitave notation, Greek numerals 1-10 may be used.

Normalized

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Diatonic	Oriole, Annapolis	18eds	13eds	21eds	8eds	19eds	11eds	14eds
La#	Α#	1\18 46.15385	1\13 63.1579	2\21 77.41935	1\8 100	3\19 124.1379	2\11 [[1]]	3\14 [[2]]
Sib	Βb	3\18 138.4615	2\13 126.3158	3\21 116.129		2\19 82.7586	1\11 63.814	1\14 50.1396
Si	Β	4\18 184.6154	3\13 189.4736	5\21 193.5484	2\8 200	5\19 206.89655	3\11 [[3]]	4\14 [[4]]
Si#	Β#	5\18 230.7692	4\13 252.6316	7\21 270.9677	3\8 300	8\19 331.0345	5\11 319.07045	7\14 [[5]]
Dob	Γb	6\18 276.9231		6\21 232.2581	2\8 200	4\19 165.5172	2\11 [[6]]	2\14 [[7]]
Do	Γ	7\18 323.0769	5\13 315.7895	8\21 309.6774	3\8 300	7\19 289.6552	4\11 [[8]]	5\14 [[9]]
Do#	Γ#	8\18 369.2308	6\13 378.9474	10\21 387.0968	4\8 400	10\19 413.7931	6\11 382.88455	8\14 [[10]]
Reb	Δb	10\18 461.5385	7\13 442.1053	11\21 425.80645		9\19 372.4138	5\11 319.07045	6\14 [[11]]
Re	Δ	11\18 507.6923	8\13 505.2632	13\21 503.2259	5\8 500	12\19 496.5517	7\11 [[12]]	9\14 [[13]]
Re#	Δ#	12\18 553.84615	9\13 568.42105	15\21 580.6452	6\8 600	15\19 620.6897	9\11 [[14]]	12\14 [[15]]
Mib	Εb	14\18 646.15385	10\13 631.57895	16\21 619.3548		14\19 579.3103	8\11 [[16]]	10\14 [[17]]
Mi	Ε	15\18 692.3077	11\13 694.7368	18\21 696.7742	7\8 700	17\19 703.4483	10\11 [[18]]	13\14 [[19]]
Mi#	Ε#	16\18 738.4615	12\13 757.8947	20\21 774.19355	8\8 800	20\19 827.5862	12\11 765.769	16\14 [[20]]
Lab	Ϛb/Ϝb	17\18 784.6154		19\21 735.4839	7\8 700	16\19 662.069	9\11 [[21]]	11\14 551.636
La	Ϛ/Ϝ	18\18 830.7692	13\13 821.0526	21\21 812.9032	8\8 800	19\19 786.2069
La#	Ϛ#/Ϝ#	19\18 876.9231	14\13 884.2105	23\21 890.3226	9\8 900	22\19 910.3448	13\11 [[22]]	17\14 [[23]]
Sib	Ζb	21\18 969.2308	15\13 947.3684	24\21 929.0323		21\19 868.9655	12\11 765.769	15\14 [[24]]
Si	Ζ	22\18 1015.3846	16\13 1010.5263	26\21 1006.4516	10\8 1000	24\19 993.10345	14\11 [[25]]	18\14 [[26]]
Si#	Ζ#	23\18 1061.5385	17\13 1071.6842	28\21 1083.871	11\8 1100	27\19 1117.2414	16\11 1021.02545	21\14 1052.9235
Dob	Ηb	24\18 1107.6923		27\21 1045.1613	10\8 1000	23\19 951.7241	13\11 [[27]]	16\14 [[28]]
Do	Η	25\18 1153.84615	18\13 1136.8421	29\21 1122.58065	11\8 1100	26\19 1075.8621	15\11 [[29]]	19\14 [[30]]
Do#	Η#	26\18 1200	19\13 1200	31\21 1200	12\8 1200	29\19 1200	17\11 1200	22\14 1200
Reb	Θb	28\18 1292.3077	20\13 1263.1579	32\21 1238.7097		28\19 1158.6207	16\11 1021.02545	20\14 1002.7929
Re	Θ	29\18 1338.4615	21\13 1326.3158	34\21 1316.129	13\8 1300	31\19 1282.7586	18\11 1148.6536	23\14 1153.2118
Re#	Θ#	30\18 1384.6154	22\13 1389.4737	36\21 1393.5484	14\8 1400	34\19 1406.8965f	20\11 1276.2818	26\14 1303.6307
Mib	Ιb	32\18 1476.9231	23\13 1452.6316	37\21 1432.2581		33\19 1365.5172	19\11 1212.5678	24\14 1203.3514
Mi	Ι	33\18 1523.0769	24\13 1515.7895	39\21 1509.6774	15\8 1500	36\19 1489.6551	21\11 1340.0959	27\14 1353.8704
Mi#	Ι#	34\18 1569.2308	25\13 1578.9474	41\21 1587.0968	16\8 1600	39\19 1613.7931	23\11 1468.724	30\14 1504.1892
Lab	Αb	35\18 1615.3846		40\21 1548.3871	15\8 1500	35\19 1448.2859	20\11 1276.2818	25\14 1253.591
La	Α	36\18 1661.5385	26\13 1642.1053	42\21 1625.80645	16\8 1600	38\19 1572.4138

ed2\3

Notation		Supersoft	Soft	Semisoft	Basic	Semihard	Hard	Superhard
Diatonic	Oriole, Annapolis	18eds	13eds	21eds	8eds	19eds	11eds	14eds
La#	Α#	1\18 38.9975	1\13 53.9965	2\21 66.8529	1\8 87.7444	3\19 110.835	2\11 [127.6282]	3\14 [150.4189]
Sib	Βb	3\18 [116.9925]	2\13 [107.9931]	3\21 [100.2793]		2\19 73.89	1\11 63.814	1\14 50.1396
Si	Β	4\18 155.99	3\13 [161.9896]	5\21 [167.1321]	2\8 175.48875	5\19 184.725	3\11 [191.4423]	4\14 [200.5586]
Si#	Β#	5\18 [194.9875]	4\13 [215.9862]	7\21 233.985	3\8 [263.2331]	8\19 295.56	5\11 319.07045	7\14 [350.9775]
Dob	Γb	6\18 233.985		6\21 [200.5586]	2\8 175.48875	4\19 147.78	2\11 [127.6282]	2\14 [100.2793]
Do	Γ	7\18 [272.9825]	5\13 [269.9829]	8\21 [267.4114]	3\8 [263.2331]	7\19 258.615	4\11 [255.2564]	5\14 [250.6982]
Do#	Γ#	8\18 311.98	6\13 [323.9792]	10\21 [334.2643]	4\8 [350.9775]	9\19 332.505	6\11 382.88455	8\14 [401.1171]
Reb	Δb	10\18 389.975	7\13 [377.9758]	11\21 [367.9607]		10\19 369.45	5\11 319.07045	6\14 [300.8379]
Re	Δ	11\18 [428.9725]	8\13 [431.9723]	13\21 [434.5436]	5\8 [438.7219]	12\19 470.285	7\11 [446.6986]	9\14 [451.2568]
Re#	Δ#	12\18 467.97	9\13 [485.9688]	15\21 [501.3964]	6\8 526.46625	15\19 554.175	9\11 [574.3268]	12\14 [601.6757]
Mib	Εb	14\18 545.965	10\13 [539.9653]	16\21 [534.8229]		14\19 516.23	8\11 [510.5128]	10\14 [501.3964]
Mi	Ε	15\18 [584.9625]	11\13 [593.9619]	18\21 [601.6757]	7\8 [614.2106]	17\19 628.065	10\11 [638.1409]	13\14 [651.8154]
Mi#	Ε#	16\18 622.96	12\13 [646.9585]	20\21 [668.5286]	8\8 701.955	20\19 738.9	12\11 765.769	16\14 [802.2343]
Lab	Ϛb/Ϝb	17\18 [662.9575]		19\21 [635.1021]	7\8 [614.2106]	16\19 591.12	9\11 [574.3268]	11\14 551.636
La	Ϛ/Ϝ	701.955
La#	Ϛ#/Ϝ#	19\18 [740.9525]	14\13 [754.9515]	23\21 [768.8021]	9\8 [789.6994]	22\19 812.79	13\11 [829.5832]	17\14 [852.3739]
Sib	Ζb	21\18 [818.9475]	15\13 [809.9481]	24\21 [802.2343]		21\19 775.845	12\11 765.769	15\14 [752.0946]
Si	Ζ	22\18 857.945	16\13 [862.9446]	26\21 [868.0871]	10\8 877.44375	24\19 886.68	14\11 [893.3973]	18\14 [902.5136]
Si#	Ζ#	23\18 [896.9425]	17\13 [917.9412]	28\21 [935.9406]	11\8 [965.1881]	27\19 997.515	16\11 1021.02545	21\14 1052.9235
Dob	Ηb	24\18 935.94		27\21 [902.5136]	10\8 877.44375	23\19 849.753	13\11 [829.5832]	16\14 [802.2343]
Do	Η	25\18 [974.9375]	18\13 [971.9379]	29\21 [969.3664]	11\8 [965.1881]	26\19 960.57	15\11 [957.2114]	19\14 [952.6532]
Do#	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
Reb	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
Re	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
Re#	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
Mib	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
Mi	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
Mi#	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
Lab	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
La	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 27/20wolf 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 = [[31]]

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 = [[32]]

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 = [[33]]

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					[[34]]	[[35]]	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					[[36]]	[[37]]	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