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⟩ ↘ |
┌╥╥┬╥┬┐ │║║│║││ │││││││ └┴┴┴┴┴┘
sLsLL
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.
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.
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.
Semiwolf
Subgroup: 3/2.7/4.5/2
POL2 generator: ~7/6 = 262.1728
Mapping: [⟨1 1 3], ⟨0 1 -2]]
Semilupine
Subgroup: 3/2.7/4.5/2.11/4
POL2 generator: ~7/6 = 264.3771
Mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]
Hemilycan
Subgroup: 3/2.7/4.5/2.11/4
POL2 generator: ~7/6 = 261.5939
Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]
Notation
Since 1-7/4-5/2 is fifth-equivalent to a tone cluster of 1-10/9-7/6, it is more convenient to notate uranian scales as repeating at multiple fifths. 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]:
| Note | 18edf | 13edf | 21edf | 8edf | 19edf | 11edf | 14edf |
|---|---|---|---|---|---|---|---|
| 1# | 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 |
| 2b | 3\18
116.9925 |
2\13
107.9931 |
3\21
100.2793 |
2\19
73.89 |
1\11
63.814 |
1\14
50.1396 | |
| 2 | 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 |
| 2# | 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 |
| 3b | 7\18
272.9825 |
5\13
269.9829 |
8\21
267.4114 |
7\19
258.615 |
4\11
255.2564 |
5\14
250.6982 | |
| 3 | 8\18
311.98 |
6\13
323.9792 |
10\21
334.2643 |
4\8
350.9775 |
10\19
369.45 |
6\11
382.88455 |
8\14
401.1171 |
| 3# | 9\18
350.9775 |
7\13
377.9758 |
12\21
401.1171 |
5\8
438.7219 |
13\19
470.285 |
8\11
510.5128 |
11\14
551.536 |
| 4b | 10\18
389.975 |
11\21
367.9607 |
4\8
350.9775 |
9\19
332.505 |
5\11
319.07045 |
6\14
300.8379 | |
| 4 | 11\18
428.9725 |
8\13
431.9723 |
13\21
434.5436 |
5\8
438.7219 |
12\19
443.34 |
7\11
446.6986 |
9\14
451.2568 |
| 4# | 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 |
| 5b | 13\18
506.9675 |
10\13
539.9653 |
16\21
534.8229 |
14\19
516.23 |
8\11
510.5128 |
10\14
501.3964 | |
| 5 | 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 |
| 5# | 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 |
| 6b | 17\18
662.9575 |
19\21
635.1021 |
7\8
614.2106 |
16\19
591.12 |
9\11
574.3268 |
11\14
551.636 | |
| 6 | 701.955 | ||||||
| 6# | 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 |
| 7b | 21\18
818.9475 |
15\13
809.9481 |
24\21
802.2343 |
21\19
775.845 |
12\11
765.769 |
15\14
752.0946 | |
| 7 | 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 |
| 7# | 23\18
896.9425 |
17\13
917.9412 |
28\21
935.94 |
11\8
965.1881 |
27\19
997.515 |
16\11
1021.02545 |
21\14
1052.9235 |
| 8b | 25\18
974.9375 |
18\13
971.9379 |
29\21
969.3664 |
26\19
960.57 |
15\11
957.2114 |
19\14
952.6532 | |
| 8 | 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 |
| 8# | 27\18
1052.9325 |
20\13
1079.9308 |
33\21
1103.0721 |
13\8
1140.7769 |
32\19
1172.24 |
19\11
1212.5678 |
25\14
1253.4911 |
| 9b | 28\18
1091.93 |
32\21
1069.9157 |
12\8
1052.9235 |
28\19
1034.46 |
16\11
1021.02545 |
20\14
1002.7929 | |
| 9 | 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 |
| 9# | 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 |
| 0b | 31\18
1208.9225 |
23\13
1241.9203 |
37\21
1236.7779 |
33\19
1218.285 |
19\11
1212.5678 |
24\14
1203.3514 | |
| 0 | 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 |
| 0# | 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 |
| 1b’ | 35\18
1364.9125 |
40\21
1337.0571 |
15\8
1316.1656 |
35\19
1293.075 |
20\11
1276.2818 |
25\14
1253.591 | |
| 1’ | 1403.91 | ||||||