8L 3s (3/1-equivalent)

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↖ 7L 2s⟨3/1⟩ ↑ 8L 2s⟨3/1⟩ 9L 2s⟨3/1⟩ ↗
← 7L 3s⟨3/1⟩ 8L 3s (3/1-equivalent) 9L 3s⟨3/1⟩ →
↙ 7L 4s⟨3/1⟩ ↓ 8L 4s⟨3/1⟩ 9L 4s⟨3/1⟩ ↘ 
┌╥╥╥┬╥╥╥┬╥╥┬┐
│║║║│║║║│║║││
│││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLLsLLs
sLLsLLLsLLL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 4\11 to 3\8 (691.6 ¢ to 713.2 ¢)
Dark 5\8 to 7\11 (1188.7 ¢ to 1210.3 ¢)
Related MOS scales
Parent 3L 5s⟨3/1⟩
Sister 3L 8s⟨3/1⟩
Daughters 11L 8s⟨3/1⟩, 8L 11s⟨3/1⟩
Neutralized 5L 6s⟨3/1⟩
2-Flought 19L 3s⟨3/1⟩, 8L 14s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 4\11 (691.6 ¢)
Supersoft (L:s = 4:3) 15\41 (695.8 ¢)
Soft (L:s = 3:2) 11\30 (697.4 ¢)
Semisoft (L:s = 5:3) 18\49 (698.7 ¢)
Basic (L:s = 2:1) 7\19 (700.7 ¢)
Semihard (L:s = 5:2) 17\46 (702.9 ¢)
Hard (L:s = 3:1) 10\27 (704.4 ¢)
Superhard (L:s = 4:1) 13\35 (706.4 ¢)
Collapsed (L:s = 1:0) 3\8 (713.2 ¢)

8L 3s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 8 large steps and 3 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 691.6 ¢ to 713.2 ¢, or from 1188.7 ¢ to 1210.3 ¢. 8L 3s⟨3/1⟩ scale pattern includes the well-known 5L 2s pattern within it.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 8L 3s⟨3/1⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Minor 1-mosstep m1ms s 0.0 ¢ to 172.9 ¢
Major 1-mosstep M1ms L 172.9 ¢ to 237.7 ¢
2-mosstep Minor 2-mosstep m2ms L + s 237.7 ¢ to 345.8 ¢
Major 2-mosstep M2ms 2L 345.8 ¢ to 475.5 ¢
3-mosstep Minor 3-mosstep m3ms 2L + s 475.5 ¢ to 518.7 ¢
Major 3-mosstep M3ms 3L 518.7 ¢ to 713.2 ¢
4-mosstep Diminished 4-mosstep d4ms 2L + 2s 475.5 ¢ to 691.6 ¢
Perfect 4-mosstep P4ms 3L + s 691.6 ¢ to 713.2 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 713.2 ¢ to 864.5 ¢
Major 5-mosstep M5ms 4L + s 864.5 ¢ to 951.0 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 951.0 ¢ to 1037.4 ¢
Major 6-mosstep M6ms 5L + s 1037.4 ¢ to 1188.7 ¢
7-mosstep Perfect 7-mosstep P7ms 5L + 2s 1188.7 ¢ to 1210.3 ¢
Augmented 7-mosstep A7ms 6L + s 1210.3 ¢ to 1426.5 ¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 1188.7 ¢ to 1383.2 ¢
Major 8-mosstep M8ms 6L + 2s 1383.2 ¢ to 1426.5 ¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 1426.5 ¢ to 1556.1 ¢
Major 9-mosstep M9ms 7L + 2s 1556.1 ¢ to 1664.2 ¢
10-mosstep Minor 10-mosstep m10ms 7L + 3s 1664.2 ¢ to 1729.1 ¢
Major 10-mosstep M10ms 8L + 2s 1729.1 ¢ to 1902.0 ¢
11-mosstep Perfect 11-mosstep P11ms 8L + 3s 1902.0 ¢

Generator chain

Generator chain of 8L 3s⟨3/1⟩
Bright gens Scale degree Abbrev.
18 Augmented 6-mosdegree A6md
17 Augmented 2-mosdegree A2md
16 Augmented 9-mosdegree A9md
15 Augmented 5-mosdegree A5md
14 Augmented 1-mosdegree A1md
13 Augmented 8-mosdegree A8md
12 Augmented 4-mosdegree A4md
11 Augmented 0-mosdegree A0md
10 Augmented 7-mosdegree A7md
9 Major 3-mosdegree M3md
8 Major 10-mosdegree M10md
7 Major 6-mosdegree M6md
6 Major 2-mosdegree M2md
5 Major 9-mosdegree M9md
4 Major 5-mosdegree M5md
3 Major 1-mosdegree M1md
2 Major 8-mosdegree M8md
1 Perfect 4-mosdegree P4md
0 Perfect 0-mosdegree
Perfect 11-mosdegree		 P0md
P11md
−1 Perfect 7-mosdegree P7md
−2 Minor 3-mosdegree m3md
−3 Minor 10-mosdegree m10md
−4 Minor 6-mosdegree m6md
−5 Minor 2-mosdegree m2md
−6 Minor 9-mosdegree m9md
−7 Minor 5-mosdegree m5md
−8 Minor 1-mosdegree m1md
−9 Minor 8-mosdegree m8md
−10 Diminished 4-mosdegree d4md
−11 Diminished 11-mosdegree d11md
−12 Diminished 7-mosdegree d7md
−13 Diminished 3-mosdegree d3md
−14 Diminished 10-mosdegree d10md
−15 Diminished 6-mosdegree d6md
−16 Diminished 2-mosdegree d2md
−17 Diminished 9-mosdegree d9md
−18 Diminished 5-mosdegree d5md

Modes

Scale degrees of the modes of 8L 3s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11
10|0 1 LLLsLLLsLLs Perf. Maj. Maj. Maj. Perf. Maj. Maj. Aug. Maj. Maj. Maj. Perf.
9|1 5 LLLsLLsLLLs Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
8|2 9 LLsLLLsLLLs Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
7|3 2 LLsLLLsLLsL Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf.
6|4 6 LLsLLsLLLsL Perf. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Perf.
5|5 10 LsLLLsLLLsL Perf. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Perf.
4|6 3 LsLLLsLLsLL Perf. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Perf.
3|7 7 LsLLsLLLsLL Perf. Maj. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Perf.
2|8 11 sLLLsLLLsLL Perf. Min. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Perf.
1|9 4 sLLLsLLsLLL Perf. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Perf.
0|10 8 sLLsLLLsLLL Perf. Min. Min. Min. Dim. Min. Min. Perf. Min. Min. Min. Perf.

Theory

By dividing the 5L 2s of LLsLLLs into A=LLs and B=LLLs, and combining them as ABBABB..., it becomes 8L 3s⟨3/1⟩. This scale has octaves that are too frequent for the listener to feel a tritave equivalence. The range of possible dark generators will likely feel sufficiently pseudo-octave. Similar to Angel, it would be good to utilize a finite-length chain of octaves and make use of existing diatonic music theory.

Low harmonic entropy scales

  • Pythagorean tuning (period = 3/1, generator = 3/2): L/s = 2.260
  • Tritave-equivalent meantone tunings:

Tuning ranges

Simple tunings

Simple Tunings of 8L 3s⟨3/1⟩
Scale degree Abbrev. Basic (2:1)
19edt 		Hard (3:1)
27edt 		Soft (3:2)
30edt
Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\19 0.0 0\27 0.0 0\30 0.0
Minor 1-mosdegree m1md 1\19 100.1 1\27 70.4 2\30 126.8
Major 1-mosdegree M1md 2\19 200.2 3\27 211.3 3\30 190.2
Minor 2-mosdegree m2md 3\19 300.3 4\27 281.8 5\30 317.0
Major 2-mosdegree M2md 4\19 400.4 6\27 422.7 6\30 380.4
Minor 3-mosdegree m3md 5\19 500.5 7\27 493.1 8\30 507.2
Major 3-mosdegree M3md 6\19 600.6 9\27 634.0 9\30 570.6
Diminished 4-mosdegree d4md 6\19 600.6 8\27 563.5 10\30 634.0
Perfect 4-mosdegree P4md 7\19 700.7 10\27 704.4 11\30 697.4
Minor 5-mosdegree m5md 8\19 800.8 11\27 774.9 13\30 824.2
Major 5-mosdegree M5md 9\19 900.9 13\27 915.8 14\30 887.6
Minor 6-mosdegree m6md 10\19 1001.0 14\27 986.2 16\30 1014.4
Major 6-mosdegree M6md 11\19 1101.1 16\27 1127.1 17\30 1077.8
Perfect 7-mosdegree P7md 12\19 1201.2 17\27 1197.5 19\30 1204.6
Augmented 7-mosdegree A7md 13\19 1301.3 19\27 1338.4 20\30 1268.0
Minor 8-mosdegree m8md 13\19 1301.3 18\27 1268.0 21\30 1331.4
Major 8-mosdegree M8md 14\19 1401.4 20\27 1408.9 22\30 1394.8
Minor 9-mosdegree m9md 15\19 1501.5 21\27 1479.3 24\30 1521.6
Major 9-mosdegree M9md 16\19 1601.6 23\27 1620.2 25\30 1585.0
Minor 10-mosdegree m10md 17\19 1701.7 24\27 1690.6 27\30 1711.8
Major 10-mosdegree M10md 18\19 1801.9 26\27 1831.5 28\30 1775.2
Perfect 11-mosdegree P11md 19\19 1902.0 27\27 1902.0 30\30 1902.0

Soft-of-basic tunings

Tunings of 8L 3s⟨3/1⟩
Scale degree Abbrev. 5:4
52edt 		Supersoft (4:3)
41edt 		Soft (3:2)
30edt 		Semisoft (5:3)
49edt
Steps ¢ Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\52 0.0 0\41 0.0 0\30 0.0 0\49 0.0
Minor 1-mosdegree m1md 4\52 146.3 3\41 139.2 2\30 126.8 3\49 116.4
Major 1-mosdegree M1md 5\52 182.9 4\41 185.6 3\30 190.2 5\49 194.1
Minor 2-mosdegree m2md 9\52 329.2 7\41 324.7 5\30 317.0 8\49 310.5
Major 2-mosdegree M2md 10\52 365.8 8\41 371.1 6\30 380.4 10\49 388.2
Minor 3-mosdegree m3md 14\52 512.1 11\41 510.3 8\30 507.2 13\49 504.6
Major 3-mosdegree M3md 15\52 548.6 12\41 556.7 9\30 570.6 15\49 582.2
Diminished 4-mosdegree d4md 18\52 658.4 14\41 649.4 10\30 634.0 16\49 621.0
Perfect 4-mosdegree P4md 19\52 694.9 15\41 695.8 11\30 697.4 18\49 698.7
Minor 5-mosdegree m5md 23\52 841.2 18\41 835.0 13\30 824.2 21\49 815.1
Major 5-mosdegree M5md 24\52 877.8 19\41 881.4 14\30 887.6 23\49 892.8
Minor 6-mosdegree m6md 28\52 1024.1 22\41 1020.6 16\30 1014.4 26\49 1009.2
Major 6-mosdegree M6md 29\52 1060.7 23\41 1067.0 17\30 1077.8 28\49 1086.8
Perfect 7-mosdegree P7md 33\52 1207.0 26\41 1206.1 19\30 1204.6 31\49 1203.3
Augmented 7-mosdegree A7md 34\52 1243.6 27\41 1252.5 20\30 1268.0 33\49 1280.9
Minor 8-mosdegree m8md 37\52 1353.3 29\41 1345.3 21\30 1331.4 34\49 1319.7
Major 8-mosdegree M8md 38\52 1389.9 30\41 1391.7 22\30 1394.8 36\49 1397.4
Minor 9-mosdegree m9md 42\52 1536.2 33\41 1530.8 24\30 1521.6 39\49 1513.8
Major 9-mosdegree M9md 43\52 1572.8 34\41 1577.2 25\30 1585.0 41\49 1591.4
Minor 10-mosdegree m10md 47\52 1719.1 37\41 1716.4 27\30 1711.8 44\49 1707.9
Major 10-mosdegree M10md 48\52 1755.7 38\41 1762.8 28\30 1775.2 46\49 1785.5
Perfect 11-mosdegree P11md 52\52 1902.0 41\41 1902.0 30\30 1902.0 49\49 1902.0

Hard-of-basic tunings

Tunings of 8L 3s⟨3/1⟩
Scale degree Abbrev. Semihard (5:2)
46edt 		Hard (3:1)
27edt 		Superhard (4:1)
35edt 		5:1
43edt
Steps ¢ Steps ¢ Steps ¢ Steps ¢
Perfect 0-mosdegree P0md 0\46 0.0 0\27 0.0 0\35 0.0 0\43 0.0
Minor 1-mosdegree m1md 2\46 82.7 1\27 70.4 1\35 54.3 1\43 44.2
Major 1-mosdegree M1md 5\46 206.7 3\27 211.3 4\35 217.4 5\43 221.2
Minor 2-mosdegree m2md 7\46 289.4 4\27 281.8 5\35 271.7 6\43 265.4
Major 2-mosdegree M2md 10\46 413.5 6\27 422.7 8\35 434.7 10\43 442.3
Minor 3-mosdegree m3md 12\46 496.2 7\27 493.1 9\35 489.1 11\43 486.5
Major 3-mosdegree M3md 15\46 620.2 9\27 634.0 12\35 652.1 15\43 663.5
Diminished 4-mosdegree d4md 14\46 578.9 8\27 563.5 10\35 543.4 12\43 530.8
Perfect 4-mosdegree P4md 17\46 702.9 10\27 704.4 13\35 706.4 16\43 707.7
Minor 5-mosdegree m5md 19\46 785.6 11\27 774.9 14\35 760.8 17\43 751.9
Major 5-mosdegree M5md 22\46 909.6 13\27 915.8 17\35 923.8 21\43 928.9
Minor 6-mosdegree m6md 24\46 992.3 14\27 986.2 18\35 978.1 22\43 973.1
Major 6-mosdegree M6md 27\46 1116.4 16\27 1127.1 21\35 1141.2 26\43 1150.0
Perfect 7-mosdegree P7md 29\46 1199.1 17\27 1197.5 22\35 1195.5 27\43 1194.3
Augmented 7-mosdegree A7md 32\46 1323.1 19\27 1338.4 25\35 1358.5 31\43 1371.2
Minor 8-mosdegree m8md 31\46 1281.8 18\27 1268.0 23\35 1249.9 28\43 1238.5
Major 8-mosdegree M8md 34\46 1405.8 20\27 1408.9 26\35 1412.9 32\43 1415.4
Minor 9-mosdegree m9md 36\46 1488.5 21\27 1479.3 27\35 1467.2 33\43 1459.6
Major 9-mosdegree M9md 39\46 1612.5 23\27 1620.2 30\35 1630.2 37\43 1636.6
Minor 10-mosdegree m10md 41\46 1695.2 24\27 1690.6 31\35 1684.6 38\43 1680.8
Major 10-mosdegree M10md 44\46 1819.3 26\27 1831.5 34\35 1847.6 42\43 1857.7
Perfect 11-mosdegree P11md 46\46 1902.0 27\27 1902.0 35\35 1902.0 43\43 1902.0

Scale tree

Scale tree and tuning spectrum of 8L 3s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
4\11 691.620 1210.335 1:1 1.000 Equalized 8L 3s⟨3/1⟩
23\63 694.365 1207.590 6:5 1.200
19\52 694.945 1207.010 5:4 1.250
34\93 695.338 1206.617 9:7 1.286
15\41 695.837 1206.118 4:3 1.333 Supersoft 8L 3s⟨3/1⟩
41\112 696.251 1205.704 11:8 1.375
26\71 696.491 1205.464 7:5 1.400
37\101 696.756 1205.199 10:7 1.429
11\30 697.384 1204.572 3:2 1.500 Soft 8L 3s⟨3/1⟩
40\109 697.965 1203.990 11:7 1.571
29\79 698.186 1203.769 8:5 1.600
47\128 698.374 1203.581 13:8 1.625
18\49 698.677 1203.278 5:3 1.667 Semisoft 8L 3s⟨3/1⟩
43\117 699.009 1202.946 12:7 1.714
25\68 699.248 1202.707 7:4 1.750
32\87 699.570 1202.385 9:5 1.800
7\19 700.720 1201.235 2:1 2.000 Basic 8L 3s⟨3/1⟩
Scales with tunings softer than this are proper
31\84 701.912 1200.043 9:4 2.250 Pythagorean tuning is around here
24\65 702.260 1199.695 7:3 2.333
41\111 702.524 1199.431 12:5 2.400
17\46 702.896 1199.059 5:2 2.500 Semihard 8L 3s⟨3/1⟩
44\119 703.244 1198.711 13:5 2.600
27\73 703.463 1198.492 8:3 2.667
37\100 703.723 1198.232 11:4 2.750
10\27 704.428 1197.527 3:1 3.000 Hard 8L 3s⟨3/1⟩
33\89 705.219 1196.736 10:3 3.333
23\62 705.564 1196.391 7:2 3.500
36\97 705.880 1196.075 11:3 3.667
13\35 706.440 1195.515 4:1 4.000 Superhard 8L 3s⟨3/1⟩
29\78 707.137 1194.818 9:2 4.500
16\43 707.704 1194.251 5:1 5.000
19\51 708.571 1193.384 6:1 6.000
3\8 713.233 1188.722 1:0 → ∞ Collapsed 8L 3s⟨3/1⟩
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