3L 8s (3/1-equivalent)

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↖ 2L 7s⟨3/1⟩ ↑ 3L 7s⟨3/1⟩ 4L 7s⟨3/1⟩ ↗
← 2L 8s⟨3/1⟩ 3L 8s (3/1-equivalent) 4L 8s⟨3/1⟩ →
↙ 2L 9s⟨3/1⟩ ↓ 3L 9s⟨3/1⟩ 4L 9s⟨3/1⟩ ↘
Scale structure
Step pattern LssLsssLsss
sssLsssLssL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 7\11 to 2\3 (1210.3 ¢ to 1268.0 ¢)
Dark 1\3 to 4\11 (634.0 ¢ to 691.6 ¢)
Related MOS scales
Parent 3L 5s⟨3/1⟩
Sister 8L 3s⟨3/1⟩
Daughters 11L 3s⟨3/1⟩, 3L 11s⟨3/1⟩
Neutralized 6L 5s⟨3/1⟩
2-Flought 14L 8s⟨3/1⟩, 3L 19s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 7\11 (1210.3 ¢)
Supersoft (L:s = 4:3) 23\36 (1215.1 ¢)
Soft (L:s = 3:2) 16\25 (1217.3 ¢)
Semisoft (L:s = 5:3) 25\39 (1219.2 ¢)
Basic (L:s = 2:1) 9\14 (1222.7 ¢)
Semihard (L:s = 5:2) 20\31 (1227.1 ¢)
Hard (L:s = 3:1) 11\17 (1230.7 ¢)
Superhard (L:s = 4:1) 13\20 (1236.3 ¢)
Collapsed (L:s = 1:0) 2\3 (1268.0 ¢)
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3L 8s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 3 large steps and 8 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1210.3 ¢ to 1268 ¢, or from 634 ¢ to 691.6 ¢.

This MOS is generated by the 17&26 temperament, which unifies the two no-twos zeta peak tunings 17edt and 26edt (double Bohlen-Pierce). That temperament has a generator in the range 658 to 671 cents, generating an 11-tone MOS with a hardness between 3.0 and 6.0.

In 17edt this generates the scale (plus its rotations):

  • 335.641
  • 447.520
  • 559.399
  • 895.040
  • 1006.919
  • 1118.798
  • 1230.677
  • 1566.318
  • 1678.197
  • 1790.076
  • 1901.955

In 26edt it generates the scale (plus its rotations):

  • 438.913
  • 512.065
  • 585.217
  • 1024.13
  • 1097.282
  • 1170.434
  • 1243.586
  • 1682.499
  • 1755.651
  • 1828.803
  • 1901.955

Modes

Modes of 3L 8s⟨3/1⟩
UDP Cyclic
order		 Step
pattern
10|0 1 LssLsssLsss
9|1 8 LsssLssLsss
8|2 4 LsssLsssLss
7|3 11 sLssLsssLss
6|4 7 sLsssLssLss
5|5 3 sLsssLsssLs
4|6 10 ssLssLsssLs
3|7 6 ssLsssLssLs
2|8 2 ssLsssLsssL
1|9 9 sssLssLsssL
0|10 5 sssLsssLssL

Scale degrees

Scale degrees of the modes of 3L 8s⟨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 LssLsssLsss Perf. Maj. Maj. Maj. Aug. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
9|1 8 LsssLssLsss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
8|2 4 LsssLsssLss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Min. Maj. Maj. Perf.
7|3 11 sLssLsssLss Perf. Min. Maj. Maj. Perf. Maj. Maj. Perf. Min. Maj. Maj. Perf.
6|4 7 sLsssLssLss Perf. Min. Maj. Maj. Perf. Min. Maj. Perf. Min. Maj. Maj. Perf.
5|5 3 sLsssLsssLs Perf. Min. Maj. Maj. Perf. Min. Maj. Perf. Min. Min. Maj. Perf.
4|6 10 ssLssLsssLs Perf. Min. Min. Maj. Perf. Min. Maj. Perf. Min. Min. Maj. Perf.
3|7 6 ssLsssLssLs Perf. Min. Min. Maj. Perf. Min. Min. Perf. Min. Min. Maj. Perf.
2|8 2 ssLsssLsssL Perf. Min. Min. Maj. Perf. Min. Min. Perf. Min. Min. Min. Perf.
1|9 9 sssLssLsssL Perf. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Perf.
0|10 5 sssLsssLssL Perf. Min. Min. Min. Perf. Min. Min. Dim. Min. Min. Min. Perf.

Intervals

Intervals of 3L 8s⟨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 634.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 345.8 ¢
Major 2-mosstep M2ms L + s 345.8 ¢ to 634.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 518.7 ¢
Major 3-mosstep M3ms L + 2s 518.7 ¢ to 634.0 ¢
4-mosstep Perfect 4-mosstep P4ms L + 3s 634.0 ¢ to 691.6 ¢
Augmented 4-mosstep A4ms 2L + 2s 691.6 ¢ to 1268.0 ¢
5-mosstep Minor 5-mosstep m5ms L + 4s 634.0 ¢ to 864.5 ¢
Major 5-mosstep M5ms 2L + 3s 864.5 ¢ to 1268.0 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 634.0 ¢ to 1037.4 ¢
Major 6-mosstep M6ms 2L + 4s 1037.4 ¢ to 1268.0 ¢
7-mosstep Diminished 7-mosstep d7ms L + 6s 634.0 ¢ to 1210.3 ¢
Perfect 7-mosstep P7ms 2L + 5s 1210.3 ¢ to 1268.0 ¢
8-mosstep Minor 8-mosstep m8ms 2L + 6s 1268.0 ¢ to 1383.2 ¢
Major 8-mosstep M8ms 3L + 5s 1383.2 ¢ to 1902.0 ¢
9-mosstep Minor 9-mosstep m9ms 2L + 7s 1268.0 ¢ to 1556.1 ¢
Major 9-mosstep M9ms 3L + 6s 1556.1 ¢ to 1902.0 ¢
10-mosstep Minor 10-mosstep m10ms 2L + 8s 1268.0 ¢ to 1729.1 ¢
Major 10-mosstep M10ms 3L + 7s 1729.1 ¢ to 1902.0 ¢
11-mosstep Perfect 11-mosstep P11ms 3L + 8s 1902.0 ¢

Scale tree

Scale tree and tuning spectrum of 3L 8s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
7\11 1210.335 691.620 1:1 1.000 Equalized 3L 8s⟨3/1⟩
37\58 1213.316 688.639 6:5 1.200
30\47 1214.014 687.941 5:4 1.250
53\83 1214.501 687.454 9:7 1.286
23\36 1215.138 686.817 4:3 1.333 Supersoft 3L 8s⟨3/1⟩
62\97 1215.683 686.272 11:8 1.375
39\61 1216.004 685.951 7:5 1.400
55\86 1216.367 685.588 10:7 1.429
16\25 1217.251 684.704 3:2 1.500 Soft 3L 8s⟨3/1⟩
57\89 1218.106 683.849 11:7 1.571
41\64 1218.440 683.515 8:5 1.600
66\103 1218.728 683.227 13:8 1.625
25\39 1219.202 682.753 5:3 1.667 Semisoft 3L 8s⟨3/1⟩
59\92 1219.732 682.223 12:7 1.714
34\53 1220.122 681.833 7:4 1.750
43\67 1220.658 681.297 9:5 1.800
9\14 1222.685 679.270 2:1 2.000 Basic 3L 8s⟨3/1⟩
Scales with tunings softer than this are proper
38\59 1224.988 676.967 9:4 2.250
29\45 1225.704 676.251 7:3 2.333
49\76 1226.260 675.695 12:5 2.400
20\31 1227.068 674.887 5:2 2.500 Semihard 3L 8s⟨3/1⟩
51\79 1227.844 674.111 13:5 2.600
31\48 1228.346 673.609 8:3 2.667
42\65 1228.956 672.999 11:4 2.750
11\17 1230.677 671.278 3:1 3.000 Hard 3L 8s⟨3/1⟩
35\54 1232.749 669.206 10:3 3.333
24\37 1233.701 668.254 7:2 3.500
37\57 1234.602 667.353 11:3 3.667
13\20 1236.271 665.684 4:1 4.000 Superhard 3L 8s⟨3/1⟩
28\43 1238.482 663.473 9:2 4.500
15\23 1240.405 661.550 5:1 5.000
17\26 1243.586 658.369 6:1 6.000
2\3 1267.970 633.985 1:0 → ∞ Collapsed 3L 8s⟨3/1⟩


Todo: expand
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