5L 3s (5/2-equivalent)

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↖4L 2s⟨5/2⟩ ↑5L 2s⟨5/2⟩ 6L 2s⟨5/2⟩↗
←4L 3s⟨5/2⟩5L 3s (5/2-equivalent) 6L 3s⟨5/2⟩→
↙4L 4s⟨5/2⟩ ↓5L 4s⟨5/2⟩ 6L 4s⟨5/2⟩↘
 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 5/2 (1586.3¢)
Period 5/2 (1586.3¢)
Generator size(ed5/2)
Bright 3\8 to 2\5 (594.9¢ to 634.5¢)
Dark 3\5 to 5\8 (951.8¢ to 991.4¢)
Related MOS scales
Parent 3L 2s⟨5/2⟩
Sister 3L 5s⟨5/2⟩
Daughters 8L 5s⟨5/2⟩
5L 8s⟨5/2⟩
Equal tunings(ed5/2)
Equalized (L:s = 1:1) 3\8 (594.9¢)
Supersoft (L:s = 4:3) 11\29 (601.7¢)
Soft (L:s = 3:2) 8\21 (604.3¢)
Semisoft (L:s = 5:3) 13\34 (606.5¢)
Basic (L:s = 2:1) 5\13 (610.1¢)
Semihard (L:s = 5:2) 12\31 (614.1¢)
Hard (L:s = 3:1) 7\18 (616.9¢)
Superhard (L:s = 4:1) 9\23 (620.7¢)
Collapsed (L:s = 1:0) 2\5 (634.5¢)

5L 3s⟨5/2⟩ is a 5/2-equivalent (non-octave) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every interval of 5/2 (1586.3¢). Generators that produce this scale range from 594.9¢ to 634.5¢, or from 951.8¢ to 991.4¢.

Theory

This scale can be thought of as a MOS generated by 5/2-equivalent jubilic temperament where 50/49 vanishes. This is one of the most simple no-threes temperaments, although in this case it is using the second-best period choice of 5/2 rather than 2/1, causing strict MOS to be generated rather than multi-MOS. The bright generator of the MOS represents 7/5~10/7 and the dark generator represents 7/4~25/14. This gives 5L 3s⟨5/2⟩ many approximations of 4:5:7 chords for no-threes harmony.

Intervals

The below uses diamond MOS notation names.

Scale degree of 5L 3s⟨5/2⟩
Scale degree On J 13ed5/2 (Basic, L:s = 2:1) 18ed5/2 (Hard, L:s = 3:1) 21ed5/2 (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-mosdegree (unison) J 0 0 0 0 0 0 1/1 (exact)
Minor 1-mosdegree K@ 1 122 1 88.1 2 151.1
Major 1-mosdegree K 2 244 3 264.4 3 226.6
Minor 2-mosdegree L 3 366.1 4 352.5 5 377.7
Major 2-mosdegree L& 4 488.1 6 528.8 6 453.2
Diminished 3-mosdegree M@ 4 488.1 5 440.6 7 528.8
Perfect 3-mosdegree M 5 610.1 7 616.9 8 604.3
Minor 4-mosdegree N@ 6 732.1 8 705 10 755.4
Major 4-mosdegree N 7 854.2 10 881.3 11 830.9
Perfect 5-mosdegree O 8 976.2 11 969.4 13 982
Augmented 5-mosdegree O& 9 1098.2 13 1145.7 14 1057.5
Minor 6-mosdegree P@ 9 1098.2 12 1057.5 15 1133.1
Major 6-mosdegree P 10 1220.2 14 1233.8 16 1208.6
Minor 7-mosdegree Q 11 1342.3 15 1321.9 18 1359.7
Major 7-mosdegree Q& 12 1464.3 17 1498.2 19 1435.2
Perfect 8-mosdegree (equave) J 13 1586.3 18 1586.3 21 1586.3 5/2 (exact)

Modes

Modes of 5L 3s⟨5/2⟩
UDP Rotational order Step pattern
7|0 1 LLsLLsLs
6|1 4 LLsLsLLs
5|2 7 LsLLsLLs
4|3 2 LsLLsLsL
3|4 5 LsLsLLsL
2|5 8 sLLsLLsL
1|6 3 sLLsLsLL
0|7 6 sLsLLsLL

Scale tree

Scale Tree and Tuning Spectrum of 5L 3s⟨5/2⟩
Generator(ed5/2) Cents Step Ratio Comments
Bright Dark L:s Hardness
3\8 594.868 991.446 1:1 1.000 Equalized 5L 3s⟨5/2⟩
17\45 599.274 987.040 6:5 1.200
14\37 600.227 986.087 5:4 1.250
25\66 600.876 985.437 9:7 1.286
11\29 601.705 984.609 4:3 1.333 Supersoft 5L 3s⟨5/2⟩
Jubilic is in this region
30\79 602.398 983.916 11:8 1.375
19\50 602.799 983.515 7:5 1.400
27\71 603.246 983.068 10:7 1.429
8\21 604.310 982.004 3:2 1.500 Soft 5L 3s⟨5/2⟩
29\76 605.304 981.010 11:7 1.571
21\55 605.683 980.630 8:5 1.600
34\89 606.007 980.306 13:8 1.625
13\34 606.532 979.782 5:3 1.667 Semisoft 5L 3s⟨5/2⟩
31\81 607.108 979.206 12:7 1.714
18\47 607.524 978.789 7:4 1.750
23\60 608.087 978.227 9:5 1.800
5\13 610.121 976.193 2:1 2.000 Basic 5L 3s⟨5/2⟩
Scales with tunings softer than this are proper
22\57 612.261 974.052 9:4 2.250
17\44 612.894 973.420 7:3 2.333
29\75 613.375 972.939 12:5 2.400
12\31 614.057 972.257 5:2 2.500 Semihard 5L 3s⟨5/2⟩
31\80 614.697 971.617 13:5 2.600
19\49 615.101 971.212 8:3 2.667
26\67 615.584 970.729 11:4 2.750
7\18 616.900 969.414 3:1 3.000 Hard 5L 3s⟨5/2⟩
23\59 618.393 967.920 10:3 3.333
16\41 619.049 967.264 7:2 3.500
25\64 619.654 966.660 11:3 3.667
9\23 620.731 965.582 4:1 4.000 Superhard 5L 3s⟨5/2⟩
20\51 622.084 964.230 9:2 4.500
11\28 623.195 963.119 5:1 5.000
13\33 624.911 961.402 6:1 6.000
2\5 634.525 951.788 1:0 → ∞ Collapsed 5L 3s⟨5/2⟩
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