10L 3s

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↖ 9L 2s ↑ 10L 2s 11L 2s ↗
← 9L 3s 10L 3s 11L 3s →
↙ 9L 4s ↓ 10L 4s 11L 4s ↘ 
┌╥╥╥╥┬╥╥╥┬╥╥╥┬┐
│║║║║│║║║│║║║││
│││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLLsLLLsLLLs
sLLLsLLLsLLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 9\13 to 7\10 (830.8 ¢ to 840.0 ¢)
Dark 3\10 to 4\13 (360.0 ¢ to 369.2 ¢)
TAMNAMS information
Related to 3L 7s (sephiroid)
With tunings 1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent 3L 7s
Sister 3L 10s
Daughters 13L 10s, 10L 13s
Neutralized 7L 6s
2-Flought 23L 3s, 10L 16s
Equal tunings
Equalized (L:s = 1:1) 9\13 (830.8 ¢)
Supersoft (L:s = 4:3) 34\49 (832.7 ¢)
Soft (L:s = 3:2) 25\36 (833.3 ¢)
Semisoft (L:s = 5:3) 41\59 (833.9 ¢)
Basic (L:s = 2:1) 16\23 (834.8 ¢)
Semihard (L:s = 5:2) 39\56 (835.7 ¢)
Hard (L:s = 3:1) 23\33 (836.4 ¢)
Superhard (L:s = 4:1) 30\43 (837.2 ¢)
Collapsed (L:s = 1:0) 7\10 (840.0 ¢)

10L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 10 large steps and 3 small steps, repeating every octave. 10L 3s is a child scale of 3L 7s, expanding it by 3 tones. Generators that produce this scale range from 830.8 ¢ to 840 ¢, or from 360 ¢ to 369.2 ¢.

A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

Luachoid is a proposed name for this scale.

Modes

Proposed names

Names based on the Hebrew calendar have been proposed as mode names.

Modes of 10L 3s
UDP Cyclic
order		 Step
pattern
12|0 1 LLLLsLLLsLLLs
11|1 10 LLLsLLLLsLLLs
10|2 6 LLLsLLLsLLLLs
9|3 2 LLLsLLLsLLLsL
8|4 11 LLsLLLLsLLLsL
7|5 7 LLsLLLsLLLLsL
6|6 3 LLsLLLsLLLsLL
5|7 12 LsLLLLsLLLsLL
4|8 8 LsLLLsLLLLsLL
3|9 4 LsLLLsLLLsLLL
2|10 13 sLLLLsLLLsLLL
1|11 9 sLLLsLLLLsLLL
0|12 5 sLLLsLLLsLLLL

Degrees

Scale degrees of the modes of 10L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13
12|0 1 LLLLsLLLsLLLs Perf. Maj. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
11|1 10 LLLsLLLLsLLLs Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
10|2 6 LLLsLLLsLLLLs Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Perf.
9|3 2 LLLsLLLsLLLsL Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Min. Perf. Maj. Maj. Min. Perf.
8|4 11 LLsLLLLsLLLsL Perf. Maj. Maj. Min. Perf. Maj. Maj. Maj. Min. Perf. Maj. Maj. Min. Perf.
7|5 7 LLsLLLsLLLLsL Perf. Maj. Maj. Min. Perf. Maj. Maj. Min. Min. Perf. Maj. Maj. Min. Perf.
6|6 3 LLsLLLsLLLsLL Perf. Maj. Maj. Min. Perf. Maj. Maj. Min. Min. Perf. Maj. Min. Min. Perf.
5|7 12 LsLLLLsLLLsLL Perf. Maj. Min. Min. Perf. Maj. Maj. Min. Min. Perf. Maj. Min. Min. Perf.
4|8 8 LsLLLsLLLLsLL Perf. Maj. Min. Min. Perf. Maj. Min. Min. Min. Perf. Maj. Min. Min. Perf.
3|9 4 LsLLLsLLLsLLL Perf. Maj. Min. Min. Perf. Maj. Min. Min. Min. Perf. Min. Min. Min. Perf.
2|10 13 sLLLLsLLLsLLL Perf. Min. Min. Min. Perf. Maj. Min. Min. Min. Perf. Min. Min. Min. Perf.
1|11 9 sLLLsLLLLsLLL Perf. Min. Min. Min. Perf. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.
0|12 5 sLLLsLLLsLLLL Perf. Min. Min. Min. Perf. Min. Min. Min. Min. Dim. Min. Min. Min. Perf.

Intervals

Intervals of 10L 3s
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 92.3 ¢
Major 1-mosstep M1ms L 92.3 ¢ to 120.0 ¢
2-mosstep Minor 2-mosstep m2ms L + s 120.0 ¢ to 184.6 ¢
Major 2-mosstep M2ms 2L 184.6 ¢ to 240.0 ¢
3-mosstep Minor 3-mosstep m3ms 2L + s 240.0 ¢ to 276.9 ¢
Major 3-mosstep M3ms 3L 276.9 ¢ to 360.0 ¢
4-mosstep Perfect 4-mosstep P4ms 3L + s 360.0 ¢ to 369.2 ¢
Augmented 4-mosstep A4ms 4L 369.2 ¢ to 480.0 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 360.0 ¢ to 461.5 ¢
Major 5-mosstep M5ms 4L + s 461.5 ¢ to 480.0 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 480.0 ¢ to 553.8 ¢
Major 6-mosstep M6ms 5L + s 553.8 ¢ to 600.0 ¢
7-mosstep Minor 7-mosstep m7ms 5L + 2s 600.0 ¢ to 646.2 ¢
Major 7-mosstep M7ms 6L + s 646.2 ¢ to 720.0 ¢
8-mosstep Minor 8-mosstep m8ms 6L + 2s 720.0 ¢ to 738.5 ¢
Major 8-mosstep M8ms 7L + s 738.5 ¢ to 840.0 ¢
9-mosstep Diminished 9-mosstep d9ms 6L + 3s 720.0 ¢ to 830.8 ¢
Perfect 9-mosstep P9ms 7L + 2s 830.8 ¢ to 840.0 ¢
10-mosstep Minor 10-mosstep m10ms 7L + 3s 840.0 ¢ to 923.1 ¢
Major 10-mosstep M10ms 8L + 2s 923.1 ¢ to 960.0 ¢
11-mosstep Minor 11-mosstep m11ms 8L + 3s 960.0 ¢ to 1015.4 ¢
Major 11-mosstep M11ms 9L + 2s 1015.4 ¢ to 1080.0 ¢
12-mosstep Minor 12-mosstep m12ms 9L + 3s 1080.0 ¢ to 1107.7 ¢
Major 12-mosstep M12ms 10L + 2s 1107.7 ¢ to 1200.0 ¢
13-mosstep Perfect 13-mosstep P13ms 10L + 3s 1200.0 ¢

Scale tree

Scale tree and tuning spectrum of 10L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
9\13 830.769 369.231 1:1 1.000 Equalized 10L 3s
52\75 832.000 368.000 6:5 1.200
43\62 832.258 367.742 5:4 1.250
77\111 832.432 367.568 9:7 1.286
34\49 832.653 367.347 4:3 1.333 Supersoft 10L 3s
93\134 832.836 367.164 11:8 1.375
59\85 832.941 367.059 7:5 1.400
84\121 833.058 366.942 10:7 1.429
25\36 833.333 366.667 3:2 1.500 Soft 10L 3s
91\131 833.588 366.412 11:7 1.571
66\95 833.684 366.316 8:5 1.600
107\154 833.766 366.234 13:8 1.625
41\59 833.898 366.102 5:3 1.667 Semisoft 10L 3s
98\141 834.043 365.957 12:7 1.714
57\82 834.146 365.854 7:4 1.750
73\105 834.286 365.714 9:5 1.800
16\23 834.783 365.217 2:1 2.000 Basic 10L 3s
Scales with tunings softer than this are proper
71\102 835.294 364.706 9:4 2.250
55\79 835.443 364.557 7:3 2.333
94\135 835.556 364.444 12:5 2.400
39\56 835.714 364.286 5:2 2.500 Semihard 10L 3s
101\145 835.862 364.138 13:5 2.600
62\89 835.955 364.045 8:3 2.667
85\122 836.066 363.934 11:4 2.750
23\33 836.364 363.636 3:1 3.000 Hard 10L 3s
76\109 836.697 363.303 10:3 3.333
53\76 836.842 363.158 7:2 3.500
83\119 836.975 363.025 11:3 3.667
30\43 837.209 362.791 4:1 4.000 Superhard 10L 3s
67\96 837.500 362.500 9:2 4.500
37\53 837.736 362.264 5:1 5.000
44\63 838.095 361.905 6:1 6.000
7\10 840.000 360.000 1:0 → ∞ Collapsed 10L 3s
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