10L 3s
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Scale structure
Step pattern
LLLLsLLLsLLLs
sLLLsLLLsLLLL
Equave
2/1 (1200.0¢)
Period
2/1 (1200.0¢)
Step size
Large
1\13 to 1\10 (92.3¢ to 120.0¢)
Small
0\10 to 1\13 (0.0¢ to 92.3¢)
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
Descends from
3L 7s (sephiroid)
Related MOS scales
Parent
3L 7s
Sister
3L 10s
Daughters
13L 10s
10L 13s
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¢)
| ↖9L 2s | ↑10L 2s | 11L 2s↗ |
| ←9L 3s | 10L 3s | 11L 3s→ |
| ↙9L 4s | ↓10L 4s | 11L 4s↘ |
┌╥╥╥╥┬╥╥╥┬╥╥╥┬┐ │║║║║│║║║│║║║││ │││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┘
sLLLsLLLsLLLL
10L 13s
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
| Circle of fifths | Ascending scale | Mode |
|---|---|---|
| I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X | L L L L s L L L s L L L s | Tishrei |
| x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI | L L L s L L L s L L L s L | Cheshvan |
| vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II | L L s L L L s L L L s L L | Kislev |
| ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI | L s L L L s L L L s L L L | Tevet |
| xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII | s L L L s L L L s L L L L | Shvat |
| vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III | L L L s L L L s L L L L s | Adar minor |
| iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII | L L s L L L s L L L L s L | Adar major |
| xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII | L s L L L s L L L L s L L | Nisan |
| viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV | s L L L s L L L L s L L L | Iyar |
| iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII | L L L s L L L L s L L L s | Sivan |
| xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX | L L s L L L L s L L L s L | Tammuz |
| ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V | L s L L L L s L L L s L L | Av |
| v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I | s L L L L s L L L s L L L | Elul |
Scale tree
| Generator (in steps of 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 | |||||