10L 3s

↖ 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 ¢)
View • Talk • Edit

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

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

Scale tree

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

3/10		360
16/53		362.264
13/43		362.791
		363.487
10/33		363.818
		363.976
	27/89	364.045
		364.112
	17/56	364.286
7/23		365.217
		365.904
	18/59	366.102
		366.256
	29/95	366.316
		366.414
11/36		366.667
15/49		367.347
19/62		367.742
4/13		369.231