12L 1s

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Todo: cleanup
Clean up lead section, find out who first proposed quasidozenal

←11L 1s12L 1s13L 1s→
↙11L 2s↓12L 2s 13L 2s↘
Brightest mode LLLLLLLLLLLLs
Period 2/1
Range for bright generator 1\13 (92.3¢) to 1\12 (100¢)
Range for dark generator 11\12 (1100¢) to 12\13 (1107.7¢)
Parent MOS 1L 11s
Sister MOS 1L 12s
Daughter MOSes 13L 12s, 12L 13s
Equal tunings
Supersoft (L:s = 4:3) 4\51 (94.1¢)
Soft (L:s = 3:2) 3\38 (94.7¢)
Semisoft (L:s = 5:3) 5\63 (95.2¢)
Basic (L:s = 2:1) 2\25 (96¢)
Semihard (L:s = 5:2) 5\62 (96.8¢)
Hard (L:s = 3:1) 3\37 (97.3¢)
Superhard (L:s = 4:1) 4\49 (98¢)
Brightest-mode tunings on xenpaper
Supersoft Soft Semisoft Basic Semihard Hard Superhard

12L 1s, also called quasidozenal, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 12 large steps and 1 small step, repeating every octave. 12L 1s is a great-grandchild scale of 1L 9s, expanding it by 3 tones. Generators that produce this scale range from 92.3¢ to 100¢, or from 1100¢ to 1107.7¢. Scales of this form are always proper because there is only one small step.

Apparently, it belongs to no particularly important temperament. However, it becomes a compressed 12edo scale when you ignore the octave (this obviously does not work when the generator is very near 12edo (within -7/24 ¢ of it), for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or 1L 11s in the 1.75 ¢ above 1\13 because the large and small steps register as identical to one another for human listeners).

Modes

Modes of 12L 1s
UDP Rotational order Step pattern
12|0 1 LLLLLLLLLLLLs
11|1 2 LLLLLLLLLLLsL
10|2 3 LLLLLLLLLLsLL
9|3 4 LLLLLLLLLsLLL
8|4 5 LLLLLLLLsLLLL
7|5 6 LLLLLLLsLLLLL
6|6 7 LLLLLLsLLLLLL
5|7 8 LLLLLsLLLLLLL
4|8 9 LLLLsLLLLLLLL
3|9 10 LLLsLLLLLLLLL
2|10 11 LLsLLLLLLLLLL
1|11 12 LsLLLLLLLLLLL
0|12 13 sLLLLLLLLLLLL

Intervals

Scale degree of 12L 1s
Scale degree 25edo (Basic, L:s = 2:1) 37edo (Hard, L:s = 3:1) 38edo (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-mosdegree (unison) 0 0 0 0 0 0 1/1 (exact)
Diminished 1-mosdegree 1 48 1 32.4 2 63.2
Perfect 1-mosdegree 2 96 3 97.3 3 94.7
Minor 2-mosdegree 3 144 4 129.7 5 157.9
Major 2-mosdegree 4 192 6 194.6 6 189.5
Minor 3-mosdegree 5 240 7 227 8 252.6
Major 3-mosdegree 6 288 9 291.9 9 284.2
Minor 4-mosdegree 7 336 10 324.3 11 347.4
Major 4-mosdegree 8 384 12 389.2 12 378.9
Minor 5-mosdegree 9 432 13 421.6 14 442.1
Major 5-mosdegree 10 480 15 486.5 15 473.7
Minor 6-mosdegree 11 528 16 518.9 17 536.8
Major 6-mosdegree 12 576 18 583.8 18 568.4
Minor 7-mosdegree 13 624 19 616.2 20 631.6
Major 7-mosdegree 14 672 21 681.1 21 663.2
Minor 8-mosdegree 15 720 22 713.5 23 726.3
Major 8-mosdegree 16 768 24 778.4 24 757.9
Minor 9-mosdegree 17 816 25 810.8 26 821.1
Major 9-mosdegree 18 864 27 875.7 27 852.6
Minor 10-mosdegree 19 912 28 908.1 29 915.8
Major 10-mosdegree 20 960 30 973 30 947.4
Minor 11-mosdegree 21 1008 31 1005.4 32 1010.5
Major 11-mosdegree 22 1056 33 1070.3 33 1042.1
Perfect 12-mosdegree 23 1104 34 1102.7 35 1105.3
Augmented 12-mosdegree 24 1152 36 1167.6 36 1136.8
Perfect 13-mosdegree (octave) 25 1200 37 1200 38 1200 2/1 (exact)

Scale tree

Scale tree and tuning spectrum of 12L 1s
Generator (in steps of edo) Cents Step ratio Comments
(Every tuning produces a proper scale)
Bright Dark L:s Hardness
1\13 92.308 1107.692 1:1 1.000 Equalized 12L 1s
6\77 93.506 1106.494 6:5 1.200
5\64 93.750 1106.250 5:4 1.250
9\115 93.913 1106.087 9:7 1.286
4\51 94.118 1105.882 4:3 1.333 Supersoft 12L 1s
11\140 94.286 1105.714 11:8 1.375
7\89 94.382 1105.618 7:5 1.400
10\127 94.488 1105.512 10:7 1.429
3\38 94.737 1105.263 3:2 1.500 Soft 12L 1s
11\139 94.964 1105.036 11:7 1.571
8\101 95.050 1104.950 8:5 1.600
13\164 95.122 1104.878 13:8 1.625
5\63 95.238 1104.762 5:3 1.667 Semisoft 12L 1s
12\151 95.364 1104.636 12:7 1.714
7\88 95.455 1104.545 7:4 1.750
9\113 95.575 1104.425 9:5 1.800
2\25 96.000 1104.000 2:1 2.000 Basic 12L 1s
9\112 96.429 1103.571 9:4 2.250
7\87 96.552 1103.448 7:3 2.333
12\149 96.644 1103.356 12:5 2.400
5\62 96.774 1103.226 5:2 2.500 Semihard 12L 1s
13\161 96.894 1103.106 13:5 2.600
8\99 96.970 1103.030 8:3 2.667
11\136 97.059 1102.941 11:4 2.750
3\37 97.297 1102.703 3:1 3.000 Hard 12L 1s
10\123 97.561 1102.439 10:3 3.333
7\86 97.674 1102.326 7:2 3.500
11\135 97.778 1102.222 11:3 3.667
4\49 97.959 1102.041 4:1 4.000 Superhard 12L 1s
9\110 98.182 1101.818 9:2 4.500
5\61 98.361 1101.639 5:1 5.000
6\73 98.630 1101.370 6:1 6.000
1\12 100.000 1100.000 1:0 → ∞ Collapsed 12L 1s