12L 1s

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

←11L 1s	12L 1s	13L 1s→
↙11L 2s	↓12L 2s	13L 2s↘

┌╥╥╥╥╥╥╥╥╥╥╥╥┬┐ │║║║║║║║║║║║║││ │││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LLLLLLLLLLLLs sLLLLLLLLLLLL
Equave	2/1 (1200.0¢)
Period	2/1 (1200.0¢)
Generator size
Bright	1\13 to 1\12 (92.3¢ to 100.0¢)
Dark	11\12 to 12\13 (1100.0¢ to 1107.7¢)
TAMNAMS information
Descends from	1L 9s (antisinatonic)
Required step ratio range	3:1 to 4:1 (parahard)
Related MOS scales
Parent	1L 11s
Sister	1L 12s
Daughters	13L 12s 12L 13s
Equal tunings
Equalized (L:s = 1:1)	1\13 (92.3¢)
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 (96.0¢)
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 (98.0¢)
Collapsed (L:s = 1:0)	1\12 (100.0¢)

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 (edo)						Cents		Step ratio		Comments(Always proper)
						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