6L 1s

← 5L 1s 6L 1s 7L 1s →

↙ 5L 2s ↓ 6L 2s 7L 2s ↘
	Scale structure
Step pattern	LLLLLLs; sLLLLLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
	Generator size
Bright	1\7 to 1\6 (171.4 ¢ to 200.0 ¢)
Dark	5\6 to 6\7 (1000.0 ¢ to 1028.6 ¢)
	TAMNAMS information
Name	archaeotonic
Prefix	arch-
Abbrev.	arc
	Related MOS scales
Parent	1L 5s
Sister	1L 6s
Daughters	7L 6s, 6L 7s
Neutralized	5L 2s
2-Flought	13L 1s, 6L 8s
	Equal tunings
Equalized (L:s = 1:1)	1\7 (171.4 ¢)
Supersoft (L:s = 4:3)	4\27 (177.8 ¢)
Soft (L:s = 3:2)	3\20 (180.0 ¢)
Semisoft (L:s = 5:3)	5\33 (181.8 ¢)
Basic (L:s = 2:1)	2\13 (184.6 ¢)
Semihard (L:s = 5:2)	5\32 (187.5 ¢)
Hard (L:s = 3:1)	3\19 (189.5 ¢)
Superhard (L:s = 4:1)	4\25 (192.0 ¢)
Collapsed (L:s = 1:0)	1\6 (200.0 ¢)
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6L 1s, named archaeotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 6 large steps and 1 small step, repeating every octave. Generators that produce this scale range from 171.4 ¢ to 200 ¢, or from 1000 ¢ to 1028.6 ¢. Scales of this form are always proper because there is only one small step.

Names

TAMNAMS suggests the temperament-agnostic name archaeotonic as the name of 6L 1s. The name was originally used as a name for the 6L 1s scale in 13edo.

Scale properties

Intervals

Intervals of 6L 1s
Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.	Steps subtended	Range in cents
0-archstep	Perfect 0-archstep	P0arcs	0	0.0 ¢
1-archstep	Diminished 1-archstep	d1arcs	s	0.0 ¢ to 171.4 ¢
1-archstep	Perfect 1-archstep	P1arcs	L	171.4 ¢ to 200.0 ¢
2-archstep	Minor 2-archstep	m2arcs	L + s	200.0 ¢ to 342.9 ¢
2-archstep	Major 2-archstep	M2arcs	2L	342.9 ¢ to 400.0 ¢
3-archstep	Minor 3-archstep	m3arcs	2L + s	400.0 ¢ to 514.3 ¢
3-archstep	Major 3-archstep	M3arcs	3L	514.3 ¢ to 600.0 ¢
4-archstep	Minor 4-archstep	m4arcs	3L + s	600.0 ¢ to 685.7 ¢
4-archstep	Major 4-archstep	M4arcs	4L	685.7 ¢ to 800.0 ¢
5-archstep	Minor 5-archstep	m5arcs	4L + s	800.0 ¢ to 857.1 ¢
5-archstep	Major 5-archstep	M5arcs	5L	857.1 ¢ to 1000.0 ¢
6-archstep	Perfect 6-archstep	P6arcs	5L + s	1000.0 ¢ to 1028.6 ¢
6-archstep	Augmented 6-archstep	A6arcs	6L	1028.6 ¢ to 1200.0 ¢
7-archstep	Perfect 7-archstep	P7arcs	6L + s	1200.0 ¢

Generator chain

Generator chain of 6L 1s
Bright gens	Scale degree	Abbrev.
12	Augmented 5-archdegree	A5arcd
11	Augmented 4-archdegree	A4arcd
10	Augmented 3-archdegree	A3arcd
9	Augmented 2-archdegree	A2arcd
8	Augmented 1-archdegree	A1arcd
7	Augmented 0-archdegree	A0arcd
6	Augmented 6-archdegree	A6arcd
5	Major 5-archdegree	M5arcd
4	Major 4-archdegree	M4arcd
3	Major 3-archdegree	M3arcd
2	Major 2-archdegree	M2arcd
1	Perfect 1-archdegree	P1arcd
0	Perfect 0-archdegree Perfect 7-archdegree	P0arcd P7arcd
−1	Perfect 6-archdegree	P6arcd
−2	Minor 5-archdegree	m5arcd
−3	Minor 4-archdegree	m4arcd
−4	Minor 3-archdegree	m3arcd
−5	Minor 2-archdegree	m2arcd
−6	Diminished 1-archdegree	d1arcd
−7	Diminished 7-archdegree	d7arcd
−8	Diminished 6-archdegree	d6arcd
−9	Diminished 5-archdegree	d5arcd
−10	Diminished 4-archdegree	d4arcd
−11	Diminished 3-archdegree	d3arcd
−12	Diminished 2-archdegree	d2arcd

Modes

Scale degrees of the modes of 6L 1s
UDP	Cyclic order	Step pattern	Scale degree (archdegree)
UDP	Cyclic order	Step pattern	0	1	2	3	4	5	6	7
6\|0	1	LLLLLLs	Perf.	Perf.	Maj.	Maj.	Maj.	Maj.	Aug.	Perf.
5\|1	2	LLLLLsL	Perf.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Perf.
4\|2	3	LLLLsLL	Perf.	Perf.	Maj.	Maj.	Maj.	Min.	Perf.	Perf.
3\|3	4	LLLsLLL	Perf.	Perf.	Maj.	Maj.	Min.	Min.	Perf.	Perf.
2\|4	5	LLsLLLL	Perf.	Perf.	Maj.	Min.	Min.	Min.	Perf.	Perf.
1\|5	6	LsLLLLL	Perf.	Perf.	Min.	Min.	Min.	Min.	Perf.	Perf.
0\|6	7	sLLLLLL	Perf.	Dim.	Min.	Min.	Min.	Min.	Perf.	Perf.

Proposed names

Temperaments

There are two notable harmonic entropy minima with this MOS pattern. The first is tetracot, in which the generator is identified with 10/9 and four generators make a 3/2. These produce very soft tunings of archaeotonic, ranging from 4:3 in 27edo to 7:6 in 48edo. The second is known as didacus, which is at a basic level the temperament in the 2.5.7 subgroup defined by 3136/3125, where two generators make 5/4 and five make 7/4, and produces very hard tunings, ranging from 4:1 in 25edo to 7:1 in 43edo; it has various extensions that span portions of this range, including roulette and mediantone to the no-twos 19-limit, and hemithirds (along with its 5-limit microtemperament restriction luna) and hemiwürschmidt to the full 7-limit.

The 6L 1s pattern also houses a temperament of the 11th and 13th harmonics, i.e. Bluebirds, where the generator is identified with 143/128; for example L = 7, s = 4 (46 edo) is such a scale.

Scales

Tetracot7 – 34edo, 41edo, and POTE tuning
Bluebirds7 – 329edo tuning
Glacial7 – 84edo tuning
Deutone7 – 44edo tuning
Leantone7 – 81edo tuning
Roulette7 – 37edo tuning

Scale tree

Scale tree and tuning spectrum of 6L 1s
Generator^(edo)						Cents		Step ratio		Comments^{(always proper)}
Generator^(edo)						Bright	Dark	L:s	Hardness	Comments^{(always proper)}
1\7						171.429	1028.571	1:1	1.000	Equalized 6L 1s
					6\41	175.610	1024.390	6:5	1.200
				5\34		176.471	1023.529	5:4	1.250	Tetracot is in this region
					9\61	177.049	1022.951	9:7	1.286	Tetracot/modus/wollemia
			4\27			177.778	1022.222	4:3	1.333	Supersoft 6L 1s
					11\74	178.378	1021.622	11:8	1.375
				7\47		178.723	1021.277	7:5	1.400
					10\67	179.104	1020.896	10:7	1.429
		3\20				180.000	1020.000	3:2	1.500	Soft 6L 1s
					11\73	180.822	1019.178	11:7	1.571
				8\53		181.132	1018.868	8:5	1.600
					13\86	181.395	1018.605	13:8	1.625	Wilson Golden 2 (181.3227 ¢)
			5\33			181.818	1018.182	5:3	1.667	Semisoft 6L 1s
					12\79	182.278	1017.722	12:7	1.714	Bluebirds
				7\46		182.609	1017.391	7:4	1.750
					9\59	183.051	1016.949	9:5	1.800
	2\13					184.615	1015.385	2:1	2.000	Basic 6L 1s
					9\58	186.207	1013.793	9:4	2.250
				7\45		186.667	1013.333	7:3	2.333
					12\77	187.013	1012.987	12:5	2.400
			5\32			187.500	1012.500	5:2	2.500	Semihard 6L 1s
					13\83	187.952	1012.048	13:5	2.600	Golden glacial (188.0298 ¢)
				8\51		188.235	1011.765	8:3	2.667
					11\70	188.571	1011.429	11:4	2.750
		3\19				189.474	1010.526	3:1	3.000	Hard 6L 1s Spell
					10\63	190.476	1009.524	10:3	3.333
				7\44		190.909	1009.091	7:2	3.500	Isra/deutone
					11\69	191.304	1008.696	11:3	3.667
			4\25			192.000	1008.000	4:1	4.000	Superhard 6L 1s Isra/leantone
					9\56	192.857	1007.143	9:2	4.500
				5\31		193.548	1006.452	5:1	5.000	Didacus/hemithirds/hemiwürschmidt
					6\37	194.595	1005.405	6:1	6.000	Didacus/mediantone/roulette
1\6						200.000	1000.000	1:0	→ ∞	Collapsed 6L 1s