6L 1s

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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 ¢)

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.
0-archstep Perfect 0-archstep P0arcs 0 0.0 ¢
1-archstep Diminished 1-archstep d1arcs s 0.0 ¢ to 171.4 ¢
Perfect 1-archstep P1arcs L 171.4 ¢ to 200.0 ¢
2-archstep Minor 2-archstep m2arcs L + s 200.0 ¢ to 342.9 ¢
Major 2-archstep M2arcs 2L 342.9 ¢ to 400.0 ¢
3-archstep Minor 3-archstep m3arcs 2L + s 400.0 ¢ to 514.3 ¢
Major 3-archstep M3arcs 3L 514.3 ¢ to 600.0 ¢
4-archstep Minor 4-archstep m4arcs 3L + s 600.0 ¢ to 685.7 ¢
Major 4-archstep M4arcs 4L 685.7 ¢ to 800.0 ¢
5-archstep Minor 5-archstep m5arcs 4L + s 800.0 ¢ to 857.1 ¢
Major 5-archstep M5arcs 5L 857.1 ¢ to 1000.0 ¢
6-archstep Perfect 6-archstep P6arcs 5L + s 1000.0 ¢ to 1028.6 ¢
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)
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

Archeotonic.png

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

Scale tree

Scale tree and tuning spectrum of 6L 1s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
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/roulette
1\6 200.000 1000.000 1:0 → ∞ Collapsed 6L 1s
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