6L 1s
| ← 5L 1s | 6L 1s | 7L 1s → |
| ↙ 5L 2s | ↓ 6L 2s | 7L 2s ↘ |
┌╥╥╥╥╥╥┬┐ │║║║║║║││ │││││││││ └┴┴┴┴┴┴┴┘
sLLLLLL
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
The intervals of 6L 1s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-archstep and perfect 7-archstep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.
| 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
A chain of bright generators, each a perfect 1-archstep, produces the following scale degrees. A chain of 7 bright generators contains the scale degrees of one of the modes of 6L 1s. Expanding the chain to 13 scale degrees produces the modes of either 7L 6s (for soft-of-basic tunings) or 6L 7s (for hard-of-basic tunings).
| 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
| 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
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
| 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 | |||||