7L 4s
Todo: review Adopt scale tree template, review entries |
↖ 6L 3s | ↑7L 3s | 8L 3s ↗ |
← 6L 4s | 7L 4s | 8L 4s → |
↙ 6L 5s | ↓7L 5s | 8L 5s ↘ |
┌╥╥┬╥╥┬╥╥┬╥┬┐ │║║│║║│║║│║││ │││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┘
sLsLLsLLsLL
7L 4s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 7 large steps and 4 small steps, repeating every octave. 7L 4s is a child scale of 4L 3s, expanding it by 4 tones. Generators that produce this scale range from 327.3¢ to 342.9¢, or from 857.1¢ to 872.7¢.
JI approximation
7L 4s fails to represent common just intonation intervals and simple temperaments, and it has no clearly discernible harmonic entropy minimum. From a purely computational perspective, 7L 4s's harmonic entropy minimum is improper and is associated with unusually large step ratios.
Near the harmonic entropy minimum, the simplest temperament of low-complexity JI supported by 7L 4s is amity and its variant hitchcock. It is unconventional to put forward this as the most common approach to this scale, because the large and steps are extremely unequal, being at least of 5:1 step ratio in 39edo, the smallest patent val supporting either of the two. However, it is still significant by virtue of it being the only 5-limit temperament representation of 7L 4s with reasonably low badness.
A temperament which spans more of the tuning range is sixix, but it is high in just intonation error relative to its step sizes. Additionally, as sixix shares the mapping for 3/2 with amity (-5 generators), the generator is of a similar size, and the most accurate tunings of sixix, 32edo and 39c-edo, have highly improper 7L 4s scales like with amity.
On the soft side of the scale, 7L 4s is a scale of the rarity temperament, with tunings like 29edo, and 69edo which are consistent in the 5-limit. However, the temperament is extremely high complexity and high badness. In fact, the third or fifth harmonics do not appear at all in the Rarity[11] 7L 4s scale, and the only common 5-limit intervals which make an appearance are 16/15 and 15/8. The comma itself is also quite complex.
Demon temperament is closer to the center of this MOS's tuning range, but it is in the uncommon subgroup 2.9.11, and like sixix it is moderately inaccurate, compressing 11/9 into a supraminor third.
7L 4s's generator range contains 17/14 and 23/19.
In the equal divisions which are in the size of hundreds, cohemimabila temperament is the first intepretation of 7L 4s of reasonable hardness (roughly semihard) through regular temperament theory. It is supported by 43edo, notable for being studied by Joseph Sauveur due to harmonic strength, and 111edo, which is uniquely consistent in the 15-odd-limit. The generator is mapped to 128/105, and in higher limits it is tempered together with 17/14.
The scale can be made by using every other generator of the tertiaschis temperament, for example in 159edo, which is realized as 2.9.5.7.33.13.17 subgroup 47 & 112 temperament, where it tempers out exactly the same commas as tertiaschis.
Nomenclature
The extended TAMNAMS name for this pattern, as proposed by Eliora, is daemotonic. The name originates in the term "daemon", an archaic spelling of demon.
The name is prescribed to 7L 4s due to the fact that among relatively simple scales it has lowest degree of adherence to regular temperament theory and just intonation (see above). In addition, daemon in ancient times didn't necessarily mean an evil entity, but it could be any kind of spirit, encapsulating that 7L 4s can be found as a useful scale by composers who do not adhere to common regular temperament or consonance-based approaches. A coincidence in the cent measuring system is that two basic (L:s = 2:1) generators stacked together are equal to 666.6666… cents.
From traditional TAMNAMS perspective, the scale may be called m-chro smitonic. Another name, which is deprecated but proposed for reinstation by Ganaram inukshuk, is suprasmitonic.
Modes
UDP | Rotational Order |
Step Pattern |
Scale Degree (degree) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |||
10|0 | 1 | LLsLLsLLsLs | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Aug. | Maj. | Maj. | Perf. |
9|1 | 4 | LLsLLsLsLLs | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
8|2 | 7 | LLsLsLLsLLs | Perf. | Maj. | Maj. | Perf. | Maj. | Min. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
7|3 | 10 | LsLLsLLsLLs | Perf. | Maj. | Min. | Perf. | Maj. | Min. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
6|4 | 2 | LsLLsLLsLsL | Perf. | Maj. | Min. | Perf. | Maj. | Min. | Maj. | Maj. | Perf. | Maj. | Min. | Perf. |
5|5 | 5 | LsLLsLsLLsL | Perf. | Maj. | Min. | Perf. | Maj. | Min. | Maj. | Min. | Perf. | Maj. | Min. | Perf. |
4|6 | 8 | LsLsLLsLLsL | Perf. | Maj. | Min. | Perf. | Min. | Min. | Maj. | Min. | Perf. | Maj. | Min. | Perf. |
3|7 | 11 | sLLsLLsLLsL | Perf. | Min. | Min. | Perf. | Min. | Min. | Maj. | Min. | Perf. | Maj. | Min. | Perf. |
2|8 | 3 | sLLsLLsLsLL | Perf. | Min. | Min. | Perf. | Min. | Min. | Maj. | Min. | Perf. | Min. | Min. | Perf. |
1|9 | 6 | sLLsLsLLsLL | Perf. | Min. | Min. | Perf. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Perf. |
0|10 | 9 | sLsLLsLLsLL | Perf. | Min. | Min. | Dim. | Min. | Min. | Min. | Min. | Perf. | Min. | Min. | Perf. |
Scale tree
Generator^{(edo)} | Cents | Step Ratio | Comments | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Bright | Dark | L:s | Hardness | |||||||
3\11 | 327.273 | 872.727 | 1:1 | 1.000 | Equalized 7L 4s | |||||
17\62 | 329.032 | 870.968 | 6:5 | 1.200 | ||||||
14\51 | 329.412 | 870.588 | 5:4 | 1.250 | ||||||
25\91 | 329.670 | 870.330 | 9:7 | 1.286 | ||||||
11\40 | 330.000 | 870.000 | 4:3 | 1.333 | Supersoft 7L 4s | |||||
30\109 | 330.275 | 869.725 | 11:8 | 1.375 | ||||||
19\69 | 330.435 | 869.565 | 7:5 | 1.400 | ||||||
27\98 | 330.612 | 869.388 | 10:7 | 1.429 | ||||||
8\29 | 331.034 | 868.966 | 3:2 | 1.500 | Soft 7L 4s | |||||
29\105 | 331.429 | 868.571 | 11:7 | 1.571 | ||||||
21\76 | 331.579 | 868.421 | 8:5 | 1.600 | ||||||
34\123 | 331.707 | 868.293 | 13:8 | 1.625 | ||||||
13\47 | 331.915 | 868.085 | 5:3 | 1.667 | Semisoft 7L 4s | |||||
31\112 | 332.143 | 867.857 | 12:7 | 1.714 | ||||||
18\65 | 332.308 | 867.692 | 7:4 | 1.750 | ||||||
23\83 | 332.530 | 867.470 | 9:5 | 1.800 | ||||||
5\18 | 333.333 | 866.667 | 2:1 | 2.000 | Basic 7L 4s Scales with tunings softer than this are proper | |||||
22\79 | 334.177 | 865.823 | 9:4 | 2.250 | ||||||
17\61 | 334.426 | 865.574 | 7:3 | 2.333 | ||||||
29\104 | 334.615 | 865.385 | 12:5 | 2.400 | ||||||
12\43 | 334.884 | 865.116 | 5:2 | 2.500 | Semihard 7L 4s | |||||
31\111 | 335.135 | 864.865 | 13:5 | 2.600 | ||||||
19\68 | 335.294 | 864.706 | 8:3 | 2.667 | ||||||
26\93 | 335.484 | 864.516 | 11:4 | 2.750 | ||||||
7\25 | 336.000 | 864.000 | 3:1 | 3.000 | Hard 7L 4s | |||||
23\82 | 336.585 | 863.415 | 10:3 | 3.333 | ||||||
16\57 | 336.842 | 863.158 | 7:2 | 3.500 | ||||||
25\89 | 337.079 | 862.921 | 11:3 | 3.667 | ||||||
9\32 | 337.500 | 862.500 | 4:1 | 4.000 | Superhard 7L 4s | |||||
20\71 | 338.028 | 861.972 | 9:2 | 4.500 | ||||||
11\39 | 338.462 | 861.538 | 5:1 | 5.000 | ||||||
13\46 | 339.130 | 860.870 | 6:1 | 6.000 | ||||||
2\7 | 342.857 | 857.143 | 1:0 | → ∞ | Collapsed 7L 4s |