7L 4s

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Todo: review
Adopt scale tree template, review entries
↖6L 3s ↑7L 3s 8L 3s↗
←6L 4s7L 4s 8L 4s→
↙6L 5s ↓7L 5s 8L 5s↘
┌╥╥┬╥╥┬╥╥┬╥┬┐
│║║│║║│║║│║││
│││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLLsLs
sLsLLsLLsLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 3\11 to 2\7 (327.3¢ to 342.9¢)
Dark 5\7 to 8\11 (857.1¢ to 872.7¢)
TAMNAMS information
Descends from 4L 3s (smitonic)
Required step ratio range 1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent 4L 3s
Sister 4L 7s
Daughters 11L 7s, 7L 11s
Neutralized 3L 8s
Equal tunings
Equalized (L:s = 1:1) 3\11 (327.3¢)
Supersoft (L:s = 4:3) 11\40 (330.0¢)
Soft (L:s = 3:2) 8\29 (331.0¢)
Semisoft (L:s = 5:3) 13\47 (331.9¢)
Basic (L:s = 2:1) 5\18 (333.3¢)
Semihard (L:s = 5:2) 12\43 (334.9¢)
Hard (L:s = 3:1) 7\25 (336.0¢)
Superhard (L:s = 4:1) 9\32 (337.5¢)
Collapsed (L:s = 1:0) 2\7 (342.9¢)

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

Modes of 7L 4s
UDP Rotational order Step pattern
10|0 1 LLsLLsLLsLs
9|1 4 LLsLLsLsLLs
8|2 7 LLsLsLLsLLs
7|3 10 LsLLsLLsLLs
6|4 2 LsLLsLLsLsL
5|5 5 LsLLsLsLLsL
4|6 8 LsLsLLsLLsL
3|7 11 sLLsLLsLLsL
2|8 3 sLLsLLsLsLL
1|9 6 sLLsLsLLsLL
0|10 9 sLsLLsLLsLL

Scale tree

Scale Tree and Tuning Spectrum of 7L 4s
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