7L 4s

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↖6L 3s↑7L 3s 8L 3s↗
←6L 4s7L 4s8L 4s→
↙6L 5s↓7L 5s 8L 5s↘
Brightest mode LLsLLsLLsLs
Period 2/1
Range for bright generator 3\11 (327.3¢) to 2\7 (342.9¢)
Range for dark generator 5\7 (857.1¢) to 8\11 (872.7¢)
Parent MOS 4L 3s
Sister MOS 4L 7s
Daughter MOSes 11L 7s, 7L 11s
Equal tunings
Supersoft (L:s = 4:3) 11\40 (330¢)
Soft (L:s = 3:2) 8\29 (331¢)
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¢)
Superhard (L:s = 4:1) 9\32 (337.5¢)

7L 4s has a generator of an augmented minor or diminished neutral third of 327.273 (3/11edo) to 342.857 (2/7edo) cents.

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. However, 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.

A temperament which spans more of the tuning range is sixix, but it is high in just intonation error relative to its step sizes.

7L 4s is still notable for representing 17/14 with tolerable accuracy for as much as that's worth.

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 Step pattern
10|0 LLsLLsLLsLs
9|1 LLsLLsLsLLs
8|2 LLsLsLLsLLs
7|3 LsLLsLLsLLs
6|4 LsLLsLLsLsL
5|5 LsLLsLsLLsL
4|6 LsLsLLsLLsL
3|7 sLLsLLsLLsL
2|8 sLLsLLsLsLL
1|9 sLLsLsLLsLL
0|10 sLsLLsLLsLL

Scale tree

Generator Cents L s L/s Comments
3\11 327.273 1 1 1.000
17\62 329.032 6 5 1.200 Mabon
14\51 329.412 5 4 1.250
25\91 329.670 9 7 1.286
11\40 330.000 4 3 1.333
30\109 330.275 11 8 1.375
19\69 330.435 7 5 1.400
27\98 330.612 10 7 1.428
8\29 331.034 3 2 1.500 L/s = 3/2
29\105 331.429 11 7 1.571
21\76 331.579 8 5 1.600
34\123 331.707 13 8 1.625 Unnamed golden tuning
13\47 331.915 5 3 1.667
31\112 332.143 12 7 1.714
18\65 332.308 7 4 1.750
23\83 332.530 9 5 1.800
5\18 333.333 2 1 2.000 Basic daemotonic
(Generators smaller than this are proper)
22\79 334.177 9 4 2.250
17\61 334.426 7 3 2.333
29\104 334.615 12 5 2.400
12\43 334.884 5 2 2.500
31\111 335.135 13 5 2.600 Cohemimabila, unnamed golden tuning
19\68 335.294 8 3 2.667
26\93 335.484 11 4 2.750
7\25 336.000 3 1 3.000 L/s = 3/1
23\82 336.585 10 3 3.333
16\57 336.842 7 2 3.500
25\89 337.079 11 3 3.667
9\32 337.500 4 1 4.000 Sixix
20\71 338.028 9 2 4.500
11\39 338.462 5 1 5.000
13\46 339.130 6 1 6.000 Amity/hitchcock↓
2\7 342.857 1 0 → inf