7L 4s: Difference between revisions
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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 and therefore can be regared as simplest interpretation in the 5-limit for 7L 4s, using edo numbers alone. However, the comma itself is quite complex. | 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 and therefore can be regared as simplest interpretation in the 5-limit for 7L 4s, using edo numbers alone. However, the comma itself is quite complex. | ||
[[Subgroup temperaments#Demon temperament]] is closer to the center of this MOS's tuning range, but it is somewhat inaccurate (compressing 11/9 into a supraminor third) | [[Subgroup temperaments#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 somewhat inaccurate (compressing 11/9 into a supraminor third). | ||
7L 4s is still notable for representing [[17/14]] and [[23/19]] with tolerable accuracy for as much as that is worth. | 7L 4s is still notable for representing [[17/14]] and [[23/19]] with tolerable accuracy for as much as that is worth. | ||
Revision as of 07:01, 3 June 2023
| ↖ 6L 3s | ↑ 7L 3s | 8L 3s ↗ |
| ← 6L 4s | 7L 4s | 8L 4s → |
| ↙ 6L 5s | ↓ 7L 5s | 8L 5s ↘ |
┌╥╥┬╥╥┬╥╥┬╥┬┐ │║║│║║│║║│║││ │││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┘
sLsLLsLLsLL
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.
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 and therefore can be regared as simplest interpretation in the 5-limit for 7L 4s, using edo numbers alone. However, the comma itself is quite complex.
Subgroup temperaments#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 somewhat inaccurate (compressing 11/9 into a supraminor third). 7L 4s is still notable for representing 17/14 and 23/19 with tolerable accuracy for as much as that is 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
| UDP | Cyclic 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
| 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 | Rarity | |||||
| 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 | ||||||