7L 4s: Difference between revisions

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 [[39edo|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.

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 [[Wikipedia:Joseph Sauveur|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.

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.

== Modes ==

{{MOS mode degrees}}

 

{{MOS intervals}}

{{Scale tree|Comments=

6/5: [[Mabon]];

7/5: [[Rarity]];

13/8: Unnamed golden tuning;

13/5: [[Cohemimabila]], unnamed golden tuning;

4/1: [[Sixix]];

6/1:↓ [[Amity]] / [[hitchcock]];