7L 4s: Difference between revisions

Line 4:

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.

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.

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 and ~~therefore can be regarded 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. 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.

[[Subgroup temperaments#Demon temperament|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.

@@ Line 4: / Line 4: @@
 L 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.
+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.
+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 and therefore can be regarded 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. 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.
 [[Subgroup temperaments#Demon temperament|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.