7L 4s: Difference between revisions
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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. | 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. | 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 amity's mapping of [[5/4]] is reached by -13 generators and is thus inaccessible, and 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. | 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. | ||
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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 {{w|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. | 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 {{w|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 {{nowrap|47 & 112}} temperament, where it tempers out exactly the same commas as tertiaschis. | 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 [[daemotertiaschis]] ({{nowrap|47 & 112}}) temperament, where it tempers out exactly the same commas as tertiaschis. | ||
== Nomenclature == | == Nomenclature == | ||
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| 6/5 = [[Mabon]] | | 6/5 = [[Mabon]] | ||
| 7/5 = [[Rarity]] | | 7/5 = [[Rarity]] | ||
| 3/2 = [[Demon]] | |||
| 13/8 = Unnamed golden tuning | | 13/8 = Unnamed golden tuning | ||
| 13/5 = [[Cohemimabila]], unnamed golden tuning | | 13/5 = [[Cohemimabila]], unnamed golden tuning | ||