9/8: Difference between revisions

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Two 9/8's stacked produce [[81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10/9]] yields [[5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is [[81/80]], the syntonic comma of about 21.5¢), such as [[19edo]], [[26edo]], and [[31edo]], are called [[meantone]] temperaments.

9/8 is well-represented in [[6edo]] and its multiples. [[Edo]]s which tune [[3/2]] close to just ([[29edo]], [[41edo]], [[53edo]], to name three) will tune 9/8 close to just as well. The difference between six ~~instances~~ of 9/8 ~~and~~ the octave is the [[Pythagorean comma]].

A stack of six intervals of 9/8 exceeds the octave by the [[Pythagorean comma]].

== History ==

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== Temperaments ==

~~When this ratio~~ is ~~taken as a comma to be~~ [[~~tempering out|tempered out~~]], ~~it produces~~ [[~~Very low accuracy temperaments #Antitonic|antitonic~~]] ~~temperament~~. ~~Edos that temper~~ it ~~out include~~ [[~~2edo~~]] ~~and~~ [[~~4edo~~]]. If it ~~is instead used as~~ a generator, it ~~produces, among others,~~ [[Subgroup temperaments #Baldy|baldy]].

In [[meantone]], 9/8 is equated with [[10/9]], so that two instances of 9/8~10/9 stack to ~[[5/4]]. [[Superpyth]] instead sharpens 9/8 to equate it with [[8/7]].

Since 9/8 is reached by stacking two instances of [[3/2]], temperaments in subgroups that include 3 cannot be generated by ~9/8. However, it can be a generator in subgroups such as [[2.9.5.7 subgroup|2.9.5.7]], where it generates [[Subgroup temperaments #Baldy|baldy]] for example.

== Approximation ==

{{~~Interval_Edo_Approximation~~ | 9/8}

9/8 is well-represented in [[6edo]] and its multiples, though only multiples of [[12edo]] (up to [[300edo]]) map 9/8 to 1\6 by [[patent val]]. [[Edo]]s which tune [[3/2]] close to just, such as [[29edo]], [[41edo]], and [[53edo]], will tune 9/8 close to just as well.

== Notation ==

In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between ~~A and B~~, ~~between C and D~~, ~~between D and E~~, ~~between F and G~~, ~~as well as between G~~ and A.

In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between the notes A–B, C–D, D–E, F–G, and G–A.

The scale is structured with the following step pattern:

@@ Line 11: / Line 11: @@
 Two 9/8's stacked produce [[81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10/9]] yields [[5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is [[81/80]], the syntonic comma of about 21.5¢), such as [[19edo]], [[26edo]], and [[31edo]], are called [[meantone]] temperaments.
-/8 is well-represented in [[6edo]] and its multiples. [[Edo]]s which tune [[3/2]] close to just ([[29edo]], [[41edo]], [[53edo]], to name three) will tune 9/8 close to just as well. The difference between six instances of 9/8 and the octave is the [[Pythagorean comma]].
+A stack of six intervals of 9/8 exceeds the octave by the [[Pythagorean comma]].
 == History ==
@@ Line 17: / Line 17: @@
 == Temperaments ==
-When this ratio is taken as a comma to be [[tempering out|tempered out]], it produces [[Very low accuracy temperaments #Antitonic|antitonic]] temperament. Edos that temper it out include [[2edo]] and [[4edo]]. If it is instead used as a generator, it produces, among others, [[Subgroup temperaments #Baldy|baldy]].
+In [[meantone]], 9/8 is equated with [[10/9]], so that two instances of 9/8~10/9 stack to ~[[5/4]]. [[Superpyth]] instead sharpens 9/8 to equate it with [[8/7]].
+Since 9/8 is reached by stacking two instances of [[3/2]], temperaments in subgroups that include 3 cannot be generated by ~9/8. However, it can be a generator in subgroups such as [[2.9.5.7 subgroup|2.9.5.7]], where it generates [[Subgroup temperaments #Baldy|baldy]] for example.
 == Approximation ==
- {{Interval_Edo_Approximation | 9/8}
+/8 is well-represented in [[6edo]] and its multiples, though only multiples of [[12edo]] (up to [[300edo]]) map 9/8 to 1\6 by [[patent val]]. [[Edo]]s which tune [[3/2]] close to just, such as [[29edo]], [[41edo]], and [[53edo]], will tune 9/8 close to just as well.
+{{Interval edo approximation|9/8}}
 == Notation ==
-In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between A and B, between C and D, between D and E, between F and G, as well as between G and A.
+In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between the notes A–B, C–D, D–E, F–G, and G–A.
 The scale is structured with the following step pattern: