9/8: Difference between revisions

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{{Infobox Interval

| Name = whole tone, major second

| Name = Pythagorean whole tone, Pythagorean major second

| Color name = w2, wa 2nd

| Sound = jid_9_8_pluck_adu_dr220.mp3

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'''9/8''' is the ~~Pythagorean~~ '''whole tone''' or '''major second''', measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context.

'''9/8''', the '''Pythagorean whole tone''' or '''major second''', is an interval measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context, though because of its relatively close proximity to the [[unison]], it is the largest [[superparticular]] interval known to cause crowding, which lends more to it being considered a type of dissonance- at least in historical Western Classical traditions and in the xenharmonic traditions derived from them.

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 as well. The difference between 6~~ intervals 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 ==

The (whole) tone as an interval measure was already known in Ancient Greece. ~~[[Wikipedia:Aristoxenus~~|Aristoxenus (fl. 335 BC)]] defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size.

The (whole) tone as an interval measure was already known in Ancient Greece. {{w|Aristoxenus}} (fl. 335 BC) defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size, which implies [[12edo]].

== Temperaments ==

~~When this ratio~~ is ~~taken as a comma~~ to be ~~tempered~~, 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~~, [[~~Chromatic_pairs#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 ==

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 ~~use~~ the ~~cycle~~ of fifths ~~and fourths along with seven note names~~, such as the [[ups and downs notation]], the whole tone is represented by the distances between ~~A and B~~, ~~C and D~~, ~~D and E~~, ~~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:

A to B: [[9/8|whole tone]]

* A to B: [[9/8|whole tone]]

B to C: [[256/243|limma]]

* B to C: [[256/243|limma]]

C to D: [[9/8|whole tone]]

* C to D: [[9/8|whole tone]]

D to E: [[9/8|whole tone]]

* D to E: [[9/8|whole tone]]

E to F: [[256/243|limma]]

* E to F: [[256/243|limma]]

F to G: [[9/8|whole tone]]

* F to G: [[9/8|whole tone]]

G to A: [[9/8|whole tone]]

* G to A: [[9/8|whole tone]]

This pattern highlights the placement of the whole tone intervals between ~~A and B, C and D, D and E, F and G, and G and A~~, distinguishing them from the [[~~whole tone~~]] that ~~occur~~ between the other note pairs.

This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] that occurs between the other note pairs.

== See also ==

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[[Category:Whole tone]]

[[Category:Ancient Greek music]]

[[Category:Commas named after their interval size]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = whole tone, major second
+| Name = Pythagorean whole tone, Pythagorean major second
 | Color name = w2, wa 2nd
 | Sound = jid_9_8_pluck_adu_dr220.mp3
@@ Line 7: / Line 7: @@
 {{Wikipedia|Major second}}
-'''9/8''' is the Pythagorean '''whole tone''' or '''major second''', measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context.
+'''9/8''', the '''Pythagorean whole tone''' or '''major second''', is an interval measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context, though because of its relatively close proximity to the [[unison]], it is the largest [[superparticular]] interval known to cause crowding, which lends more to it being considered a type of dissonance- at least in historical Western Classical traditions and in the xenharmonic traditions derived from them.
 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 as well. The difference between 6 intervals 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 ==
-The (whole) tone as an interval measure was already known in Ancient Greece. [[Wikipedia:Aristoxenus|Aristoxenus (fl. 335 BC)]] defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size.
+The (whole) tone as an interval measure was already known in Ancient Greece. {{w|Aristoxenus}} (fl. 335 BC) defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size, which implies [[12edo]].
 == Temperaments ==
-When this ratio is taken as a comma to be tempered, 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, [[Chromatic_pairs#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 ==
+/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 use the cycle of fifths and fourths along with seven note names, such as the [[ups and downs notation]], the whole tone is represented by the distances between A and B, C and D, D and E, 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:
-    A to B: [[9/8|whole tone]]
+* A to B: [[9/8|whole tone]]
-    B to C: [[256/243|limma]]
+* B to C: [[256/243|limma]]
-    C to D: [[9/8|whole tone]]
+* C to D: [[9/8|whole tone]]
-    D to E: [[9/8|whole tone]]
+* D to E: [[9/8|whole tone]]
-    E to F: [[256/243|limma]]
+* E to F: [[256/243|limma]]
-    F to G: [[9/8|whole tone]]
+* F to G: [[9/8|whole tone]]
-    G to A: [[9/8|whole tone]]
+* G to A: [[9/8|whole tone]]
-This pattern highlights the placement of the whole tone intervals between A and B, C and D, D and E, F and G, and G and A, distinguishing them from the [[whole tone]] that occur between the other note pairs.
+This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] that occurs between the other note pairs.
 == See also ==
@@ Line 45: / Line 52: @@
 [[Category:Whole tone]]
 [[Category:Ancient Greek music]]
+[[Category:Commas named after their interval size]]