9/8: Difference between revisions

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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, 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.

'''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 to just as well. The difference between ~~6 intervals~~ of 9/8 and the octave is the [[Pythagorean comma]].

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]].

== 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, [[Subgroup temperaments#Baldy|~~Baldy~~]].

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]].

== Notation ==

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

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

This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] 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 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, 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.
+'''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 to just as well. The difference between 6 intervals of 9/8 and the octave is the [[Pythagorean comma]].
+/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]].
 == 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, [[Subgroup temperaments#Baldy|Baldy]].
+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]].
 == Notation ==
@@ Line 30: / Line 30: @@
 * F to G: [[9/8|whole tone]]
 * G to A: [[9/8|whole tone]]
-This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] 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 ==