9/8: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = Pythagorean whole tone, Pythagorean major second | |||
| Name = whole tone | |||
| Color name = w2, wa 2nd | | Color name = w2, wa 2nd | ||
| Sound = jid_9_8_pluck_adu_dr220.mp3 | | Sound = jid_9_8_pluck_adu_dr220.mp3 | ||
| Comma = yes | |||
}} | }} | ||
{{Wikipedia|Major second}} | |||
'''9/8''' | '''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 | 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. [[ | 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. {{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 [[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 == | |||
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. | |||
The scale is structured with the following step pattern: | |||
* A to B: [[9/8|whole tone]] | |||
* B to C: [[256/243|limma]] | |||
* C to D: [[9/8|whole tone]] | |||
* D to E: [[9/8|whole tone]] | |||
* E to F: [[256/243|limma]] | |||
* 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 occurs between the other note pairs. | |||
== See also == | == See also == | ||
* [[Gallery of | * [[16/9]] – its [[octave complement]] | ||
* [[ | * [[4/3]] – its [[fifth complement]] | ||
* [ | * [[32/27]] – its [[fourth complement]] | ||
* [[Gallery of just intervals]] | |||
* [[List of superparticular intervals]] | |||
== External links == | |||
* [http://www.tonalsoft.com/monzo/aristoxenus/aristoxenus.aspx The measurement of Aristoxenus's Divisions of the Tetrachord] on [[Tonalsoft Encyclopedia]] | |||
[[Category:Second]] | [[Category:Second]] | ||
[[Category:Whole tone]] | [[Category:Whole tone]] | ||
[[Category: | [[Category:Ancient Greek music]] | ||
[[Category: | [[Category:Commas named after their interval size]] | ||