9/8

Interval information

Ratio	9/8
Factorization	2-3 × 32
Monzo	[-3 2⟩
Size in cents	203.91¢
Names	Pythagorean whole tone, Pythagorean major second
Color name	w2, wa 2nd
FJS name	[math]\displaystyle{ \text{M2} }[/math]
Special properties	square superparticular, reduced, reduced harmonic
Tenney height (log2 nd)	6.16993
Weil height (log2 max(n, d))	6.33985
Wilson height (sopfr(nd))	12
Comma size	large
S-expression	S3
[sound info]
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Major second

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. Edos 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. Aristoxenus (fl. 335 BC) defined the tone as the difference between the just fifth (3/2) and the 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 out, it produces antitonic temperament. Edos that temper it out include 2edo and 4edo. If it is instead used as a generator, it produces, among others, baldy.

Notation

In musical notations that employ the diatonic 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: whole tone
• B to C: limma
• C to D: whole tone
• D to E: whole tone
• E to F: limma
• F to G: whole tone
• G to A: whole tone

This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the limma that occurs between the other note pairs.

See also

• 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

• The measurement of Aristoxenus's Divisions of the Tetrachord on Tonalsoft Encyclopedia