9/8
Ratio | 9/8 |
Factorization | 2^{-3} × 3^{2} |
Monzo | [-3 2⟩ |
Size in cents | 203.91¢ |
Names | whole tone, major second |
Color name | w2, wa 2nd |
FJS name | [math]\text{M2}[/math] |
Special properties | square superparticular, reduced, reduced harmonic |
Tenney height (log_{2} nd) | 6.16993 |
Weil height (log_{2} max(n, d)) | 6.33985 |
Wilson height (sopfr (nd)) | 12 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.19837 bits |
Comma size | large |
S-expression | S3 |
[sound info] | |
open this interval in xen-calc |
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.
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 as well. The difference between 6 intervals 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.
Temperaments
When this ratio is taken as a comma to be tempered, 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 occur 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