Diabolus in musica

The name

The page Intervals describes the tritone as the “notorious” diabolus in musica: three whole tones stacked. Because of its tense, unstable sound it was often avoided in composition in the Middle Ages; the name “diabolus in musica” is gladly ascribed to the Middle Ages, but is documented only from the 18th century. In equal temperament it divides the octave into exactly two equal halves: its frequency ratio is √2 ≈ 1.4142, which is 600 cents, and no other interval does this. The just ratio 45:32 = 1.40625 lies slightly below, at about 590 cents.

The devil becomes useful

Precisely its tension makes it indispensable. In the dominant seventh chord the major third and the minor seventh form a tritone that insistently asks for resolution into the tonic. This built-in point of tension, the page says, is the driving force of all Western tonal music. In the 20th century, according to the page, the tritone was fully rehabilitated by classical music and jazz.

Symmetrical, and so unique

The page Intervals names a second peculiarity: the inversion of the tritone is again a tritone, since 6 + 6 = 12 semitones, an octave. It is thus the only interval that is symmetrical with itself. On the guitar it corresponds to a distance of six frets; on the high E string (329.63 Hz) the 6th fret sounds B♭ (466.16 Hz).

You will not find it as a simple natural interval: the ratio 45:32 does not occur among the lower partials of the overtone series; only approximations such as 7:5 (about 583 cents) and 11:8 (about 551 cents) lie in it. It is therefore above all an interval of scale construction.