Split an octave EXACTLY in half and you get the tritone — three whole tones, a frequency ratio of √2. Medieval musicians called it diabolus in musica, the devil in music, and banned it for its restless tension; jazz built whole harmonies on it. Because it’s its own mirror (half of 12 is 6), two dominant chords a tritone apart can swap — the tritone substitution. Slide around the octave to its midpoint.
The tritone spans six semitones — exactly half of the twelve-semitone octave — so its frequency ratio is 2^(6/12) = √2 ≈ 1.4142, an irrational number that no pair of small integers approximates (its restless, unresolved sound). Splitting the octave in half makes it uniquely SYMMETRIC: the tritone above a note equals the tritone below, and two tones a tritone apart map to each other, which is why two dominant seventh chords whose roots are a tritone apart share the same tritone of guide-tones and can substitute for one another (the jazz ‘tritone sub’). A fail-loud self-check throws unless six semitones give √2 and two tritones make an octave. ◆ real music theory, node-verified.
Equal-tempered tritone = √2 exactly (in just intonation the ear splits it into 45:32 or 7:5, slightly different); the half-octave symmetry and tritone substitution are exact in 12-tone equal temperament.