Definition:Musical Interval
Definition
A musical interval is a ratio of frequencies of musical notes which is recognised by convention as being particularly aesthetically pleasing or meaningful.
Octave
An octave is a musical interval whose components have frequencies in the ratio $2 : 1$.
Fifth
A fifth is a musical interval whose components have frequencies in the ratio $3 : 2$.
Fourth
A fourth is a musical interval whose components have frequencies in the ratio $4 : 3$.
Tone
A tone is a musical interval whose components have frequencies in the ratio $9 : 8$.
Semitone
A semitone is a musical interval whose components have frequencies in the ratio $2^{1/12} : 1$.
Also see
Historical Note
The Pythagoreans noticed that aesthetically pleasing musical intervals were obtained by sounding notes from strings whose lengths were in ratios of small natural numbers: $1 : 2$, $2 : 3$, $3 : 4$ and so on.
This may have been one of the inspirations behind their mystical appreciation of the numerical nature of the Universe.