# 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 $2^{1/12} : 1$.

## 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.