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2.4.5 Musical acoustics

Standard musical pitch is based on a frequency of 440 Hz for A in the treble stave (a′) (see ISO 16: 1975). Wind instruments should be constructed to conform to this frequency at 20 °C room temperature after being blown long enough to reach thermal equilibrium. Tuning C for keyboard instruments (c″) is a minor third of equal temperament above 440 Hz, namely 523.25 Hz.

Diatonic major scale. The frequency ratios of musical intervals in the diatonic major scale are shown below; the upper row shows the ratio between adjacent steps, the lower row the ratio to the keynote.

Diatonic major scale

The interval between a major and a minor tone, 81/80, is called a comma.
   A chromatic semitone = 135/128 or 25/24 according to whether it divides a major or minor tone.
   A minor interval is a major interval diminished by a chromatic semitone. An augmented interval is a major interval increased by a chromatic semitone. A diminished interval is a perfect or minor interval diminished by a chromatic semitone.

Equally tempered major scale. A semitone of equal temperament = 2^1/12 = 1.059 46. A tone of equal temperament = 2 semitones of ET = 2^1/6 = 1.122 46. The table below gives the frequencies (Hz) of musical notes in equal temperament for the octave starting at c'.

Frequencies of notes in equal temperament (Hz)

Any pitch interval may be expressed in cents, a cent being defined as the ratio 2^1/1200:1. Thus there are 100 cents to the ET semitone (see Wood and Bowsher, The Physics of Music, 1975).

References

ISO 16: 1975 Acoustics–standard tuning frequency (standard musical pitch).
A. B. Wood, revised J. M. Bowsher (1975) The Physics of Music, 7th edn, Chapman & Hall, London.

D.R.Bacon

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