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Chapter: 2 General physics
    Section: 2.4 Acoustics
        SubSection: 2.4.5 Musical acoustics

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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

 

Major
tone
9/8

Minor
tone
10/9

Diatonic
semitone
16/15

Major
tone
9/8

Minor
tone
10/9

Major
tone
9/8

Diatonic
semitone
16/15

 

1
Keynote

1.1250
Major
2nd

1.2500
Major
3rd

1.3333
Perfect
4th

1.5000
Perfect
5th

1.6667
Major
6th

1.8750
Major
7th

2
Octave

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 = 21/12 = 1.059 46. A tone of equal temperament = 2 semitones of ET = 21/6 = 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)

c´#

d´#

f´#

g´#

a´#

261.63

277.18

293.66

311.13

329.63

349.23

369.99

392.00

415.30

440.00

466.16 493.88

Any pitch interval may be expressed in cents, a cent being defined as the ratio 21/1200:1. 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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