Intervals, Pitch & Frequency Page 3
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5Sounding Number
Dr. Rachel Hall
Pitch and Frequency
Converting between frequency and pitch. Let’s make a table of frequencies and pitches, using MIDI
numbers:
frequency pitch
440 · 2
4 · 12
4
27.5 Hz
21
69
440 · 2
3 · 12
3
55 Hz
33
69
440 · 2
2 · 12
2
110 Hz
45
69
440 · 2
1 · 12
1
220 Hz
57
69
440 · 2
69 + 0 · 12
0
440 Hz
69
440 · 2
69 + 1 · 12
1
880 Hz
81
440 · 2
69 + 2 · 12
2
1760 Hz
93
440 · 2
69 + 3 · 12
3
3520 Hz
105
This conversion table suggests that the frequency 440 · 2
x
Hz corresponds to MIDI pitch number
69 + 12x. In fact, this relationship is valid even if x is not a whole number. For example, let’s calculate
the frequency for pitch 60 (middle C). First, we ind the value of x. Since pitch = 60 = 69 + 12x,
3/4. Then
x =
9/12 =
frequency = 440 · 2
= 440 · 2
≈ 261.63 Hz.
x
3/4
Checking on the piano chart, we see that the frequency of middle C is indeed 261.63 Hz.
The general pitch-to-frequency conversion formula is
the frequency corresponding to pitch p is f = 440 · 2
(p 69)/12
Hz.
In order to convert from frequency to pitch, we need a way to write any given frequency f in the form
440 · 2
x
x
(that is, we need to ind x in terms of f ). Since f /440 = 2
, x is the exponent of 2 that produces
(f /440). This exponent is called the logarithm base 2 of (f /440) and written
x = log
(f /440).
2
The general frequency-to-pitch conversion formula is
the pitch corresponding to frequency f is p = 69 + 12 log
(f /440).
2
Note: most calculators don’t compute log
directly, so you have to use the fact that log
x = log x/ log 2.
2
2
Example. Approximate the pitch corresponding to 660 Hz to two decimal places.
p = 69 + 12 log
(660/440)
2
= 69 + 12 log(3/2)/ log(2) ≈ 76.02
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