Intervals, Pitch & Frequency

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Sounding Number
Dr. Rachel Hall
Intervals and Pitch
A musical interval is a relationship between two frequencies. The most fundamental musical intervals
are the octave, which corresponds to a 2:1 frequency ratio, and the perfect ifth, which corresponds to a
3:2 frequency ratio. Suppose we start with a frequency f . Then
(1/2)f = an octave below f
(2/3)f = a ifth below f
2f = an octave above f
(3/2)f = a ifth above f
4f = two octaves above f
(9/4)f = two ifths above f
8f = three octaves above f
(27/8)f = three ifths above f
1
2
3
1
Notice that 2 = 2
, 4 = 2
, 8 = 2
, and (1/2) = 2
. So we have the formulas:
n
n
f = n octaves above f
f = n ifths above f
2
(3/2)
This formula makes sense for any whole number n if we interpret a negative value of n to mean “octaves
below.”¹
So ( 3) octaves above f is the same as 3 octaves below f , and the frequency is
3
3
)f = (1/8)f .
2
f = (1/2
We can combine octaves and ifths to form new intervals. For example, if two frequencies are in a 3:1
ratio, then the interval between them is an octave plus a ifth. Let’s take this step-by-step: the frequency
an octave above f is 2f ; an ifth above 2f is (3/2)(2f ) = 3f . Notice that it doesn’t matter whether we go
up a ifth, then an octave, or vice versa. This gives the formulas
n
m
The frequency n octaves plus m ifths above f is 2
f .
(3/2)
n
m
The frequency n octaves plus m ifths below f is 2
f .
(3/2)
Negative values of n or m are interpreted to mean octaves or ifths below.
Examples.
1. The frequency two octaves plus one ifth above 900 Hz is found by setting n = 2 and m = 1, so the
2
1
answer is 2
900 = 5400 Hz.
(3/2)
2. The frequency two octaves plus one ifth below 900 Hz is found by setting n =
2 and m =
1, so
2
1
the answer is 2
900 = (1/4)(2/3)900 = 150 Hz.
(3/2)
3. The frequency two octaves minus three ifths above 900 Hz is found by setting n = 2 and m =
3,
2
2
3
3
so the answer is 2
Hz.
(3/2)
900 = (4)(2/3)
900 = 1066
3
4. The frequency two ifths minus one octave above 900 Hz is found by setting n =
1 and m = 2, so
1
2
2
1
the answer is 2
Hz.
(3/2)
900 = (1/2)(3/2)
900 = 1012
2
1
n
0
¹Negative exponents indicate division: a
. A zero exponent results in a value of 1; that is, a
=
= 1.
n
a

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