Page 37 - Master Handbook of Acoustics
P. 37
FIGURE 1-12 Comparison of harmonics and octaves. Harmonics are linearly related; octaves are
logarithmically related.
The interval from 100 to 200 Hz is an octave, as is the interval from 200 to 400 Hz. The interval
from 100 to 200 Hz is perceived as being larger than the interval from 200 to 300 Hz; this
demonstrates that the ear perceives intervals as ratios rather than arithmetic differences. In particular,
our perception of frequency is logarithmic. Because the octave is important in acoustical work, it is
useful to consider the mathematics of the octave.
As the ratio of 2:1 is defined as the octave, its mathematical expression is
where f =
2
frequency of the upper edge of the octave interval, Hz
f =
1
frequency of the lower edge of the octave interval, Hz
n =
number of octaves
For one octave, n = 1 and Eq. (1-4) becomes f /f = 2, which is the definition of the octave. Other
2 1
applications of Eq. (1-4) are as follows:
Example 1 The low-frequency limit of a band is 20 Hz, what is the high-frequency limit of a band
that is 10 octaves wide?
Example 2 If 446 Hz is the lower limit of a 1/3-octave band, what is the frequency of the upper
limit?
Example 3 What is the lower limit of a 1/3-octave band centered on 1,000 Hz? The f is 1,000 Hz
1
