Digital Sound & Music: Concepts, Applications, & Science, Chapter 2, last updated 6/25/2013
Figure 2.4 One cycle of a sine wave
The frequency of a wave, f, is the number of cycles per unit time, customarily the
number of cycles per second. A unit that is used in speaking of sound frequency is Hertz,
defined as 1 cycle/second, and abbreviated Hz. In Figure 2.4, the time units on the x-axis are
seconds. Thus, the frequency of the wave is 6 cycles/0.0181 seconds = 331 Hz. Henceforth,
we‟ll use the abbreviation s for seconds and ms for milliseconds.
Frequency is related to pitch in human perception. A single frequency sound is perceived
as a single pitch. For example, a sound wave of 440 Hz sounds like the note A on a piano (just
above middle C). Humans hear in a frequency range of approximately 20 Hz to 20,000 Hz. The
frequency ranges of most musical instruments fall between about 50 Hz and 5000 Hz. The range
of frequencies that an individual can hear varies with age and other individual factors.
The period of a wave, T, is the time it takes for the wave to complete one cycle,
measured in s/cycle. Frequency and period have an inverse relationship, given below.
Let the frequency of a sine wave be and the period of a sine wave be . Then
Equation 2.2
The period of the wave in Figure 2.4 is about three milliseconds per cycle. A 440 Hz
wave (which has a frequency of 440 cycles/s) has a period of 1 s/440 cycles, which is about
0.00227 s/cycle. There are contexts in which it is more convenient to speak of period only in
units of time, and in these contexts the "per cycle" can be omitted as long as units are handled
consistently for a particular computation. With this in mind, a 440 Hz wave would simply be
said to have a period of 2.27 milliseconds.
The phase of a wave,
, is its offset from some specified starting
position at x = 0. The sine of 0 is 0, so the blue graph in Figure 2.5 represents a
sine function with no phase offset. However, consider a second sine wave with
exactly the same frequency and amplitude, but displaced in the positive or
negative direction on the x-axis relative to the first, as shown in Figure 2.5. The
extent to which two waves have a phase offset relative to each other can be
measured in degrees. If one sine wave is offset a full cycle from another, it has
a 360 degree offset (denoted
if it is offset a half cycle, is has a 180
offset; if it is offset a
0 0.0023 0.0045 0.0068 0.0091 0.0113 0.0136 0.0159 0.0181
time in seconds
one cycle
period of wave,
measured in s/cycle
Max Demo:
Phase and
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