o/Graphs of loudness ver-
sus frequency for the vowel “ah,”
sung as three different musical
notes. G is consonant with D,
since every overtone of G that
is close to an overtone of D
(marked “*”) is at exactly the
same frequency. G and C]are
dissonant together, since some
of the overtones of G (marked
“x”) are close to, but not right on
top of, those of C].
waves, with frequenciesfo, 2fo, 3fo,...Except for some technical
details, to be discussed below, this analysis applies to a vast range of
sound-producing systems, including the air column within the hu-
man vocal tract. Because sounds composed of this kind of pattern
of frequencies are so common, our ear-brain system has evolved so
as to perceive them as a single, fused sensation of tone.Musical applications
Many musicians claim to be able to pick out by ear several of the
frequencies 2fo, 3fo, ..., called overtones orharmonicsof the funda-
mentalfo, but they are kidding themselves. In reality, the overtone
series has two important roles in music, neither of which depends
on this fictitious ability to “hear out” the individual overtones.
First, the relative strengths of the overtones is an important
part of the personality of a sound, called its timbre (rhymes with
“amber”). The characteristic tone of the brass instruments, for ex-
ample, is a sound that starts out with a very strong harmonic series
extending up to very high frequencies, but whose higher harmonics
die down drastically as the attack changes to the sustained portion
of the note.
Second, although the ear cannot separate the individual harmon-
ics of a single musical tone, it is very sensitive to clashes between
the overtones of notes played simultaneously, i.e., in harmony. We
tend to perceive a combination of notes as being dissonant if they
have overtones that are close but not the same. Roughly speaking,
strong overtones whose frequencies differ by more than 1% and less
than 10% cause the notes to sound dissonant. It is important to re-
alize that the term “dissonance” is not a negative one in music. No
matter how long you search the radio dial, you will never hear more
than three seconds of music without at least one dissonant combi-
nation of notes. Dissonance is a necessary ingredient in the creation
of a musical cycle of tension and release. Musically knowledgeable
people do not usually use the word “dissonant” as a criticism of mu-
sic, and if they do, what they are really saying is that the dissonance
has been used in a clumsy way, or without providing any contrast
between dissonance and consonance.Standing waves
Figure p shows sinusoidal wave patterns made by shaking a rope.
I used to enjoy doing this at the bank with the pens on chains, back
in the days when people actually went to the bank. You might think
that I and the person in the photos had to practice for a long time
in order to get such nice sine waves. In fact, a sine wave is the only
shape that can create this kind of wave pattern, called astanding
wave, which simply vibrates back and forth in one place without386 Chapter 6 Waves