Harmonics
Suppose
that we have two tones, one with a fundamental frequency of f1 Hz and
the other with a fundamental frequency of f2 Hz.
The harmonics of the first tone will fall at the frequencies 1f1,
2f1, 3f1, . . . ,
while the harmonics of
the second tone will fall at frequencies lf2, 2f2, 3f2,
and so on. Obviously, two tones with exactly the same pitch (f1 = f2)
will sound harmonious, because the frequencies of all harmonics of the first
tone will exactly match the frequencies
of all harmonics from the second tone precisely. The harmonics of the two tones need not have the same strengths.
Furthermore
it is obvious that if the fundamental frequencies of the two tones are related
by an octave (f1 = 2f2
or f2 = 2f1), all the harmonics of the upper pitch
will be identical to the even-numbered harmonics of the lower tone.
In fact two tones separated by any number of octaves (f1 = 2n
f2), where n is an integer, will line up in this way, because all
harmonics of the upper tone will invariably coincide precisely with harmonics of
the lower tone. For this reason,
different frequencies are considered to have
the same pitch, or, more precisely, two pitches are said to belong to the
same pitch-class if they are separated only by an arbitrary number of octaves.
Finding
harmonious combinations of octaves is therefore no problem, which is fortunate
because the typically strong second harmonic of any complex tone falls one
octave higher than the fundamental frequency.
Furthermore
the second and third harmonics, however, are also generally quite strong in
amplitude and have a pitch that is not the same as that of the fundamental. The
presence of frequency components near but not equal to these harmonics would be
very likely to cause strong beating or roughness interaction and therefore a
perceived sensation of dissonance. The frequency of the third harmonic is 3f Hz.
The frequency of the second harmonic is 2f Hz.
The ratio between the third and the second harmonics is 3f/2f , or simply
3/2. The musical name for this
interval is the perfect fifth.
If
we sound two complex tones together,
one with a fundamental of f1 Hz, the other with a fundamental
frequency of f2 = 3f1 Hz.
Clearly all the harmonics of the second tone again fall precisely on top
of harmonics of the first tone just as in the case of the octave.
This
is easy to understand if we allow the fundamental frequency f1 to be
l Hz to simplify the mathematics (even though this would be below our hearing
range). The harmonics of 1 are 1,
2, 3, . . ., while the harmonics of 3 are 3, 6, 9, . . ., clearly the harmonics
of 3, are also harmonics of 1. To generalize this principle even further
harmonics of harmonics are also harmonics. Thus if we choose two pitches so that
the fundamental of one is a harmonic of the other, we are guaranteed perfect
agreement, or musical consonance.
If,
however, the two fundamental frequencies are related as 3 is related to 2 and 3
is clearly not a harmonic of 2. Since every other harmonic of 3 is even then
every other harmonic of 3 is a harmonic of 2.
How the two fundamentals interact will depend on their absolute
frequencies. Somewhere around 100 to 200 Hz and below the two fundamental
frequencies will fall within the same critical band, causing interaction to
occur.
Higher-numbered
harmonic pairs from the two tones
will tend to be separated by more than a critical band, however, until fairly
high numbered harmonics are reached. By the time this happens, the harmonics
will have become fairly weak in amplitude.
Even
more important is the fact that the combination of two tones a perfect fifth
(3/2 frequency ratio) apart will result in a set of frequency components all of
which are harmonics of the
frequency one
octave below
the lower tone.
Once
again the harmonics of 2 are 2, 4, 6, 8, . . ., while the harmonics of 3 are 3,
6, 9, 12, . . . The combination of both tones together (assuming that they
are perfectly tuned) results in frequency components at 2, 3, 4, 6, 8, 9, 10,
12, . . ., all of which are harmonics of 1.
Thus
the combination sounds as harmonious as a single
tone with a fundamental frequency of
1 that is missing
a few of its harmonics,
including the fundamental!
Another
psychophysical phenomenon, that of residue pitch, makes the combination tend to
sound as if it has a single pitch associated with the "implied"
fundamental, which lies one octave lower than the lower tone.
Experiments
have shown that the brain interpolates the harmonic components of a composite
musical sound and replaces the fundamental pitch. This phenomenon is used by
organ manufacturers to improve the
bass frequencies.
TOM SCARFF
1 Martello Court
Portmarnock
Dublin
Ireland.
Email: tscarff@eircom.net
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