Spectra and Harmonics
The way you're probably most familiar with spectral modification is by modifying a sound source using a tone control, or a graphic equalizer. These modify the spectral balance of an input sound by selectively emphasizing some frequency components and de-emphasizing others. Figure 3.1 shows a typical graphic equalizer setting from a software editing package.
FIGURE 3.1. The
infamous "disco curve": lots of bass and some treble emphasis.
FIGURE 3.2. Producing
different harmonics on a guitar string by stopping the string at points
corresponding to simple rations (left); the equivalent musical pitches
that make up the overtone series (right). Click on each string or the
note to hear the pitch.
FIGURE 3.3. A square wave (in red) superimposed over sine waves that represent its first five odd-numbered partials (250, 750, 1250, 1750, and 2250 Hz). An actual square wave would be composed of successive odd-numbered harmonics up to the highest frequency capable of being produced by the audio system (in a digital system, almost half the sampling rate).
A sawtooth wave contains both even and odd harmonics; like the square
wave, each harmonic’s amplitude is the reciprocal of the harmonic
number. Figure 3.4 shows two ways of describing a sawtooth wave. In the
inset, the time display of the sawtooth waveform is shown as a function
of intensity. Figure 3.4 also shows a graph of the relative intensity
of each of the first five harmonics.
FIGURE 3.4 Sawtooth wave. Inset: time display with the x axis is used for indicating time. Below, the frequency of each harmonic, instead of time, is shown on the x axis.
If we decompose the sawtooth wave into sine wave components, you’ll
end up with a graph something like that shown in Figure 3.5:
FIGURE 3.5. Sine wave decomposition of the first 6 partials of a sawtooth wave.
In Figure 3.6, we add each of the partials of the sawtooth wave progressively. Notice how the waveform looks more and more like the sawtooth shown in Figure 3.4, above. We would need to add together many more partials to get the perfect-looking sawtooth shape.
FIGURE 3.6. Progressive
addition of the partials of a sawtooth wave. Click each waveform to hear
Normally, the bow is played against the string so that its hairs activate
the natural harmonics of the violin string to a significant intensity
relative to inharmonic partials. In the second example the violin was
played col legno, where the wood part of
the bow instead of the hair is rubbed against the string. This causes
the intensity of the non-harmonic components to be relatively more intense.
FIGURE 3.7. A waveform plot of the spoken word “shut.” A noise-like (aperiodic) portion for the “sh” sound precedes the more pitched “u” sound, while the “t” is transient.
For certain types of noise it is easiest to describe its frequency in
a statistical manner. Figure 3.8 shows a plot of white
noise, a non-periodic waveform of the type discussed earlier in
Chapter 1. White noise can be thought of as
the complete opposite of a sine wave: a sine wave has a single deterministic
frequency with a predictable intensity, while white noise has no deterministic
frequency with random amplitudes. It has a "flat" spectrum,
meaning that in contains all frequency components at equal intensity.
Click here to listen to a second of white noise.
Click here to listen to a second of pink noise.
FIGURE 3.8. White noise.
Amplitude Modulation (AM) and Amplitude Envelopes
FIGURE 3.9. The ADSR (Attack-Decay-Sustain-Release) amplitude envelope, a simplified way of describing the overall intensity of a complex sound over time. It is commonly used on sound synthesizers and samplers to describe what occurs with a single keystroke, or digital “note on” command. Specifically, the ADS portion of the sound is what happens when a key is pushed down (note on), and the R is what occurs when the key is let up (note off).
Click here to listen to the sound of a violin being plucked with the fingers (a pizzicato note). The waveform shown in Figure 3.10. Notice the amplitude envelope has greatest intensity when the string is initially plucked, how the intensity is reduced considerably after this point, and then how the sound dies away steadily as the intensity of the vibration of the string (and its resonance within the body o the violin) diminishes. Note also how the waveform looks noisy at first, and then how periodic frequency can be detected later in time.
FIGURE 3.10. A
waveform of a plucked violin string, along with its amplitude envelope.
FIGURE 3.11. The amplitude envelope of a zipper being open.
Figure 3.11 shows the amplitude envelope of a zipper being pulled open.
Click here to listen to its sound. Note
the envelope is loudest when pulling harder on the zipper, and then dies
down quickly once the zipper is going smoothly. The amplitude envelope
is irregular since the resistance of pulling open a zipper will also be
Click here to listen to the unaltered sound.
Click here to listen to only the attack portion of the harmonic. This is where the “bite” of the sound is produced in the excitation of the string, and is characteristic of a guitar attack.
Click here to listen to only the decay and sustain portion of the harmonic. Amazingly, the sound has lost any characteristic of the guitar; it has the rich content of partials and the decay of the harmonic; but without the attack, the sound is almost like a French Horn.
The amplitude envelope is a form of amplitude modulation.
If you multiply the output of a digital device by a time-varying function,
that would be a form of amplitude modulation. Similarly if you wiggle
the volume control on an amplifier back and forth, you've got hand-operated
amplitude modulation. Amplitude modulation means “varying intensity
over time.” Figure 3.13 shows the amplitude modulation of a sine
wave by two cycles of a triangle wave, shown below in Figure 3.12. The
triangle wave oscillates at a much slower frequency than the sine wave;
while the sine wave is within the audio frequency range, the triangle
wave used here has a frequency of about 0.5 Hz, well below the lowest
frequency of human hearing. We hear the effect of the triangle
wave, demonstrated by the up-and-down ramping of the sine wave's
FIGURE 3.12. The
triangle wave used for amplitude modulation in Figure 3.13.
FIGURE 3.13. The
amplitude modulation of a 100 Hz sine wave by a 10 Hz triangle wave as
shown in Figure 3.12 results in a constant, synthetic amplitude envelope.
FIGURE 3.14. Frequency modulation. The plus signs indicate when the modulation is maximal and the minus signs when the modulation is minimal, corresponding to the peaks of the modulating sine wave.
Low-frequency modulation is usually termed vibrato. It is a variation in the frequency of a pitch above and below a fundamental pitch, usually, no more than within a musical half-step. It is a feature that most instrumentalists include in their music, if it is possible to create the effect. For instance, a singer almost always uses vibrato. There are different styles of vibrato; a “wide” vibrato can sound “schmaltzy” or overly romantic, while music of the baroque era (from 1600—1750; e.g., J. S. Bach) was performed in its day with very little vibrato. Every performer uses vibrato a little bit differently, which contributes to the unique character of an individual performance.
For instance, click here to hear a violin tone played without vibrato (indicated as “non vibrato” in a music score); then
click hereto hear a violin tone played as we usually expect to hear it. The second example has a moderate amount of vibrato, and either no special indication or the word “normale” would be indicated in a musical score. It is interesting that in spite of the variation in frequency, we associate pitch with the center frequency of the modulation.
Figures 3.15—16 show the relationship between a modulating wave
(the modulator) and the wave affected by it (the carrier),
for both frequency and amplitude modulation. Two parameters of the modulating
wave are relevant: the frequency of the modulating waveform, and the intensity
of the waveform (sometimes referred to as modulation depth). In Figure
3.15, the modulation is applied to the frequency of the carrier, while
in Figure 3.16, the modulation is applied to the intensity. Figures 3.15-3.16
use a sine wave carrier and modulator.
FIGURE 3.15. The
effect of frequency modulation (FM) of a carrier oscillator.
F is the frequency control input of the oscillator; G is the gain of the
FIGURE 3.16. The effect of amplitude modulation (AM) of a carrier oscillator. F is the frequency control input of the oscillator; G is the gain of the oscillator.
FIGURE 3.17. Two 250-Hz sine waves, A and B. Wave B is delayed by a quarter of a cycle (90��), i.e., by .001 seconds (or 1 millisecond).
When listening to a periodic waveform, it’s impossible to distinguish between a version with all of the harmonics “in-phase” and one with some of the harmonics completely “out-of-phase.” Figure 3.18 shows two waveforms consisting of the first six harmonics of a triangle wave. The blue and red waveforms are identical except that every other harmonic is out-of-phase in
FIGURE 3.18. Triangle wave with in-phase (blue) and out-of-phase (red) harmonics.
the red version. Although the two waveforms in Figure 3.18 look different,
they sound identical: click hereto listen
to the in-phase version, and click here
to listen to the out-of phase version. This is a simple proof that high-end
audio system guaranteed to have “linear phase” are more smoke
than fire. The absolute phase of individual
harmonic components is inaudible to the ear.
FIGURE 3.19. Constructive and destructive interference. The blue waveform is an in-phase sine wave. By adding this waveform to an in-phase copy of itself, the green waveform with twice the intensity would result (constructive interference). But if the blue waveform were added to a 180?degree, out-of-phase copy of itself (the red waveform), destructive interference results (represented by the black waveform with 0 intensity).
FIGURE 3.20 See text below. You can also click on each part of the Figure to listen to the waveform.
The following example shows an extreme case of how destructive interference
occurs when signals are out-of-phase. Click here to listen to the lower waveform on the left of Figure 3.20. This
is an in-phase triangle wave. Click here
to listen to the upper waveform at the left of Figure 3.20. This is the
same triangle wave, 180 degrees out-of-phase, with a steadily increasing
amplitude envelope; eventually it reaches the same intensity as the lower
waveform. At the right side of Figure 3.20 is the result of adding the
two waveforms on the left; click here
to listen to the result. You can also click on each of the waveforms in
the figure as well. In this example, the destructive interference increases
as a function of the intensity of the out-of-phase waveform.
The in-phase triangle wave should create a stable image localized in-between
the speakers; the out-of-phase version should sound split between two
locations in the speakers, and sound less loud in one speaker. If the
opposite occurs, your speakers are wired out-of-phase; reverse the leads
on one of your speakers, if you know how. In Chapter
8 there are additional tests for determining loudspeaker phase and
details on how to reverse the phase of your loudspeakers.