A sine wave has another important characteristic besides frequency and amplitude : It has phase. The phase changes from moment to moment as the waveform moves up and down. If you’d like to know more about phase, consult a good book on trigonometry. Without getting too technical, we can say that a single cycle of a waveform goes through 360 degrees of phase, as shown in Figure 2-3.
KICK ME: Although the ear is not good at detecting the phase of individual sine waves, I’ve read that some listeners can distinguish the polarity of extremely low-pitched, percussive sounds — kick drum samples, in other words. When a drummer hits the kick drum head with the beater, the head begins its vibration by moving toward the listener, so the waveform begins with a peak. If the polarity of the sample is flipped, the speaker cone begins the kick sound by moving away from the listener, which results in a dip rather than a peak. If a kick sample feels hollow or unsatisfying to you even though it has plenty of bottom, try reversing its polarity.
When two sine waves are added together — mixed, to use the musical term — the result depends on their phase relationship. If their phase is the same, they’re said to be “in phase.” Basically, two waves are in phase when their peaks occur at the same moment in time. If the peaks of one wave occur at the same moment as the dips in the other wave, they’re said to be “out of phase.” These relationships are shown in Figure 2-4.
The human ear is not good at distinguishing the phase of individual sine waves. Because of this, strange as it may seem, two waveforms that appear quite different on a computer screen can sound identical, if their partials all have the same frequencies and amplitudes. But when two tones are sounding at once, their phase relationship can become not only perceptible but a crucial element in sound design. This is because of the phenomenon called phase cancellation. If two sine waves have the same amplitude but are 180 degrees out of phase, when they’re mixed the result is silence. If they’re in phase, however, mixing will produce a louder sound.
This is true whether or not the sine waves have the same frequency. If their frequencies are far apart, it doesn’t make much sense to talk about their relative phase, because the phase relationship will change quite rapidly. If their frequencies are close but not identical, however, the phase relationship will change slowly. At this point, our ears will perceive the changes in phase as beating. As two sine waves of equal amplitude and slightly different frequencies are mixed, the amplitude of the mixed signal will fluctuate between zero and twice the amplitude of the individual waves. This effect is shown in Figure 2-5. The frequency of the fluctuations in amplitude will equal the difference between the frequencies of the two sine waves. For instance, if one sine wave has a frequency of 250Hz and the other a frequency of 252Hz, when they’re mixed they’ll beat at a frequency of 2Hz.
Figure 2-4. When two sine waves that have the same frequency and amplitude are added together (mixed), the result depends on their relative phase. If they’re in phase (left), the peaks and dips in the wave reinforce one another, so the result of mixing is a wave with twice the amplitude of either source wave. If they’re 180 degrees out of phase, however (right), the peaks and dips cancel out, and the result is silence.
If the beating is below 20Hz, as in this example, it will be perceived as a rhythm or pulse within the mixed sound. If it’s above 20Hz, it’s perceived as a new frequency component, which is called a difference tone. For instance, if we mix a 500Hz sine wave with a 600Hz sine wave, the mix will include a difference tone at 100Hz.
The musical result of beating depends very largely on the context. If you mix two bass synth sounds that are slightly out of tune with one another, phase cancellation will cause some of the bass notes to be much lower in volume than others. This is most likely a bad thing. But in a higher frequency range, detuning two oscillators to produce beating is a common and useful technique, as discussed in Chapter Four.
Figure 2-5. If two sine waves have slightly different frequencies, they will move in and out of phase.
When they’re mixed, the result will be a series of peaks and dips. The frequency of the peaks and dips will be the difference between the two waves. This phenomenon is called beating.
Up to now, we’ve been discussing beating between sine waves. Most sounds, however, are not simple sine waves. When each sound has a number of partials (all of which are sine waves, as discussed above), beating will occur between each pair of partials. To give a simple example, let’s suppose that one sound has partials at 100Hz, 200Hz, 300Hz, and 400Hz, while the other has partials at 102Hz, 204Hz, 306Hz, and 408Hz. The mix will contain a number of difference tones, as Table 2-1 shows.
Table 2-1. Difference tones produced by two tones each of which comprises four partials.
The amplitudes of the difference tones will depend on the relative amplitudes of the various partials.
Most of them will be soft enough that you won’t perceive them directly. What you’ll perceive is a kind of spreading of the pitch. In addition to the clear sine waves at 100Hz and 102Hz, you’ll also hear quieter sine waves at 94, 96, 98, 104, 106, and 108Hz. And with complex tones like these, it’s unlikely that all of the partials will be 180 degrees out of phase at the same time. Thus, even though phase cancellation is causing individual partials to fade in and out, the tone as a whole won’t fade in and out. Instead, you’ll hear a kind of animation in the tone that can be quite pleasing.
Phase vs. Polarity. One way to put a sine wave 180 degrees out of phase is to flip it upside down, so that the peaks in the waveform become dips and vice-versa.
This is easy to do electrically. In fact, if you look at a good mixer you’ll probably find switches labelled
“phase,” which do exactly that to the signals passing through mixer channels. Technically, however, such a switch doesn’t reverse the phase of the signal; it reverses its polarity. This distinction may or may not matter. In effect, the switch is changing the phase of all of the partials at the same time. But if you like nice academic distinctions, you might want to know that phase and polarity are not the same thing.