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Unit 10

Harmonious Math

10.3 Sound and Waves

SOMETHING IN THE AIR

As we have seen, the Greeks recognized connections between harmonic intervals and rational numbers. As it turns out, they also had a rudimentary understanding of the most basic musical concept of all…sound. Thinkers such as Aristotle suspected that sound was some sort of "disturbance" that is propagated through the air.

waves

We are all familiar with waves of one sort or another. You may have seen them at the beach, or felt them in an earthquake, or heard about them, perhaps when someone has spoken of "airwaves" in relation to TV or radio broadcasts. Each of these waves is different, but they all share some unifying characteristics. Let's look at some of the characteristics of ideal, simple waves, waves that we will later use as "atoms" to construct more-realistic, complicated waves.

Imagine the smooth surface of a pond on a still day. If you throw a pebble into the pond, you will see ripples emanating from the point at which the pebble enters the water. These ripples consist of areas where the surface of the water is heightened, called crests, followed by areas that are depressed, called troughs. A cross-section of a few of these ripples might look like this:

waves

Notice that both the crests and the troughs reach equally above and below, respectively, the still surface line. This shows us that waves travel by some sort of displacement in a medium. The amount of displacement, as measured from the still surface line, is called a wave's amplitude.

To be precise, a rock hitting a pond creates an impulse, a temporary disturbance. Over time, the effects of the disturbance dissipate and the surface of the pond becomes smooth again. To explore the concept of waves further, let's instead imagine some sort of regular disturbance, or ongoing pulsation, such as a child slapping the surface of the water in a rhythmic fashion.

child slapping water

If the child's mother is fishing in the same pond, her bob will move up and down with the crests and troughs of the ripples. The bob will not move horizontally, only vertically. This is an important point concerning waves: the medium through which a wave travels has no net movement when the wave passes through it. That is, there is no net horizontal displacement.

The bob stays in the same place as a wave passes under it.

To carry this concept from our pond example back to sound waves traveling through the air, this means that the air molecules that transmit the disturbance that we interpret as sound do not, on average, travel any net distance. For instance, a loudspeaker does not push a stream of air towards me. Rather, it compresses air molecules to form a region of high pressure that travels away from the source. Assuming that the air is of uniform density and pressure to begin with, this region of high pressure will be balanced by a region of low pressure, called rarefaction, immediately following the compression. Remember, air molecules do move forth and back, but after the wave has passed, they are, on average, in the same place they were before the wave came through.

waves

As these groups of molecules alternately experience compressions and rarefactions, a pulse is created, and this is what "reaches" our ears. Whether or not we hear the waves as sound has everything to do with their frequency, or how many times every second the molecules switch from compression to rarefaction and back to compression again, and their intensity, or how much the air is compressed.

In our graph above, the vertical axis represents air pressure and the horizontal axis represents time. The crests correspond to times of high pressure, (compression) and the troughs represent times of low pressure (rarefaction). The height of a crest corresponds to the degree of compression of the air, which, when measured from the baseline, is another way to think about amplitude. We perceive amplitude as a sound wave's loudness.

To determine the frequency of the wave from our graph, we first look at how much time elapses between successive crests or successive troughs. This peak-to-peak or trough-to-trough time, which is called the period of the wave, is usually measured in seconds. If we take the inverse of the period, we get a value expressed in units of inverse seconds (i.e., "per second"). This is the frequency of the wave. Frequency is most often measured in cycles per second, also called "hertz" (Hz). If the frequency of a wave is greater than approximately 20 Hz (20 wave crests, or pulses, pass a given point in one second), then humans generally perceive this phenomenon as a sound. The frequencies that an average human being perceives as sound range from 20 Hz on the low end to 20,000 Hz on the high end. Frequency in the music world is known as "pitch." The greater the frequency, the higher the pitch.

Frequency and amplitude are two of the mathematical concepts necessary for understanding a "pure" sound wave. The third, and last, basic concept related to waves is phase. Phase has to do with the position in the cycle of compressions or rarefactions at which a wave starts. For example, if the cone of a loudspeaker—the part that vibrates back and forth—starts out moving away from you, the sound wave that eventually reaches you will begin with a rarefaction. If, on the other hand, the cone starts by moving towards you, the wave will first hit you with a compression. The speaker doesn't have to start at one of these extremes, however; it can start at any point in the cycle. Different starting points mean different phases.

phases Item 1858 / Oregon Public Broadcasting, created for Mathematics Illuminated, SINE WAVES IN DIFFERENT PHASES (2008). Courtesy of Oregon Public Broadcasting.


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THE SOUND OF MUSIC

Now that we have some understanding of how a wave can be thought of strictly on a physical basis, let's return to the Greek idea of intervals. Recall that the Greeks considered harmonious the sounds of plucked strings whose lengths were in ratios of whole numbers. In general, strings of different lengths produce sound of different frequencies. Without considering such things as string thickness or tension, longer strings tend to produce lower frequencies than do shorter strings. So, when two strings of different lengths are plucked together, the resulting sound is a combination of frequencies. Surprisingly though, even the sound produced by a single string is not made up entirely of one frequency.

A string vibrates with some fundamental frequency, 440 Hz for an "A" note, for example, but there are other frequencies present as well. These are known as either partials or overtones, and they give each instrument its characteristic sound, or timbre. Timbre helps explain why a tuba sounds different than a cello, even though you can play a "middle C" on both instruments.

soundwaves Item 2044 / Oregon Public Broadcasting, created for Mathematics Illuminated, TWO DIFFERENT SOUNDWAVES, EACH PLAYING THE NOTE "A" (2008). Courtesy of Oregon Public Broadcasting.


soundwaves I tem 3076 / Oregon Public Broadcasting, created for Mathematics Illuminated, TWO DIFFERENT SOUNDWAVES, EACH PLAYING THE NOTE "A" (2008). Courtesy of Oregon Public Broadcasting.


For a single plucked string, the overtones occur at frequencies that are whole number multiples of the fundamental frequency. So, a string vibrating at 440 Hz (an "A") will also have some vibration at 880 Hz (440 × 2), 1320 Hz (440 × 3), and so on. These additional frequencies have smaller amplitudes than does the fundamental frequency and are, thus, more noticeable as added texture in a sound rather than as altered pitch.

Every instrument has its own timbre. If you play a middle A, corresponding to 440 Hz, on a piano, the note will have a much different sound than the same note played on a trumpet. This is due to the fact that, although both notes are based on the fundamental frequency of 440 Hz, they have different combinations of overtones attributable to the unique makeup of each instrument. If you've ever heard "harmonics" played on a guitar, you have some sense of how a tone can be made of different parts. When a guitarist plays "harmonics," he or she dampens a string at a very precise spot corresponding to some fraction of the string's length, thereby effectively muting the fundamental frequency of the vibrating string. The only sounds remaining are the overtones, which sound "thinner" than the fundamental tones and almost ethereal.

Up until this point, the connections we have drawn between music and math have been mainly physical, with a few somewhat philosophical ideas thrown in as well. There is much more to the story, however. In order to take our discussion to a deeper level, we first need to understand how waves can be combined mathematically. Before we can combine waves mathematically, however, we need a universal way to describe them. In the next section, we will see how a simple wave can be expressed mathematically using the power of triangles and trigonometry.

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Next: 10.4 Mathematics of Waves


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