Doppler Dictionary

In the summer months, our neighborhood is regularly trawled by an ice cream truck, loudly blaring music to announce its arrival. When the kids hear the song, they’re up and running, asking for money. My strategy is to stall, suggesting, for example, that the truck only sells dog treats, or that it is that very ice cream truck that took away their older sister whom we never talk about. But soon they’re out the door, listening intently for it. “It’s through the woods behind the Johnsons’,” my daughter yells. “No, it’s at the park playground,” my son responds. As the ice cream truck navigates the maze of streets, the kids can hear that it is sometimes headed toward them, only to turn at a cross street, and the kids’ hearts drop. I try to allay their heartache by telling them they weren’t getting ice cream even if the truck had come, but then they perk up, hearing it headed this way yet again.

The moral of this story about my forlorn kids is not just how to be a good parent, but how kids can hear the comings and goings of ice cream trucks. There are a variety of cues they could be using for their ice cream–truck sense, but one of the best cues is the truck’s pitch , the entire envelope of pitches that modulates as it varies in direction relative to my kids’ location, due to the Doppler effect.

What exactly is the Doppler effect? To understand it, we must begin with a special speed: 768 miles per hour. That’s the speed of sound in the Earth’s atmosphere, a speed Superman must keep in mind because passing through that speed leads to a sonic boom, something sure to flatten the soufflé he baked for the Christmas party. We, on the other hand, move so slowly that we can carry soufflés with ever so slightly less fear. But even though the speed of sound is not something we need to worry about, it nevertheless has important consequences for our lives. In particular, the speed of sound is crucial for comprehending the Doppler effect, wherein moving objects have different pitches depending on their direction of movement relative to the listener.

Let’s imagine a much slower speed of sound: say, two meters per second. Now let’s suppose I stand still and clap 10 times in one second. What will you hear (supposing you also are standing still)? You will hear 10 claps in a second, the wave fronts of a 10 Hertz sound. It also helps to think about how the waves from the 10 claps are spread out over space. Because I’m pretending that the speed of sound is two meters per second, the first clap’s wave has moved two meters by the time the final clap occurs, and so the 10 claps are spread out over two meters of space. (See Figure 23a.)

Figure 23 . (a) A stationary speaker is shown making 10 clap sounds in a second. The top indicates that the wave from the clap has just occurred, not having moved beyond the speaker. In the lower part of the panel, the speaker is in the same location, but one second of time has transpired. The first wave has moved two meters to the right, and the final wave has just left the speaker. A listener on the right will hear a 10 Hz sound. (b) Now the speaker is moving in the same direction as the waves and has moved one meter to the right after one second. The 10 claps are thus spread over one meter of space, not two meters as in (a) . All 10 waves wash over the listener’s ears in half a second, or at 20 Hz, twice as fast as in (a) . (c) In this case the speaker is moving away from the listener, or leftward. By the time the tenth clap occurs, the speaker has moved one meter leftward, and so the 10 claps are spread over three meters, not two as in (a) . Their frequency is thus lower, or 6.7 Hz rather than the 10 Hz in (a) .

Now suppose that, instead of me standing still, I am moving toward you at one meter per second. That doesn’t sound fast, but remember that the speed of sound in this pretend example is two meters per second, so I’m now moving at half the speed of sound! By the time my first clap has gone two meters toward you, my body and hands have moved one meter toward you, and so my final clap occurs one meter closer to you than my first clap. Whereas my 10 claps were spread over two meters when I was stationary, in this moving‑toward‑you scenario my 10 claps are spread over only one meter of space. These claps will thus wash over your ears in only half a second, rather than a second, and so you will hear a pitch that is 20 Hz, twice what it was before. (See Figure 23b.) If I were moving away from you instead, then rather than my 10 claps being spread over two meters as in the stationary scenario, they would be spread over three meters. The 10 claps would thus take 1½ seconds to wash over you, and be heard as a 6.66 Hz pitch–a lower pitch than in the baseline case. (See Figure 23c.)

The speed of sound is a couple hundred times faster than the two‑meter‑per‑second speed I just pretended it was, but the same principles apply: when I move toward you my pitches are upshifted, and when I move away from you my pitches are downshifted. The shifts in pitch will be much smaller than those in my pretend example, but in real life they are often large enough to be detectable by the auditory system, as we will discuss later. The Doppler effect is just the kind of strong ecological universal one expects the auditory system to have been selected to latch onto, because from it a listener’s brain can infer the direction of motion of a mover, such as an ice cream truck.

To illustrate the connection between pitch and directedness toward you, let’s go back to our generic train example and assume the track is straight. When a train is far away but approaching the station platform where you are standing, it is going almost directly toward you, as illustrated in Figure 24i. This is when its pitch will be Doppler shifted upward the most. (High and constant pitch is, by the way, the signature of an impending collision.) As the train nears, it gets less and less directed toward you, eventually to pass you by. Its pitch thus drops to an intermediate, or baseline, value when it reaches its nearest point to you and is momentarily moving neither toward nor away from you (see Figure 24ii). As the train begins to move away from you, its pitch falls below its intermediate value and continues to go lower and lower until it reaches its minimum, when headed directly away (see Figure 24iii). (If, by the way, you were unwisely standing on the tracks instead of on the platform, then the train’s pitch would have remained at its maximum the entire period of time it approached. Then, just after the sound of your body splatting, the train’s pitch would instantaneously drop to its lowest pitch. Of course, you would be in no condition to hear this pitch drop.)

Figure 24 . Illustration that the pitch of a mover (relative to the baseline pitch) indicates the mover’s directedness toward you. When the train is headed directly toward the observer, pitch is at its maximum (i) , and is at its lowest when headed directly away (iii) ; in between the pitch is in between (ii) .

As a further illustration of the relationship between pitch and mover direction, suppose that a mover is going around in a circle out in front of you (not around you). At (a) in Figure 25 the mover is headed directly away, and so has minimum pitch. The mover begins to turn around for a return, and pitch accordingly rises to a baseline, or intermediate, level at (b). The mover now begins veering toward you, raising the pitch higher, until the mover is headed directly toward you at (c), at which point the pitch is at its maximum. Now the mover begins veering away from you so as not to collide, and pitch falls back to baseline at position (d), only to fall further as the mover moves away to (a) again.

Figure 25 . The upper section shows a mover moving in a circle out in front of the listener (the ear), indicating four specific spots along the path. The lower part of the figure shows the pitch at these four spots on the path. (a) When moving directly away, pitch is at its minimum. (b) Pitch rises to baseline when at the greatest distance and moving neither toward nor away. (c) Pitch rises further to its maximum when headed directly toward the listener. (d) Pitch then falls back to baseline when passing tangentially nearby. The pitch then falls back to its minimum again at (a) , completing the circle.

From our experience with the train and looping‑mover illustrations, we can now build the simple “dictionary” of pitches shown in Figure 26. Given a pitch within a range of pitches, the figure tells us the pitch’s meaning: a direction of the mover relative to the listener. In the dictionary of nature, pitch means degree of directedness toward you.

Figure 26. Summary of the “meaning” of pitch, relative to baseline pitch. (The actual mapping from direction to pitch is non‑linear, something we discuss later in the upcoming section.)

This pitch dictionary is useful, but only to a limited extent. Doppler pitches tend to be fluctuating when you hear them, whether because movers are merely going straight past you (as in Figure 24), or because movers are turning (as in Figure 25). These dynamic pitch changes, in combination with the pitch dictionary, are a source of rich information for a listener. Whereas pitches above and below baseline mean an approaching or receding mover, respectively, changing pitch tells us about the mover’s turning and veering behavior. A rising pitch means that the mover is becoming increasingly directed toward the listener; the mover is veering more toward you. And falling pitch means that the mover is becoming decreasingly directed toward the listener; the mover is veering more away from you. One can see this in Figure 25. From (a) through (c) the mover is veering more toward the listener, and the pitch is rising throughout. In the other portion of the circular path, from (c) to (a) via (d), the mover is veering away from the listener, and the pitch is falling .

To summarize, pitch informs us of the mover’s direction relative to us, and pitch change informs us of change of direction–the mover’s veering behavior. High and low pitches mean an approaching and a receding mover, respectively; rising and falling pitches mean a mover who is veering toward or away from the listener, respectively. We have, then, the following two fundamental pitch‑related meanings:

Pitch : Low pitchmeans a recedingmover. High pitchmeans an approachingmover.

Pitch change : Falling pitchmeans a mover veering more away. Rising pitchmeans a mover veering more toward.

Because movers can be approaching or receding and at the same time veering toward or away, there are 2 × 2 = 4 qualitatively distinct cases, each defining a distinct signature of the mover’s behavior, as enumerated below and summarized in Figure 27.

(A) Moving away, veering toward.

(B) Moving toward, veering toward.

(C) Moving toward, veering away.

(D) Moving away, veering away.

Figure 27 . Four qualitatively distinct categories of movement given that a mover may move toward or away, and may veer toward or away. (I have given them alphabet labels starting at the bottom right and moving counterclockwise to the other three squares of the table, although my reason for ordering them in this way won’t be apparent until later in the chapter. I will suggest later that the sequence A‑B‑C‑D is a generic, or most common, kind of encounter.)

These four directional arcs can be thought of as the fundamental “atoms” of movement out of which more complex trajectories are built. The straight‑moving train of Figure 24, for example, can be described as C followed by D, that is, veering away over the entire encounter, but first nearing, followed by receding. (As I will discuss in more detail in the Encore section titled “Newton’s First Law of Music,” straight‑moving movers passing by a listener are effectively veering away from the listener.)

These four fundamental cases of movement have their own pitch signatures, enumerated below and summarized in Figure 28.

(E) Low, rising pitch means moving away, veering toward .

(F) High, rising pitch means moving toward, veering toward .

(G) High, falling pitch means moving toward, veering away .

(H) Low, falling pitch means moving away, veering away .

Figure 28 . Summary of the movement meaning of pitch, for low and high pitch, and rising and falling pitch. (Note that I am not claiming people move in circles as shown in the figure. The figure is useful because all movements fall into one of these four categories, which I am illustrating via the circular case.)

These four pitch categories amount to the auditory atoms of a mover’s trajectory. Given the sequence of Doppler pitches of a mover, it is easy to decompose it into the fundamental atoms of movement the mover is engaged in. Let’s walk through these four kinds of pitch profiles, and the four respective kinds of movement they indicate, keeping our eye on Figure 28.

(A) The bottom right square in Figure 28 shows a situation where the pitch is low and rising. Low pitch means my neighborhood ice cream truck is directed away from me and the kids, but the fact that the pitch is rising means the truck is turning and directing itself more toward us. Intuitively, then, a low and rising pitch is the signature of an away‑moving mover noticing you and deciding to begin to turn around and come see you. To my snack‑happy children, it means hope–the ice cream truck might be coming back!

(B) The upper right square concerns cases where the pitch is higher than baseline and is rising. The high pitch means the truck is directed at least somewhat toward us, and the fact that the pitch is rising means the truck is further directing itself toward us. Intuitively, the truck has seen my kids and is homing in on them. My kids are ecstatic now, screaming, “It’s coming! It sees us!”

(C) The top left square is where the pitch is still high, but now falling. That the pitch is high means the truck is headed in our direction; but the pitch is falling, meaning it is directing itself less and less toward us. “Hurry! It’s here!” my kids cry. This is the signature of a mover arriving , because when movers arrive at your destination, they either veer away so as not to hit you, or come to a stop; in each case, it causes a lowering pitch, moving toward baseline.

(D) The bottom left, and final, square of the matrix is where the pitch is low and falling. This means the truck is now directed away, and is directing itself even farther away. Now my kids’ faces are purple and drenched with tears, and I am preparing a plate of carrots.

Figure 28 amounts to a second kind of ecological pitch‑movement dictionary (in addition to Figure 26). Now, if melodic pitch contours have been culturally selected to mimic Doppler shifts, then the dictionary categorizes four fundamentally different meanings for melody . For example, when a melody begins at the bottom of the pitch range of a piece and rises, it is interpreted by your auditory system as an away‑moving mover veering back toward the listener (bottom right of Figure 28). And if the melody is high in pitch and falling, it means the fictional mover is arriving (upper left of Figure 28). At least, that’s what these melodic contours mean if melody has been selected over time to mimic Doppler shifts of movers. With some grounding in the ecological meaning of pitch, we are ready to begin asking whether signatures of the Doppler effect are actually found in the contours of melody. We begin by asking how many fingers one needs to play a melody.