Encore
Although Chapter 4 presented a variety of evidence that the structure of music has the signature of human movers, there is additional evidence that couldn’t reasonably be fit into that chapter, and so it appears here in the Encore.
1 THE LONG AND SHORT OF HIT
The mysterious approaching monster from the section titled “Backbone” in Chapter 4 was mysterious because you mistakenly perceived a hit sound rapidly following the footstep; that is, you perceived the between-the-steps interval to be split into a short interval (from step to rapidly following hit) and a long interval (from that quick post-step hit to the next footstep). The true gait of the approaching lilting lady had its between-the-steps interval broken, instead, into a long interval followed by a short interval. My attribution of mystery to the “short-long” gait, not the “long-short,” was not arbitrary. “Short-long” is a strange human gait pattern, whereas “long-short” is commonplace.
Your legs are a pair of 25-pound pendulums that swing forward as you move, and are the principal sources of your between-the-steps hit sounds. A close look at how your legs move when walking (see Figure 41) will reveal why between-the-step hits are more likely to occur just before a footstep than just after. Get up and take a few steps. Now try it in slow motion. Let your leading foot hit the ground in front of you for its step. Stop there for a moment. This is the start of a step-to-step interval, the end of which will occur when your now-trailing foot makes its step out in front of you. Before continuing your stride, ask yourself what your trailing foot is doing. It isn’t doing anything. It is on the ground. That is, at the start of a step-to-step interval, both your feet are planted on the ground. Very slowly continue your walk, and pay attention to your trailing foot. As you move forward, notice that your trailing foot stays planted on the ground for a while before it eventually lifts up. In fact, your trailing foot is still touching the ground for about the first 30 percent of a step-to-step interval. And when it finally does leave the ground, it initially has a very low speed, because it is only just beginning to accelerate. Therefore, for about the first third of a step, your trailing foot is either not moving or moving so slowly that any hit it does take part in will not be audible. Between-the-footsteps hit sounds are thus relatively rare immediately after a step. After this slow-moving trailing-foot period, your foot accelerates to more than twice your body speed (because it must catch up and pass your body). It is during this central portion of a step cycle that your swinging leg has the energy to really bang into something. In the final stage of the step cycle, your forward-swinging leg is decelerating, but it still possesses considerable speed, and thus is capable of an audible hit.
Figure 41. Human gait. Notice that once the black foot touches the ground (on the left in this figure), it is not until the next manikin that the trailing (white) foot lifts. And notice how even by the middle figure, the trailing foot has just begun to move. During the right half of the depicted time period, the white leg is moving quickly, ready for an energetic between-the-steps hit on something.
We see, then, that there is a fundamental temporal asymmetry to the human step cycle. Between-the-steps hits by our forward-swinging leg are most probable at the middle of the step cycle, but there is a bias toward times nearer to the later stages of the cycle. In Figure 41, this asymmetry can be seen by observing how the distance between the feet changes from one little human figure to the next. From the first to the second figure there is no change in the distance between the feet. But for the final pair, the distance between the feet changes considerably. For human gait, then, we expect between-the-steps gangly hits as shown in Figure 42a: more common in mid-step than the early or late stages, and more common in the late than the early stage.
Figure 42. (a) Because of the nature of human gait, our forward-swinging leg is most likely to create an audible between-the-steps bang near the middle of the gait cycle, but with a bias toward the late portions of the gait cycle, as illustrated qualitatively in the plot. (b) The relative commonness of between-the-beat notes occurring in the first half (“early”), middle, or second half (“late”) portions of a beat cycle. One can see the qualitative similarity between the two plots.
Does music show the same timing of when between-the-beat notes occur? In particular, are between-the-beat notes most likely to occur at about the temporal center of the interval, with notes occurring relatively rarely at the starts and ends of the beat cycle? And, additionally, do we find the asymmetry that off-beat notes are more likely to occur late than early (i.e., are long-shorts more common than short-longs)? This is, indeed, a common tendency in music. One can see this in the classical themes as well, where I measured intervals from the first 550 themes in Barlow and Morgenstern’s dictionary, using only themes in 4⁄4 time. There were 1078 cases where the beat interval had a single note directly in the center, far more than the number of beat intervals where only the first or second half had a note in it. And the gaitlike asymmetry was also found: there were 33 cases of “short-longs” (beat intervals having an off-beat note in the first half of the interval but not the second half, such as a sixteenth note followed by a dotted eighth note), and 131 cases of “long-shorts” (beat intervals having a note in the second half of the interval but not the first half, like a dotted eighth note followed by a sixteenth note). That is, beat intervals were four times more likely to be long-short than short-long, but both were rare compared to the cases where the beat interval was evenly divided. Figure 42b shows these data.
Long-shorts are more common in music because they perceptually feel more natural for movement—because they are more natural for movement. And, more generally, the time between beats in music seems to get filled in a manner similar to the way ganglies fill the time between steps. In the Chapter 4 section titled “The Length of Your Gangly,” we saw that beat intervals are filled with a human-gait-like number of notes, and now we see that those between-the-beat notes are positioned inside the beat in a human-gait-like fashion.
Thus far in our discussion of rhythm and beat, we have concentrated on the temporal pattern of notes. But notes also vary in their emphasis. As we mentioned earlier, on-beat notes typically have greater emphasis than between-the-beat notes, consistent with human movers typically having footsteps more energetic than their other gangly bangs. But even notes on the beat vary in their emphasis, and we take this up next.
2 MEASURE OF WHAT?
Thus far we have discussed beats as footsteps, and between-the-beat notes as between-the-footsteps banging ganglies. But there are other rhythmic features of music that occur at the scale of multiple beats. In particular, music rarely treats each and every beat as equal. Some beats are special. In ¾ time, for example, every third beat gets a little emphasis, and in 4⁄4 time every fourth beat gets an emphasis. This is the source of the measure in music, where the first beat in each measure gets the greatest emphasis. (And there are additional patterns: in 4⁄4 time, for instance, the third beat gets a little extra oomph, too, roughly half that of the first.) If you keep the notes of a piece of music the same, but modify which beats are emphasized, the song can often sound nearly unrecognizable. For example, here is “Twinkle, Twinkle Little Star,” but with some unusual syllables emphasized to help you sing it in ¾ time rather than the appropriate 4⁄4 time. “TWI-nkle, twi-NKLE, lit-tle STAR, <silent beat>, how I won-der WHAT you are.” As you can see, it is very challenging to even get yourself to sing it in the wrong time signature. And when you eventually manage to do it, it is a quite different song from the original.
Why should a difference in the pattern of emphasis on beats make such a huge difference in the way music sounds to us? With the movement theory of music in hand, the question becomes: does a difference in the pattern of emphasis of a mover’s footsteps make a big difference in the meaning of the underlying behavior? For example, is a mover with a ¾ time gait signature probably doing a different behavior than a mover with a 4⁄4 time gait signature?
The answer is, “Of course.” A different pattern in footstep emphasis means the mover is shifting his body weight in a different pattern. The ¾ time mover has an emphasis on every third step, and thus alternates which foot gets the greater emphasis. The 4⁄4 time mover, on the other hand, has emphasis on every other step, with extra emphasis on every fourth step. These are the gait sounds of distinct behaviors. Real movements by people may not stay within a single time signature for prolonged periods, as music often does, but, instead, change more dynamically as the mover runs, spins, and goes up for a layup. Time-signature differences in movement imply differences in behavior, and so we expect that our auditory system is sensitive to these time signatures . . . and that music may have come to harness this sensitivity, explaining why time signature matters in music.
And notice that when we hear music with a time signature, we want to move consistently not only with the beat and the temporal pattern of notes, but also with the time signature. People could waltz to music with a 4⁄4 time signature, but it just does not feel right. People not only want to step to the beat, (something we discussed early in Chapter 4); they want to step extra hard on the emphasized beat.
This and the previous Encore section concerned rhythm. The upcoming two also concern rhythm, and how it interacts with melody and with loudness, respectively.
3 FANCY FOOTWORK
When the kids and I are doing donuts in the parking lot at the dollar store—that is, driving the minivan in such tight circles that the wheels begin to screech and squeal—we are making minivan gangly banging sounds. Such behavior leads to especially complex rubber-meets-road hits and slides, sounds we revel in as we’re doing it. But the patrons at the dollar store hear an additional feature. The patrons hear Doppler shifts, something that the kids and I do not hear because we are stationary relative to the minivan. For the dollar store patrons, the pitch of the envelope of minivan gangly bangings rises and falls as we approach and recede from them in our donuts. In fact, it is because my minivan is veering so sharply that its ganglies begin banging in a more complex fashion. Compared to minivans not doing donuts, minivans doing donuts change pitch faster and have more complex “gaits.” Greater pitch changes therefore tend to be accompanied by more complex gait patterns.
This pitch-rhythm connection is also found among human movers. When we turn, we are likely to have a more complex gait and gangly pattern than when we are simply moving straight ahead. For example, when you turn left, you must lean left, lest you fall over on your right side; and your legs can no longer simply swing straight past each other, but must propel the body leftward via a push or pivot. And many turns involve more complex footwork, such as sidestepping, trotting, twists, and other maneuvers we acrobatic apes regularly carry out. For example, when a basketball player crosses the court, his or her path is roughly straight, and the resultant gait sounds are a simple beat. Once a player has crossed the court, however, his or her movements tend to be curvy, not straight, as players on offense try to free themselves up for a pass, or players on defense loom in for a steal or shadow the offense to prevent a pass, in each case setting off a richer pattern of gangly sounds.
Does music behave in this way? When melodic pitches change—a signal that the depicted mover is turning, as we discussed in Chapter 4—does the rhythm tend to get more complex? As a test for this, I sampled 713 two-beat intervals having at least two notes each from the Dictionary of Musical Themes, and for each recorded whether the pitch was varying or unvarying, and whether the rhythm was simple (one note on each beat, or “just the footsteps”) or complex (more than “just the footsteps”). (Data were sampled from 2⁄4 and 4⁄4 time signature pieces, and from every tenth theme up to “D400” in the Dictionary.) When pitch changed over the two-beat intervals, the probability was 0.66 that the beat was complex, whereas when pitch did not change the probability was only 0.35 that the beat was complex. Consistent with the prediction from real-world turners, then, these data suggest that when music changes pitch—the Doppler signature of a mover changing direction—its rhythm tends to become more complex. That is, as with people, when music “turns,” the ganglies start flying.
We see, then, that melody interacts with rhythm in the way Doppler interacts with gait. Now let’s ask whether loudness also interacts with rhythm, as expected from the ecology of human movers. We take that up in the next Encore section.
4 DISTANT BEAT
As I write this I am on the (inner) window ledge of my office at RPI, overlooking downtown Troy and the Hudson River. I’m on the fifth floor (of the city side of the building), with a steep, sloping hill at the bottom, so everything I hear is either fairly far away, or very far away. Because of my extreme distance from nearly everything, I end up hearing only a small sample of the sounds occurring in the city. Mainly, I hear the very energetic events. If a sound were not very energetic, then it would be inaudible by the time the sound waves reached me. I can see a tractor dumping rocks, but I hear only the boom of a particularly large one, missing out on the sounds of the many smaller rock hits I can see but cannot hear. Generally, when something makes complex sounds, whether it is a car, a washing machine, or a tornado, some of the noises composing the whole are more energetic than others. If it is far away from you, then you will only hear the most energetic parts of the sound. But if you are close, you’ll be able to hear the full panoply of sounds.
As with most complex sound makers, human movers make sounds of varying energy and frequency. The most energetic sounds tend to be our footsteps. Accordingly, the first thing we hear when someone is approaching from afar tends to be their footsteps. The other gait-related sounds from clanging limbs are difficult to hear when far away, but they get progressively more audible as the mover nears us. That is, as a mover gets closer to us and the loudness of his gait sounds thereby rises, the number of audible gait sounds per footstep tends to increase.
If music has culturally evolved to sound like human movement, then we accordingly expect that the louder parts of songs should have more notes per beat (i.e., more fictional gangly bangs per step). Do they? Do fortissimo passages have greater note density than pianissimo? Caitlin Morris, as an undergraduate at RPI, set out to test this among scores in An Anthology of Piano Music, Vol. II: The Classical Period, by Denes Agay (New York: Music Sales America, 1992), and found that this is indeed the case. Figure 43 shows how the density of notes (the number of notes per beat) varies with loudness over 60 classical pieces. One can see that note density increases with loudness, as predicted. Music doesn’t have to be like this. Music could pack more notes in per beat in soft parts, and have only on-the-beat notes for the loud parts. Music has this louder-is-denser characteristic because, I submit, that’s a fundamental ecological regularity our auditory systems have evolved to expect for human (and any) movers.
This result is, by the way, counter to what one might expect if loudness were due not to spatial proximity but to the energy level (or “stompiness,” as we discussed in the Chapter 4 section titled “Nearness versus Stompiness”) of the mover. Louder stomps typically require longer gaps between each stomp. “Tap, tap, tap, tap, tap” versus “BANG! . . . . . . . . BANG!”
Now that we have expanded on rhythm, we will move on to further evidence that melodic contour acts as Doppler pitch.
Figure 43. Data from 60 pieces in Denes Agay’s An Anthology of Piano Music, Vol. II: The Classical Period showing that louder portions of music tend to be packed with more notes. Each of 234 contiguous segments of constant loudness were sampled, counting the total number of notes and beats; averages are over these 234 segments. Data collected and analyzed by Caitlin Morris. (Standard errors shown.)
5 HOME PITCH
We have discussed how melody consists of one pitch at a time (in the Chapter 4 section titled “Only One Finger at a Time”), which is exactly what we expect if melody has its origins in Doppler shifts. We also mentioned earlier in that chapter (in the section titled “Why Pitch Seems Spatial”) that melodic pitch tends to change in a fairly continuous fashion. At any one time, then, melody is at one pitch, and changes pitch roughly as if it is “moving” through pitches. Thinking of melody as if it is an unknown creature we wish to better understand, it is natural to ask about melody’s home and where it roams within the space of pitches. In this Encore section we’ll discuss three facets of melody’s home range and foraging behavior.
Let’s begin with one of the most salient features about the Doppler effect, which is that for a mover at constant speed there is a maximum and minimum pitch the mover can attain, these occurring when the mover is headed directly toward and directly away from the listener, respectively. Doppler pitches for any mover are therefore bound to a fixed home range. If Doppler pitches are confined in this sense to a fixed home range, then the music-is-movement theory predicts that melodies, too, should tend to confine themselves to a fixed home range. Melodies should tend to behave as if there is an upper or lower boundary to pitch. Does melody move around as if bound within an invisible fence, as predicted, or does melody move more freely? Although melody is highly variable, it has long been noticed that any given melody tends to confine itself to a fairly fixed window of pitches called its tessitura. The notion of tessitura allows that the melody may occasionally punch through a barrier, but the barriers are still worthy of recognition because of their tendency to hold the pitch inside. The tessitura is, I submit, music’s implicit recognition that a single constant-speed mover has a fixed range of Doppler pitches it can express.
Melody, then, has a fixed home range, consistent with Doppler shifts. Let’s now look into how melody spends its time within its home range. If melody really is acting like Doppler shifts, then melody should distribute itself within its home range in a similar manner to Doppler pitches. How do Doppler pitches distribute themselves within their home range? In particular, for a mover in your vicinity carrying out behaviors, in which directions does the mover tend to go? Many of the movers around you are just doing their own thing, carrying out actions that do not involve the fact that you are there. These movers will tend to go in any direction relative to you. But even movers who are interacting in some way with you will tend not to strongly favor some directions over others. For example, performers on a stage will, over the course of the show, move in all directions relative to any audience member. In fact, although their actions onstage may be highly intricate, they will often be very roughly summarized as moving in circles out in front of the listener, an illustrative case we had used earlier in Figure 25 of Chapter 4. Such circle-like behavior tends to sample broadly from all directions. Individual short bouts of behavior, then, can be anywhere in the Doppler pitch range. Across bouts of behavior, then, Doppler pitches tend to occur with fairly uniform probability over the Doppler range. For melody, then, we expect that individual melodic themes will be highly variable in their pitch distribution, but we also expect that, on average, these themes will sample pitches within their tessitura fairly uniformly.
That is, in fact, what we found among the classical themes. Despite wide variability from theme to theme, across the 10,000 classical themes the average distribution of notes across the tessitura is fairly flat, as shown in Figure 44. One might have expected to find that, say, melody strongly favors a single central pitch, and meanders away from this pitch as if tied to it by an elastic band, in which case the expected distribution would be disproportionately found on or near that pitch, and would fall steeply lower and lower the farther away a pitch is from that central one. Melody does not, however, behave like this. Melody is more Doppler-like, sampling pitches within its home range in a fairly egalitarian fashion.
Both Doppler and melodic pitches have a fixed home range, and both tend to roam fairly uniformly over their home. Let’s now ask whether some regions within the home range are “stickier.” That is, are there regions of the tessitura where, if the melody goes there, it takes longer to get out? This differs from what we just finished discussing—that concerned the total amount of time (actually, the total number of notes) spent in regions of the tessitura, whereas we are now asking how long in duration each singular visit to a region tends to be. To understand sticky pitches in melody, we look to the Doppler shifts of movers and ask whether any Doppler pitches are sticky.
Figure 44. Distribution showing where within a theme’s tessitura notes tend to lie, among the nearly 200,000 notes in the 10,000 classical themes. (That is, for each note, its position within its theme’s tessitura was determined and its quintile recorded. The plot shows the distribution of these values.) One can see that themes tend to have pitches sampling widely across the tessitura, with little tendency to favor some parts of the tessitura over others. (The shape looks identical if the distribution for each theme is separately determined, and all 10,000 distributions averaged, and the error bars on such a plot are far too small to discern.) Note again that this analysis uses the tessitura for each piece, rather than measuring the number of semitones away from the average pitch in the song; the latter analysis would lead to a more normal distribution, falling quickly in probability away from the average, something researchers Tierney, Russo, and Patel found in 2008. In light of the result here, the normal distribution they found is due to the distribution of tessitura widths, not the distribution of pitches within the tessitura.
There are, indeed, sticky Doppler pitches; they are the pitches near the top and bottom of the pitch range. To see why, imagine again a mover who is running in circles out in front of you, as depicted in Figure 45. Even though the mover is going through all directions uniformly, the pitches tend to change most quickly when the mover is whizzing horizontally by, either dropping quickly in pitch when passing nearby, or rising quickly in pitch when whizzing by at the far side of the circular path. When the mover is in the approach or withdrawal parts of the path, on the other hand, the pitch is fairly stable and high or low, respectively. Figure 45 shows these four segments of the circular path of the mover, and one can see that the pitch in the “toward” and “away” segments is much more stable than in the two “whiz by” segments. Doppler pitches vary less quickly near the top and bottom of their home range. The prediction, then, is that melodic pitch tends to change more slowly near the top and bottom of the tessitura. Does it? To test this, Sean Barnett measured the durations for all notes among the 10,000 classical themes. Each note was classified as a bottom, intermediate, or top note for its theme, and the average duration was computed for each of the three categories. Figure 46 shows these average durations, and one can see that lowest and highest notes in themes tend to be 17 percent longer in duration than notes with intermediate pitches.
Figure 45. Doppler pitch changes slowly when near the maximum and minimum of the Doppler pitch range. Melodies also share this, as shown in Figure 46.
Figure 46. Across the 10,000 classical themes, this plot shows the average duration of the minimum, intermediate, and maximum pitch in a theme. One can see that the minimum and maximum pitch in a theme each tend to be longer in duration than intermediate pitches. (Averages of durations were computed in logarithmic space. Error bars are too small to see.)
In this section we looked at three facets of melody’s home range. We saw that (i) melodies typically have a fixed home range, called the tessitura; (ii) melodies tend to distribute themselves fairly uniformly within their home range; and (iii) melodies tend to dwell longer at the edges of their range. Melody behaves in these ways, I am suggesting, because Doppler shifts behave in these ways. Melody is broadly Doppler-like in the home it keeps and the manner in which it distributes its movements and time throughout its home.
6 FAST TEMPO, WIDE PITCH
In the previous section we examined melody’s home range—its size, and melody’s hangouts within it. One facet of melody we discussed was that it tends to remain in a cage, called the tessitura, and I am suggesting that the top and bottom of the tessitura correspond to the Doppler pitches when the fictional mover is directed toward and away from you, respectively. But remember that Doppler shifts are greater when the mover has greater speed. A car driving past you at a crawl will have a small difference between its high approaching pitch and its low moving-away pitch. But if you stand at the side of the freeway, the difference in pitch as the cars pass you will be much greater. It follows from these simple observations that faster-tempo pieces of music should have bigger home ranges for their melodies. That is, if melodic contour has been culturally selected to mimic the Doppler shifts of movers, then the prediction is that music with a faster tempo (more beats per minute) should have a wider tessitura.
To test this, Sean Barnett and I measured the tempo and tessitura width of the melodies of all the pieces in the Classical Fake Book (Hal Leonard Corp.). (We did not use the Dictionary of Musical Themes here because it does not include tempo data.) Figure 47 shows how tessitura width varies with tempo (for just those pieces originally intended for keyboard). One can see that although tessitura width does not change for the several low tempos, it rises among the faster tempos. Tessitura width increases with greater tempo, as predicted from the fact that the Doppler pitch home range widens as mover speed increases. This is particularly striking because themes with wider tessituras tend to be more difficult to play, and so one might predict that wider-tessitura music would go with a slower tempo, but this is the opposite of what we in fact find.
One might wonder whether this result could be due, instead, to a general phenomenon in which faster-tempo music tends simply to amplify musical qualities, whatever they may be. Caitlin Morris measured the range of loudness levels—the “loudness-tessitura” width—and the tempo for a sample of 55 pieces in Denes Agay’s piano anthology, The Classical Period. Figure 48 shows how the width of the loudness range varies with tempo, and one can see that there is no trend. The pitch tessitura width does not, then, increase in Figure 47 merely because of some general proclivity to amplify musical qualities at higher tempos. In fact, the lack of change in “loudness-tessitura” width as a function of tempo is something the music-is-movement theory does predict, assuming that loudness in music is primarily driven by proximity, as we discussed in detail in the “Nearness versus Stompiness” section of Chapter 4. Imagine that a mover carries out a bout of behavior in your vicinity at low speed. Now imagine this mover is asked to repeat the same bout of behavior, but this time moving much more quickly—that is, at a higher tempo. In each case the mover is, we presume, going through the same sequence of spatial coordinates, and thus the same sequence of distances from the listener. And so it immediately follows that the mover courses through the same sequence of loudnesses no matter whether moving slowly or quickly. The music-is-movement theory predicts, then, that, unlike pitch, the range of loudnesses should not change as a function of the music’s tempo—faster music, same loudness range—and that’s what we found.
Figure 47. Tessitura (of melody) width versus tempo, among all 92 pieces for keyboard in the Classical Fake Book for which tempo data could be acquired. One can see that faster music tends to have wider tessituras, consistent with the Doppler interpretation of melodic pitch. (We found the same result when we used the data for all pieces.) The Classical Fake Book was used for two reasons. First, it is helpful because fake books are not cluttered with the notes from the chords (chords are notated via letter labels). Second, it is the only classical fake book I possess, so it amounted to an easy-to-get, unbiased sample.
Figure 48. “Loudness-tessitura” width (i.e., the total range of loudness levels) versus tempo, sampled from 55 pieces in Denes Agay’s An Anthology of Piano Music, Vol. II: The Classical Period (used instead of the Dictionary of Musical Themes because the latter does not possess loudness information). Unlike (pitch) tessitura width, which is expected—and does—increase with increasing tempo, the loudness-tessitura is expected—and does—remain constant. This anthology was chosen because it was the only proper non-fake, non-lesson book I possessed at home.
We see, then, that faster-tempo music behaves like faster-tempo movers: in each case the range of pitches increases with tempo, and the range of loudnesses does not change. Essentially, these results show us that the physics of movers is found in the structure of music. The upcoming Encore section continues the search for physics in music, and concerns momentum and Newton’s First Law.
7 NEWTON'S FIRST LAW OF MUSIC
Objects at rest stay at rest unless pushed. And objects moving continue moving in the same direction unless pushed. This is Newton’s First Law of Motion, which concerns inertia. This is a fundamental law of physics, and applies to any object with mass. Humans have mass, so it applies to us as well. And if music sounds like hulking humans moving about, then even music should adhere to Newton’s First Law of Motion. Does it?
Before attempting to answer this, let’s make sure we steer clear of one of the psychological handicaps I talked about in Chapter 4: the tendency to interpret musical pitch as spatial. As musical notes rise and fall on the page, or as your hands move hither and thither on the piano, it’s hard to resist the feeling that inertia should show up in the musical domain as a tendency for a moving pitch to keep on moving in the same direction. But this is a pitfall. Recall that I am claiming that pitch is about the direction of the mover, not about position in space. Changes in pitch are therefore about changes in the mover’s direction, not about changes of position in space.
With our memory jogged about the meaning of pitch, what are the expected musical consequences of Newton’s First Law? A change in melody’s pitch means the fictional mover’s direction of motion has changed. Let’s ask ourselves, then: if a mover changes direction, is there any physical tendency for the mover to continue changing direction? Purely physically, Newton’s First Law tells us no. Any subsequent turn would require yet more force, without which the mover will continue going in whatever direction it was going. When a moving object for some reason makes a 30-degree change in direction, the inertial tendency is precisely not to continue turning, but to continue going straight in the new direction. (The same is true if a change in Doppler pitch is due to a change in speed—a change in speed does not lead to a further change in speed—but I’ll always presume movers are staying at constant speed, the relevant case for music at constant tempo.)
The pitch of a mover, then, following the physics of movement itself, tends to stay the same. And if the pitch does change, it will have a tendency to stay at the new pitch—the mover’s new direction—not to continue changing pitch. Newton’s First Law for the pitches of movers is, therefore, that pitches have no inertia. Inertia is about how spatial changes tend to continue, not about how velocity (speed and direction) changes tend to continue. And because pitch is about velocity (i.e., speed and direction), not spatial location, pitch changes do not tend to continue. (If one were to imagine a spatial metaphor for how pitch changes, it would be movement of a bead in thick syrup: it moves if pushed with a fork, but immediately halts when no longer pushed.)
If melody’s pitch contour acts like the Doppler pitches of a mover, then musical pitch is expected to have no “inertia” to continue moving in the same direction—“up” or “down.” I had Sean Barnett carry out an analysis on the entire data set of 10,000 classical themes, and we found that indeed, there was little or no inertia for pitch, just as is expected if melodic pitch contours sound like Doppler pitches from moving humans. In particular, for a one-semitone change, the probability of continuing up after a semitone up was 49.14 percent, and the probability of continuing down after a semitone down was 51.33 percent. For two semitones, the values were 47.06 percent and 56.17 percent. Pitches therefore do not act like spatial location: if pitch were spatial, then a change in pitch would tend to lead to more of the same kind of change due to inertia, and those percentages I just mentioned should have all been much greater than 50 percent. Instead, and as predicted, pitches have little or no tendency to continue changing the same way they have been. Pitches act like Doppler shifts, following the expectations of Newton’s First Law of Motion by not exhibiting pitch inertia (because inertia does not apply to directions of motion).
Although our data showed no strong bias toward pitch changes continuing in the same direction (which is the signature of true spatial momentum), note that there was, for both one- and two-semitone changes, a slightly greater tendency for pitches to go down—a small degree of downward momentum. To further examine this, I need to discuss some subtleties I have glossed over so far.
The ecological interpretation of pitch is the mover’s direction of motion, but more carefully expressed, it is the mover’s direction of motion relative to the listener. With this in mind, there are actually two fundamentally different ways for a mover’s Doppler pitch to change. The first is what I have assumed in this section thus far: the mover turns. But Doppler pitch can change even when the mover does not turn, and this second source of changing Doppler pitch you are very familiar with, because it happens every time a mover passes you, including the generic passing train. When movers pass listeners, their pitch falls. In fact, whenever an object simply moves straight its pitch falls (unless the object is directed perfectly toward or away from the listener). It is not, then, quite right to say that Doppler pitches have no pitch momentum. Straight-moving movers have falling pitch, and straight-moving movers tend to keep going straight (because of inertia), and therefore falling pitch in these circumstances does tend to keep on falling.
If only we could tell the difference between the pitch changes due to a mover actually turning and the pitch changes due to a mover simply going straight! We could then predict a lack of pitch momentum for the former, but predict the presence of pitch momentum for the latter. Actually, we can tell them apart. When a mover turns, it is intentional and occurs fairly quickly (on average about 45 degrees per step, as discussed earlier in the section of Chapter 4 called “Human Curves”), and it can be a change in direction either more or less toward the listener. Intentional turning behavior therefore tends to lead to large pitch changes that can be upward or downward. But the second source of Doppler pitch change is the one due to movers going straight (and going by). In this case there tend to be a lot of steps over which pitch falls—because now the falling pitch is, in essence, due to continuous change of position in space—and so the pitch change per step is small, and is always downward.
Here, then, is how we can distinguish the straight-moving mover from the turning mover. When pitch falls by only a small amount, it tends to be the signature of a straight-moving mover passing. But when the pitch change is not consistently small and downward, it is typically due to the mover turning. Thus, a turning mover is given away by either of two pitch cues: (i) a large pitch change, whether upward or downward, implicates a turning mover, and (ii) any pitch change upward at all, small or large, implicates a turning mover (because straight-moving movers only have falling pitch, not rising pitch).
We do therefore expect Doppler pitch to possess inertia in just one circumstance: when pitch falls by a small amount. Small drops in pitch are more probably attributable to a straight-moving mover. Because straight movers have inertia and are thus likely to continue moving straight, small pitch drops do tend to have inertia. Small pitch upswings do not, however, have inertia, and neither do large pitch changes, whether up or down.
Is this what we find in music?
We already saw evidence for this earlier in this section. Recall that there was generally little or no inertia for pitch—the probability of a pitch change continuing in the same direction was near 50 percent. But let’s look at the pitch momentum numbers again, more carefully this time. For a semitone pitch change, the probability of continuing in the same direction was 49.14 percent and 51.33 percent for upward and downward, respectively. (Their standard errors are small—0.005 and 0.004, respectively—because these are averages across many thousands of instances.) The same asymmetry was found when considering whole-step changes of pitch, but now with respective values of 47.06 percent and 56.17 percent. (Standard errors are each 0.004.) These results are consistent with those of Paul von Hippel that David Huron discusses in his book Sweet Anticipation: no momentum following small steps upwards, but significant momentum downwards. The signature of pitch momentum is a value greater than 50 percent, and only the downward pitch change has this. (Upward pitch change is below 50 percent, meaning that a little more than half of the time a semitone upward is followed by either no change in pitch or a downward change in pitch.) For larger pitch changes, we found that neither upward nor downward pitch changes had any pitch momentum (i.e., the probability of continuing to change in the same direction was below 50 percent). Not only, then, does melodic pitch contour have a counterintuitive tendency to have no inertia, like the pitches of movers—but it breaks this tendency exactly when movers do. Consistent with melody’s meaning coming from the Doppler shifts of movers, melody conforms to Newton’s law of inertia.
In addition to the issue of whether pitch changes continue to change in the same direction, we can make a simpler observation. Let’s ask ourselves what the baseline expectation is for the Doppler pitch change of a mover. One’s first intuition might be that, in the absence of any information otherwise, we should expect a mover’s Doppler pitch to remain unchanged from one step to the next. Doppler pitches, however, do not typically hold still. Instead, the most fundamental baseline expectation (inertia) is that movers continue moving in whatever direction they were going. People tend (though not as strongly as inanimate objects!) to keep going straight, and thus the baseline, or generic expectation, for pitch change is that pitch falls, and generally by a small amount (compared to the pitch changes of a turner).
Now consider another observation about movers. Suppose that a mover is carrying out bouts of behavior around you, and is directing those behaviors toward you. Notice that this mover will have to make intentional turns toward you to keep orienting his behavior toward you. But also notice that he or she never has to deliberately turn away from you. This is because once a mover is directed roughly but not exactly toward you, going straight inevitably leads to a movement like veering away. Simply going straight will cause the mover to pass by you and depart. Turning is necessary to go toward someone, but not to go away. (This is essentially because a listener is in one location, and all the other locations are where that listener is not.) Thus, if musical melody is about listener-directed bouts of behavior, not only do we expect small pitch changes to more commonly be downward, we expect large pitch changes to more commonly be upward.
Do we find this in music? Do we find that melodic contours have a general tendency to fall gradually? And do we find that pitch drops tend to be smaller than pitch rises? Piet G. Vos and Jim M. Troost of the University of Nijmegen indeed found this among a sample from the Dictionary of Musical Themes. We carried out our own measurements over the entire data set: Sean Barnett measured the relative probability that a pitch changes upward (so that a probability greater than 0.5 means an upward tendency, and a probability less than 0.5 means a downward tendency) as a function of the size of the pitch change. One can see these results in Figure 49. For small pitch changes, the probabilities are mostly below 0.5, meaning that small pitch changes tend to be downward, as expected. And for large pitch changes the probabilities are mostly above 0.5, meaning that large pitch changes tend to be upward, also as expected.
Figure 49. The y-axis shows the relative probability that a pitch will go up on the next note, among the 10,000 classical themes in Barlow and Morgenstern’s Dictionary. A value of 0.5 means it is equally likely to go up or down in pitch. The x-axis represents how far the pitch changes (in number of semitones). One can see that for small pitch changes the probability tends to be below 0.5, meaning pitches tend to fall. But for large pitch changes, they tend to rise. Sean Barnett, then a graduate student at RPI, made these measurements.
Another characteristic difference between rising and falling Doppler pitches from movers is that when a mover passes by, going straight, the pitch doesn’t just fall slowly with “inertia,” but continues to fall over an extended portion of the pitch range. The train that has reached your position has, for example, dropped from its maximum pitch to its baseline pitch, and will then drop through the lower half of the pitch range as it goes past and away. These kinds of long Doppler pitch runs, then, are more commonly downward than upward for movers in the physical world. Are long pitch runs for melodies more commonly downward than upward? Sean Barnett measured runs among the 10,000 classical themes. In particular, he recorded runs spanning the bottom or top half of the theme’s tessitura. Setting a low bar for what counts as a run—two or more notes approximately filling (more than 80 percent) the upper or lower half of the tessitura—51.86 percent of the 212,542 runs were downward. A two-note run is not very runlike, and our expectation is that if we create a more stringent standard for what counts as a run, then we should find an even greater asymmetry between up and down, with an even greater share of runs being down. Indeed, when Sean required a run to have five or more notes in the same direction, 54.22 percent of the 11,119 runs that qualified were in the downward direction.
Consider now yet another ecological regularity in this vein. Let’s ask ourselves: are these falling-pitch runs due to straight-moving movers more likely to occur when movers are near or far? When a mover is far away, in order for that mover to implement a long downward run, the mover must continue straight for a great many steps without turning. It is quite likely that the mover will turn somewhere over the course of that long walk. But if the mover is close by, the mover need only move straight for a relatively small number of steps to engender a substantial downward pitch run. Big downward Doppler pitch runs are therefore disproportionately probable when near. Do we find this in music? As we discussed in Chapter 4, distance from the listener is encoded in music by loudness, and so our expectation here is that louder segments of music (i.e., passages depicting a more proximal mover) are more likely to have good-sized downward pitch runs. RPI graduate student Romann Weber measured runs spanning at least half the tessitura from Denes Agay’s The Classical Period, and calculated the probability that such pitch runs are downward as a function of loudness. As can be seen in Figure 50, the probability of a large downward pitch run rises with loudness, consistent with the ecological expectation.
Figure 50. Pitch runs spanning at least half the tessitura width among a sample of 37 pieces from Denes Agay’s An Anthology of Piano Music, Vol. II: The Classical Period. Forty such runs were found, the loudness during the run measured, and the relative probability that the run is up or down computed. Louder segments of music have a greater probability of long downward runs, consistent with expectations from human movement.
Newton’s First Law is found in music where it should be found if melodic pitch is about Doppler shifts. Melodic pitch acts like a mover’s direction, and thus has no pitch momentum, just as inertia predicts. But as we looked more closely, we realized that there are fundamental asymmetries between upward and downward Doppler pitch changes, asymmetries also found in melody. Melody does show pitch momentum in the special case of small downward changes in pitch, as expected from the dynamics of movers. And melody generally drifts downward gradually, as expected from the fact that all straight-moving movers have slowly falling pitch (unless moving directly toward or away from the listener). And melody takes larger jumps upward than downward in pitch, also something expected from movers orienting their bouts of behavior toward you. Melody also favors longer runs downward than upward, something we also expected from the sounds of movers. And finally, like closer movers, louder segments of music tend to have disproportionately more large downward pitch runs.
The Encore sections thus far have mostly concerned rhythm and melody. Loudness did come up in Encores 4, 6, and 7. The next and last Encore section is about loudness, providing further evidence that loudness in music behaves like loudness due to the proximity of the mover.
8 MEDIUM ENCOUNTERS
In the Chapter 4 section titled “Slow Loudness, Fast Pitch,” we saw that loudness varies slowly, consistent with the time scales required for movers to vary their distance from you, the listener. We must be more careful, though. If a mover were a “close talker,” tending to move about uncomfortably close to you, then even small changes in distance could lead to large changes in loudness, due to the inverse square law for loudness and distance. But in real life, more than close encounters, we tend to have medium encounters: the movers we typically listen to tend to be in the several- to ten-meter range, not in the centimeter range, and not in the tens or hundreds of meters range. At “medium” distances, large loudness modulations don’t occur over just one or several steps. They require more steps, plausibly in the range of the approximately 10 beats we found for the average loudness duration in Chapter 4.
Not only are our experiences of movers usually at a “medium” distance, but it seems reasonable to expect that individual bouts of behavior tend to occur at an average “medium” distance. Recall our generic encounters from the section titled “Musical Encounters” in Chapter 4: the “center of mass” of the A-B-C-D cycle of movement would be representative of the average distance of a generic encounter. We see, then, that loudnesses of movers will tend to have a typical value. We therefore expect any piece of music to have a baseline loudness level it spends a disproportionate amount of time at, spending less time at loudness levels farther away from this average. Unlike Doppler pitches, which have a distribution that is fairly broad and flat, the distribution of mover loudnesses tends to be more peaked. Is music like this? Does music spend most of its time at an average loudness level, relatively rarely venture out of that loudness zone, and more rarely still pursue greater loudness deviations from the average? Music is indeed roughly like this. Music tends to use mezzo forte as this baseline, with lesser and greater loudness levels happening progressively more rarely. RPI students Caitlin Morris and Eric Jordan measured the average percentage of a song spent at each of its loudness levels, and the results are shown in Figure 51. One can see that there is a strong “mountain” shape to the plot: pieces tend to spend more time at intermediate loudness levels than at loudness levels deviating far from the central values. (Although our data were broadly consistent with our expectation, there was a slight downward divot at mezzo forte relative to piano and forte, with the greatest percentage of time spent in piano.)
Figure 51. For each song, the total percentage of time spent at each loudness level was determined. These distributions were then averaged together across 43 pieces in Denes Agay’s An Anthology of Piano Music, Vol. II: The Classical Period.
We can say more. Consider the obvious fact that there is less real estate—less space—near you than far from you. This asymmetry means that a mover has more chances to be farther than average from you than to be nearer than average to you. There should not only be, then, a roughly mountain shape to Figure 51, but the below-average levels of loudness should be more common than the above-average levels of loudness. The mountain should have a higher level at lower-than-average levels of loudness. The distribution we just plotted in Figure 51 has, in fact, this expected asymmetry.
We can say something further still. Not only should movers spend a greater proportion of their time relatively far away than relatively nearby, but when they do get near, and thus relatively loud, this should be more transient. Why? Because the mover will more quickly leave the near region, for the simple reason that “the near” is an inherently smaller piece of land than “the far.” This is indeed the case, as shown in Figure 52, also obtained by Caitlin Morris and Eric Jordan.
Figure 52. For each song, the average duration of each loudness level was computed, and then these per-song average normalized so that the sum across the levels equaled one. Then, these were averaged across 43 pieces measured in Denes Agay’s An Anthology of Piano Music, Vol. II: The Classical Period. One can see the asymmetry. As predicted from the spatial asymmetries of near and far, music should tend to have longer durations at lower-than-average loudness levels compared to higher-than-average loudness levels.
We see, then, that loudnesses distribute themselves as expected if they are about proximity. Encounters have a typical distance; more cumulative time is spent farther than nearer; and nearer segments of encounters tend to be short-lived relative to farther segments.