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1. What Is Music?

From Pitch to Timbre

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What is music? To many, “music” can only mean the great masters—Beethoven, Debussy, and Mozart. To others, “music” is Busta Rhymes, Dr. Dre, and Moby. To one of my saxophone teachers at Berklee

College of Music—and to legions of “traditional jazz” aficionados—

anything made before 1940 or after 1960 isn’t really music at all. I had

friends when I was a kid in the sixties who used to come over to my

house to listen to the Monkees because their parents forbade them to lis-

ten to anything but classical music, and others whose parents would

only let them listen to and sing religious hymns. When Bob Dylan dared

to play an electric guitar at the Newport Folk Festival in 1965, people

walked out and many of those who stayed, booed. The Catholic Church

banned music that contained polyphony (more than one musical part

playing at a time), fearing that it would cause people to doubt the unity

of God. The church also banned the musical interval of an augmented

fourth, the distance between C and F-sharp and also known as a tritone

(the interval in Leonard Bernstein’s West Side Story when Tony sings the

name “Maria”). This interval was considered so dissonant that it must

have been the work of Lucifer, and so the church named it Diabolus in

musica. It was pitch that had the medieval church in an uproar. And it

was timbre that got Dylan booed.

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The music of avant-garde composers such as Francis Dhomont,

Robert Normandeau, or Pierre Schaeffer stretches the bounds of what

most of us think music is. Going beyond the use of melody and harmony,

and even beyond the use of instruments, these composers use record-

ings of found objects in the world such as jackhammers, trains, and wa-

terfalls. They edit the recordings, play with their pitch, and ultimately

combine them into an organized collage of sound with the same type of

emotional trajectory—the same tension and release—as traditional mu-

sic. Composers in this tradition are like the painters who stepped out-

side of the boundaries of representational and realistic art—the cubists,

the Dadaists, many of the modern painters from Picasso to Kandinsky to

Mondrian.

What do the music of Bach, Depeche Mode, and John Cage funda-

mentally have in common? On the most basic level, what distinguishes

Busta Rhymes’s “What’s It Gonna Be?!” or Beethoven’s “Pathétique”

Sonata from, say, the collection of sounds you’d hear standing in the

middle of Times Square, or those you’d hear deep in a rainforest? As

the composer Edgard Varèse famously defined it, “Music is organized

sound.”

This book drives at a neuropsychological perspective on how music

affects our brains, our minds, our thoughts, and our spirit. But first, it is

helpful to examine what music is made of. What are the fundamental

building blocks of music? And how, when organized, do they give rise to

music? The basic elements of any sound are loudness, pitch, contour, du-

ration (or rhythm), tempo, timbre, spatial location, and reverberation.

Our brains organize these fundamental perceptual attributes into higher-

level concepts—just as a painter arranges lines into forms—and these

include meter, harmony, and melody. When we listen to music, we are ac-

tually perceiving multiple attributes or “dimensions.” Here is a brief sum-

mary of them.

~ A discrete musical sound is usually called a tone. The word note is also used, but scientists reserve that word to refer to something

that is notated on a page or score of music. The two terms, tone

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and note, refer to the same entity in the abstract, where the word

tone refers to what you hear, and the word note refers to what you

see written on a musical score.

~ Pitch is a purely psychological construct, related both to the actual frequency of a particular tone and to its relative position in the musi-

cal scale. It provides the answer to the question “What note is that?”

(“It’s a C-sharp.”) I’ll define frequency and musical scale below.

~ Rhythm refers to the durations of a series of notes, and to the way that they group together into units. For example, in the “Alphabet

Song” (the same as “Twinkle, Twinkle Little Star”) the notes of the

song are all equal in duration for the letters A B C D E F G H I J K

(with an equal duration pause, or rest, between G and H), and then

the following four letters are sung with half the duration, or twice

as fast per letter: L M N O (leading generations of schoolchildren

to spend several early months believing that there was a letter in

the English alphabet called ellemmenno).

~ Tempo refers to the overall speed or pace of the piece. ~ Contour describes the overall shape of a melody, taking into ac-

count only the pattern of “up” and “down” (whether a note goes up

or down, not the amount by which it goes up or down).

~ Timbre is that which distinguishes one instrument from another— say, trumpet from piano—when both are playing the same written

note. It is a kind of tonal color that is produced in part by over-

tones from the instrument’s vibrations.

~ Loudness is a purely psychological construct that relates (nonlin- early and in poorly understood ways) to the physical amplitude of

a tone.

~ Spatial location is where the sound is coming from. ~ Reverberation refers to the perception of how distant the source is

from us in combination with how large a room or hall the music is

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in; often referred to as “echo” by laypeople, it is the quality that

distinguishes the spaciousness of singing in a large concert hall

from the sound of singing in your shower. It has an underappreci-

ated role in communicating emotion and creating an overall pleas-

ing sound.

These attributes are separable. Each can be varied without altering

the others, allowing the scientific study of one at a time, which is why we

can think of them as dimensions. The difference between music and a

random or disordered set of sounds has to do with the way these funda-

mental attributes combine, and the relations that form between them.

When these basic elements combine and form relationships with one an-

other in a meaningful way, they give rise to higher-order concepts such

as meter, key, melody, and harmony.

~ Meter is created by our brains by extracting information from rhythm and loudness cues, and refers to the way in which tones

are grouped with one another across time. A waltz meter orga-

nizes tones into groups of three, a march into groups of two or four.

~ Key has to do with a hierarchy of importance that exists between tones in a musical piece; this hierarchy does not exist in-the-world,

it exists only in our minds, as a function of our experiences with a

musical style and musical idioms, and mental schemas that all of

us develop for understanding music.

~ Melody is the main theme of a musical piece, the part you sing along with, the succession of tones that are most salient in your

mind. The notion of melody is different across genres. In rock mu-

sic, there is typically a melody for the verses and a melody for the

chorus, and verses are distinguished by a change in lyrics and

sometimes by a change in instrumentation. In classical music, the

melody is a starting point for the composer to create variations on

that theme, which may be used throughout the entire piece in dif-

ferent forms.

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~ Harmony has to do with relationships between the pitches of dif- ferent tones, and with tonal contexts that these pitches set up that

ultimately lead to expectations for what will come next in a musi-

cal piece—expectations that a skillful composer can either meet

or violate for artistic and expressive purposes. Harmony can mean

simply a parallel melody to the primary one (as when two singers

harmonize) or it can refer to a chord progression—the clusters of

notes that form a context and background on which the melody

rests.

The idea of primitive elements combining to create art, and of the im-

portance of relationships between elements, also exists in visual art and

dance. The fundamental elements of visual perception include color

(which can be decomposed into the three dimensions of hue, saturation,

and lightness), brightness, location, texture, and shape. But a painting is

more than these—it is not just a line here and another there, or a spot of

red in one part of the picture and a patch of blue in another. What makes

a set of lines and colors into art is the relationship between this line and

that one; the way one color or form echoes another in a different part of

the canvas. Those dabs of paint and lines become art when form and

flow (the way in which your eye is drawn across the canvas) are created

out of lower-level perceptual elements. When they combine harmoni-

ously they ultimately give rise to perspective, foreground and back-

ground, emotion, and other aesthetic attributes. Similarly, dance is not

just a raging sea of unrelated bodily movements; the relationship of

those movements to one another is what creates integrity and integrality,

a coherence and cohesion that the higher levels of our brain process.

And as in visual art, music plays on not just what notes are sounded, but

which ones are not. Miles Davis famously described his improvisational

technique as parallel to the way that Picasso described his use of a can-

vas: The most critical aspect of the work, both artists said, was not the

objects themselves, but the space between objects. In Miles’s case, he

described the most important part of his solos as the empty space be-

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tween notes, the “air” that he placed between one note and the next.

Knowing precisely when to hit the next note, and allowing the listener

time to anticipate it, is a hallmark of Davis’s genius. This is particularly

apparent in his album Kind of Blue.

To nonmusicians, terms such as diatonic, cadence, or even key and

pitch can throw up an unnecessary barrier. Musicians and critics some-

times appear to live behind a veil of technical terms that can sound pre-

tentious. How many times have you read a concert review in the

newspaper and found you have no idea what the reviewer is saying? “Her

sustained appoggiatura was flawed by an inability to complete the

roulade.” Or, “I can’t believe they modulated to C-sharp minor! How

ridiculous!” What we really want to know is whether the music was per-

formed in a way that moved the audience. Whether the singer seemed to

inhabit the character she was singing about. You might want the re-

viewer to compare tonight’s performance to that of a previous night or a

different ensemble. We’re usually interested in the music, not the techni-

cal devices that were used. We wouldn’t stand for it if a restaurant re-

viewer started to speculate about the precise temperature at which the

chef introduced the lemon juice in a hollandaise sauce, or if a film critic

talked about the aperture of the lens that the cinematographer used; we

shouldn’t stand for it in music either.

Moreover, many of those who study music—even musicologists and

scientists—disagree about what is meant by some of these terms. We

employ the term timbre, for example, to refer to the overall sound or

tonal color of an instrument—that indescribable character that distin-

guishes a trumpet from a clarinet when they’re playing the same written

note, or what distinguishes your voice from Brad Pitt’s if you’re saying

the same words. But an inability to agree on a definition has caused the

scientific community to take the unusual step of throwing up its hands

and defining timbre by what it is not. (The official definition of the

Acoustical Society of America is that timbre is everything about a sound

that is not loudness or pitch. So much for scientific precision!)

What is pitch? This simple question has generated hundreds of scien-

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tific articles and thousands of experiments. Pitch is related to the fre-

quency or rate of vibration of a string, column of air, or other physical

source. If a string is vibrating so that it moves back and forth sixty times

in one second, we say that it has a frequency of sixty cycles per second.

The unit of measurement, cycles per second, is often called Hertz (ab-

breviated Hz) after Heinrich Hertz, the German theoretical physicist

who was the first to transmit radio waves (a dyed-in-the-wool theoreti-

cian, when asked what practical use radio waves might have, he report-

edly shrugged, “None”). If you were to try to mimic the sound of a fire

engine siren, your voice would sweep through different pitches, or fre-

quencies (as the tension in your vocal folds changes), some “low” and

some “high.”

Keys on the left of the piano keyboard strike longer, thicker strings

that vibrate at a relatively slow rate. Keys to the right strike shorter, thin-

ner strings that vibrate at a higher rate. The vibration of these strings dis-

places air molecules, and causes them to vibrate at the same rate—with

the same frequency as the string. These vibrating air molecules are what

reach our eardrum, and they cause our eardrum to wiggle in and out at

the same frequency. The only information that our brains get about the

pitch of sound comes from that wiggling in and out of our eardrum; our

inner ear and our brain have to analyze the motion of the eardrum in or-

der to figure out what vibrations out-there-in-the-world caused the

eardrum to move that way.

By convention, when we press keys nearer to the left of the keyboard,

we say that they are “low” pitch sounds, and ones near the right side of

the keyboard are “high” pitch. That is, what we call “low” are those

sounds that vibrate slowly, and are closer (in vibration frequency) to the

sound of a large dog barking. What we call “high” are those sounds that

vibrate rapidly, and are closer to what a small yip-yip dog might make.

But even these terms high and low are culturally relative—the Greeks

talked about sounds in the opposite way because the stringed instru-

ments they built tended to be oriented vertically. Shorter strings or pipe

organ tubes had their tops closer to the ground, so these were called

the “low” notes (as in “low to the ground,”) and the longer strings and

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tubes—reaching up toward Zeus and Apollo—were called the “high”

notes. Low and high—just like left and right—are effectively arbitrary

terms that ultimately have to be memorized. Some writers have argued

that “high” and “low” are intuitive labels, noting that what we call high-

pitched sounds come from birds (who are high up in trees or in the sky)

and what we call low-pitched sounds often come from large, close-to-

the-ground mammals such as bears or the low sounds of an earthquake.

But this is not convincing, since low sounds also come from up high

(think of thunder) and high sounds can come from down low (crickets

and squirrels, leaves being crushed underfoot).

As a first definition of pitch, let’s say it is that quality that primarily

distinguishes the sound that is associated with pressing one piano key

versus another.

Pressing a piano key causes a hammer to strike one or more strings

inside the piano. Striking a string displaces it, stretching it a bit, and its

inherent resiliency causes it to return toward its original position. But it

overshoots that original position, going too far in the opposite direction,

and then attempts to return to its original position again, overshooting it

again, and in this way it oscillates back and forth. Each oscillation cov-

ers less distance, and, in time, the string stops moving altogether. This is

why the sound you hear when you press a piano key gets softer until it

trails off into nothing. The distance that the string covers with each os-

cillation back and forth is translated by our brains into loudness; the rate

at which it oscillates is translated into pitch. The farther the string trav-

els, the louder the sound seems to us; when it is barely traveling at all,

the sound seems soft. Although it might seem counterintuitive, the dis-

tance traveled and the rate of oscillation are independent. A string can

vibrate very quickly and traverse either a great distance or a small one.

The distance it traverses is related to how hard we hit it—this corre-

sponds to our intuition that hitting something harder makes a louder

sound. The rate at which the string vibrates is principally affected by its

size and how tightly strung it is, not by how hard it was struck.

It might seem as though we should simply say that pitch is the same

as frequency; that is, the frequency of vibration of air molecules. This is

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almost true. Mapping the physical world onto the mental world is sel-

dom so straightforward. However, for most musical sounds, pitch and

frequency are closely related.

The word pitch refers to the mental representation an organism has

of the fundamental frequency of a sound. That is, pitch is a purely psy-

chological phenomenon related to the frequency of vibrating air mole-

cules. By “psychological,” I mean that it is entirely in our heads, not in

the world-out-there; it is the end product of a chain of mental events that

gives rise to an entirely subjective, internal mental representation or

quality. Sound waves—molecules of air vibrating at various frequen-

cies—do not themselves have pitch. Their motion and oscillations can

be measured, but it takes a human (or animal) brain to map them to that

internal quality we call pitch.

We perceive color in a similar way, and it was Isaac Newton who first

realized this. (Newton, of course, is known as the discoverer of the the-

ory of gravity, and the inventor, along with Leibniz, of calculus. Like

Einstein, Newton was a very poor student, and his teachers often com-

plained of his inattentiveness. Ultimately, Newton was kicked out of

school.)

Newton was the first to point out that light is colorless, and that con-

sequently color has to occur inside our brains. He wrote, “The waves

themselves are not colored.” Since his time, we have learned that light

waves are characterized by different frequencies of oscillation, and

when they impinge on the retina of an observer, they set off a chain of

neurochemical events, the end product of which is an internal mental

image that we call color. The essential point here is: What we perceive as

color is not made up of color. Although an apple may appear red, its

atoms are not themselves red. And similarly, as the philosopher Daniel

Dennett points out, heat is not made up of tiny hot things.

A bowl of pudding only has taste when I put it in my mouth—when it

is in contact with my tongue. It doesn’t have taste or flavor sitting in my

fridge, only the potential. Similarly, the walls in my kitchen are not

“white” when I leave the room. They still have paint on them, of course,

but color only occurs when they interact with my eyes.

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Sound waves impinge on the eardrums and pinnae (the fleshy parts of

your ear), setting off a chain of mechanical and neurochemical events,

the end product of which is an internal mental image we call pitch. If a

tree falls in a forest and no one is there to hear it, does it make a sound?

(The question was first posed by the Irish philosopher George Berkeley.)

Simply, no—sound is a mental image created by the brain in response to

vibrating molecules. Similarly, there can be no pitch without a human or

animal present. A suitable measuring device can register the frequency

made by the tree falling, but truly it is not pitch unless and until it is

heard.

No animal can hear a pitch for every frequency that exists, just as the

colors that we actually see are a small portion of the entire electromag-

netic spectrum. Sound can theoretically be heard for vibrations from just

over 0 cycles per second up to 100,000 cycles per second or more, but

each animal hears only a subset of the possible sounds. Humans who are

not suffering from any kind of hearing loss can usually hear sounds from

20 Hz to 20,000 Hz. The pitches at the low end sound like an indistinct

rumble or shaking—this is the sound we hear when a truck goes by out-

side the window (its engine is creating sound around 20 Hz) or when a

tricked-out car with a fancy sound system has the subwoofers cranked

up really loud. Some frequencies—those below 20 Hz—are inaudible to

humans because the physiological properties of our ears aren’t sensitive

to them.

The range of human hearing is generally 20 Hz to 20,000 Hz, but this

doesn’t mean that the range of human pitch perception is the same; al-

though we can hear sounds in this entire range, they don’t all sound mu-

sical; that is, we can’t unambiguously assign a pitch to the entire range.

By analogy, colors at the infrared and ultraviolet ends of the spectrum

lack definition compared to the colors closer to the middle. The figure on

page 23 shows the range of musical instruments, and the frequency as-

sociated with them. The sound of the average male speaking voice is

around 110 Hz, and the average female speaking voice is around 220 Hz.

The hum of fluorescent lights or from faulty wiring is 60 Hz (in North

America; in Europe and countries with a different voltage/current stan-

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A27.500 30.863 32.703 36.708 41.203 43.654 48.999 55.000 61.735 65.406 73.416 82.407 87.307 97.999 110.00 123.47 130.81 146.83 164.81 174.61 196.00 220.00 246.94 261.63 293.66 329.63 349.23 392.00 440.00 493.88 523.55 587.33 659.26 698.46 783.99 880.00 987.77 1046.5 1174.7 1318.5 1396.9 1568.0 1760.0 1975.5 2093.0 2349.3 2637.0 2793.0 3136.0 3520.0 3951.1 4186.0

A A

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A-440

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dard, it can be 50 Hz). The sound that a singer hits when she causes a

glass to break might be 1000 Hz. The glass breaks because it, like all

physical objects, has a natural and inherent vibration frequency. You can

hear this by flicking your finger against its sides or, if it’s crystal, by run-

ning your wet finger around the rim of the glass in a circular motion.

When the singer hits just the right frequency—the resonant frequency of

the glass—it causes the molecules of the glass to vibrate at their natural

rate, and they vibrate themselves apart.

A standard piano has eighty-eight keys. Very rarely, pianos can have a

few extra ones at the bottom and electronic pianos, organs, and synthe-

sizers can have as few as twelve or twenty-four keys, but these are spe-

cial cases. The lowest note on a standard piano vibrates with a frequency

of 27.5 Hz. Interestingly, this is about the same rate of motion that con-

stitutes an important threshold in visual perception. A sequence of still

photographs—slides—displayed at or about this rate of presentation

will give the illusion of motion. “Motion pictures” are a sequence of still

images alternating with pieces of black film presented at a rate (one

forty-eighth of a second) that exceeds the temporal resolving properties

of the human visual system. We perceive smooth, continuous motion

when in fact there is no such thing actually being shown to us. When

molecules vibrate at around this speed we hear something that sounds

like a continuous tone. If you put playing cards in the spokes of your bi-

cycle wheel when you were a kid, you demonstrated to yourself a related

principle: At slow speeds, you simply hear the click-click-click of the

card hitting the spokes. But above a certain speed, the clicks run to-

gether and create a buzz, a tone you can actually hum along with; a pitch.

When this lowest note on the piano plays, and vibrates at 27.5 Hz, to

most people it lacks the distinct pitch of sounds toward the middle of the

keyboard. At the lowest and the highest ends of the piano keyboard, the

notes sound fuzzy to many people with respect to their pitch. Composers

know this, and they either use these notes or avoid them depending on

what they are trying to accomplish compositionally and emotionally.

Sounds with frequencies above the highest note on the piano keyboard,

around 6000 Hz and more, sound like a high-pitched whistling to most

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people. Above 20,000 Hz most humans don’t hear a thing, and by the age

of sixty, most adults can’t hear much above 15,000 Hz or so due to a stiff-

ening of the hair cells in the inner ear. So when we talk about the range

of musical notes, or that restricted part of the piano keyboard that con-

veys the strongest sense of pitch, we are talking about roughly three

quarters of the notes on the piano keyboard, between about 55 Hz and

2000 Hz.

Pitch is one of the primary means by which musical emotion is con-

veyed. Mood, excitement, calm, romance, and danger are signaled by a

number of factors, but pitch is among the most decisive. A single high

note can convey excitement, a single low note sadness. When notes are

strung together, we get more powerful and more nuanced musical state-

ments. Melodies are defined by the pattern or relation of successive

pitches across time; most people have no trouble recognizing a melody

that is played in a higher or lower key than they’ve heard it in before. In

fact, many melodies do not have a “correct” starting pitch, they just float

freely in space, starting anywhere. “Happy Birthday” is an example of

this. One way to think about a melody, then, is as an abstract prototype

that is derived from specific combinations of key, tempo, instrumenta-

tion, and so on. A melody is an auditory object that maintains its identity

in spite of transformations, just as a chair maintains its identity when

you move it to the other side of the room, turn it upside down, or paint it

red. So, for example, if you hear a song played louder than you are ac-

customed to, you still identify it as the same song. The same holds for

changes in the absolute pitch values of the song, which can be changed

so long as the relative distances between them remain the same.

The notion of relative pitch values is seen readily in the way that we

speak. When you ask someone a question, your voice naturally rises in

intonation at the end of the sentence, signaling that you are asking. But

you don’t try to make the rise in your voice match a specific pitch. It is

enough that you end the sentence somewhat higher in pitch than you be-

gan it. This is a convention in English (though not in all languages—we

have to learn it), and is known in linguistics as a prosodic cue. There are

similar conventions for music written in the Western tradition. Certain

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sequences of pitches evoke calm, others, excitement. The brain basis for

this is primarily based on learning, just as we learn that a rising intona-

tion indicates a question. All of us have the innate capacity to learn the

linguistic and musical distinctions of whatever culture we are born into,

and experience with the music of that culture shapes our neural path-

ways so that we ultimately internalize a set of rules common to that mu-

sical tradition.

Different instruments use different parts of the range of available

pitches. The piano has the largest range of any instrument, as you can

see from the previous illustration. The other instruments each use a sub-

set of the available pitches, and this influences the ways that instruments

are used to communicate emotion. The piccolo, with its high-pitched,

shrill, and birdlike sound, tends to evoke flighty, happy moods regardless

of the notes it’s playing. Because of this, composers tend to use the pic-

colo for happy music, or rousing music, as in a Sousa march. Similarly, in

Peter and the Wolf, Prokofiev uses the flute to represent the bird, and

the French horn to indicate the wolf. The characters’ individuality in

Peter and the Wolf is expressed in the timbres of different instruments

and each has a leitmotiv—an associated melodic phrase or figure that

accompanies the reappearance of an idea, person, or situation. (This is

especially true of Wagnerian music drama.) A composer who picks so-

called sad pitch sequences would only give these to the piccolo if he

were trying to be ironic. The lumbering, deep sounds of the tuba or

double bass are often used to evoke solemnity, gravity, or weight.

How many unique pitches are there? Because pitch comes from a

continuum—the vibration frequencies of molecules—there are techni-

cally an infinite number of pitches: For every pair of frequencies you

mention, I could always come up with one between them, and a theoret-

ically different pitch would exist. But not every change in frequency

gives rise to a noticeable difference in pitch, just as adding a grain of

sand to your backpack will not change the weight perceptibly. So not all

frequency changes are musically useful. People differ in their ability to

detect small changes in frequency; training can help, but generally

speaking, most cultures don’t use distances much smaller than a semi-

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tone as the basis for their music, and most people can’t reliably detect

changes smaller than about one tenth of a semitone.

The ability to detect differences in pitch is based on physiology, and

varies from one animal to another. The basilar membrane of the human

inner ear contains hair cells that are frequency selective, firing only in

response to a certain band of frequencies. These are stretched out

across the membrane from low frequencies to high; low-frequency

sounds excite hair cells on one end of the basilar membrane, medium

frequency sounds excite the hair cells in the middle, and high-frequency

sounds excite them at the other end. We can think of the membrane as

containing a map of different pitches very much like a piano keyboard

superimposed on it. Because the different tones are spread out across

the surface topography of the membrane, this is called a tonotopic map.

After sounds enter the ear, they pass by the basilar membrane, where

certain hair cells fire, depending on the frequency of the sounds. The

membrane acts like a motion-detector lamp you might have in your gar-

den; activity in a certain part of the membrane causes it to send an elec-

trical signal on up to the auditory cortex. The auditory cortex also has a

tonotopic map, with low to high tones stretched out across the cortical

surface. In this sense, the brain contains a “map” of different pitches, and

different areas of the brain respond to different pitches. Pitch is so im-

portant that the brain represents it directly; unlike almost any other mu-

sical attribute, we could place electrodes in the brain and be able to

determine what pitches were being played to a person just by looking at

the brain activity. And although music is based on pitch relations rather

than absolute pitch values, it is, paradoxically, these absolute pitch val-

ues that the brain is paying attention to throughout its different stages of

processing.

A scale is just a subset of the theoretically infinite number of pitches, and

every culture selects these based on historical tradition or somewhat ar-

bitrarily. The specific pitches chosen are then anointed as being part of

that musical system. These are the letters that you see in the figure

above. The names “A,” “B,” “C,” and so on are arbitrary labels that we as-

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sociate with particular frequencies. In Western music—music of the Eu-

ropean tradition—these pitches are the only “legal” pitches; most instru-

ments are designed to play these pitches and not others. (Instruments

like the trombone and cello are an exception, because they can slide be-

tween notes; trombonists, cellists, violinists, etc., spend a lot of time

learning how to hear and produce the precise frequencies required to

play each of the legal notes.) Sounds in between are considered mis-

takes (“out of tune”) unless they’re used for expressive intonation (in-

tentionally playing something out of tune, briefly, to add emotional

tension) or in passing from one legal tone to another.

Tuning refers to the precise relationship between the frequency of a

tone being played and a standard, or between two or more tones being

played together. Orchestral musicians “tuning up” before a performance

are synchronizing their instruments (which naturally drift in their tuning

as the wood, metal, strings, and other materials expand and contract

with changes in temperature and humidity) to a standard frequency, or

occasionally not to a standard but to each other. Expert musicians often

alter the frequency of tones while they’re playing for expressive pur-

poses (except, of course, on fixed-pitch instruments such as keyboards

and xylophones); sounding a note slightly lower or higher than its nomi-

nal value can impart emotion when done skillfully. Expert musicians

playing together in ensembles will also alter the pitch of tones they play

to bring them more in tune with the tones being played by the other mu-

sicians, should one or more musicians drift away from standard tuning

during the performance.

The note names in Western music run from A to G, or, in an alterna-

tive system, as Do - re - mi - fa - sol - la - ti - do (the alternate system is

used as lyrics to the Rodgers and Hammerstein song “Do-Re-Mi” from

The Sound of Music: “Do, a deer, a female deer, Re, a drop of golden

sun . . .”). As frequencies get higher, so do the letter names; B has a

higher frequency than A (and hence a higher pitch) and C has a higher

frequency than either A or B. After G, the note names start all over again

at A. Notes with the same name have frequencies that are multiples of

each other. One of the several notes we call A has a frequency of 55 Hz

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and all other notes called A have frequencies that are two, three, four,

five (or a half) times this frequency.

Here is a fundamental quality of music. Note names repeat because of

a perceptual phenomenon that corresponds to the doubling and halving

of frequencies. When we double or halve a frequency, we end up with a

note that sounds remarkably similar to the one we started out with. This

relationship, a frequency ratio of 2:1 or 1:2, is called the octave. It is so

important that, in spite of the large differences that exist between musi-

cal cultures—between Indian, Balinese, European, Middle Eastern, Chi-

nese, and so on—every culture we know of has the octave as the basis

for its music, even if it has little else in common with other musical tra-

ditions. This phenomenon leads to the notion of circularity in pitch per-

ception, and is similar to circularity in colors. Although red and violet

fall at opposite ends of the continuum of visible frequencies of electro-

magnetic energy, we see them as perceptually similar. The same is true

in music, and music is often described as having two dimensions, one

that accounts for tones going up in frequency (and sounding higher and

higher) and another that accounts for the perceptual sense that we’ve

come back home again each time we double a tone’s frequency.

When men and women speak in unison, their voices are normally an

octave apart, even if they try to speak the exact same pitches. Children

generally speak an octave or two higher than adults. The first two notes

of the Harold Arlen melody “Somewhere Over the Rainbow” (from the

movie The Wizard of Oz) make an octave. In “Hot Fun in the Summer-

time” by Sly and the Family Stone, Sly and his backup singers are singing

in octaves during the first line of the verse “End of the spring and here

she comes back.” As we increase frequencies by playing the successive

notes on an instrument, there is a very strong perceptual sense that

when we reach a doubling of frequency, we have come “home” again. The

octave is so basic that even some animal species—monkeys and cats, for

example—show octave equivalence, the ability to treat as similar, the

way that humans do, tones separated by this amount.

An interval is the distance between two tones. The octave in Western

music is subdivided into twelve (logarithmically) equally spaced tones.

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The intervallic distance between A and B (or between “do” and “re”) is

called a whole step or a tone. (This latter term is confusing, since we call

any musical sound a tone; I’ll use the term whole step to avoid ambiguity).

The smallest division in our Western scale system cuts a whole step per-

ceptually in half: This is the semitone, which is one twelfth of an octave.

Intervals are the basis of melody, much more so than the actual

pitches of notes; melody processing is relational, not absolute, meaning

that we define a melody by its intervals, not the actual notes used to cre-

ate them. Four semitones always create the interval known as a major

third regardless of whether the first note is an A or a G# or any other

note. Here is a table of the intervals as they’re known in our (Western)

musical system:

The table could continue on: Thirteen semitones is a minor ninth,

Distance in semitones Interval name

0 unison

1 minor second

2 major second

3 minor third

4 major third

5 perfect fourth

6 augmented fourth, diminished fifth, or tritone

7 perfect fifth

8 minor sixth

9 major sixth

10 minor seventh

11 major seventh

12 octave

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fourteen semitones is a major ninth, etc., but these names are typically

used only in more advanced discussions. The intervals of the perfect

fourth and perfect fifth are so called because they sound particularly

pleasing to many people, and since the ancient Greeks, this particular

feature of the scale is at the heart of all music. (There is no “imperfect

fifth,” this is just the name we give the interval.) Ignore the perfect fourth

and fifth or use them in every phrase, they have been the backbone of

music for at least five thousand years.

Although the areas of the brain that respond to individual pitches

have been mapped, we have not yet been able to find the neurological ba-

sis for the encoding of pitch relations; we know which part of the cortex

is involved in listening to the notes C and E, for example, and for F and

A, but we do not know how or why both intervals are perceived as a ma-

jor third, or the neural circuits that create this perceptual equivalency.

These relations must be extracted by computational processes in the

brain that remain poorly understood.

If there are twelve named notes within an octave, why are there only

seven letters (or do-re-mi syllables)? After centuries of being forced to

eat in the servants’ quarters and to use the back entrance of the castle,

this may just be an invention by musicians to make nonmusicians feel in-

adequate. The additional five notes have compound names, such as E♭ pronounced “E-flat”) and F# (pronounced “F-sharp”). There is no reason

for the system to be so complicated, but it is what we’re stuck with.

The system is a bit clearer looking at the piano keyboard. A piano has

white keys and black keys spaced out in an uneven arrangement—some-

times two white keys are adjacent, sometimes they have a black key

between them. Whether the keys are white or black, the perceptual dis-

tance from one adjacent key to the next always makes a semitone, and a

distance of two keys is always a whole step. This applies to many West-

ern instruments; the distance between one fret on a guitar and the next

is also a semitone, and pressing or lifting adjacent keys on woodwind in-

struments (such as the clarinet or oboe) typically changes the pitch by a

semitone.

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The white keys are named A, B, C, D, E, F, and G. The notes be-

tween—the black keys—are the ones with compound names. The note

between A and B is called either A-sharp or B-flat, and in all but formal

music theoretic discussions, the two terms are interchangeable. (In fact,

this note could also be referred to as C double-flat, and similarly, A could

be called G double-sharp, but this is an even more theoretical usage.)

Sharp means high, and flat means low. B-flat is the note one semitone

lower than B; A-sharp is the note one semitone higher than A. In the par-

allel do-re-mi system, unique syllables mark these other tones: di and ra

indicate the tone between do and re, for example.

The notes with compound names are not in any way second-class mu-

sical citizens. They are just as important, and in some songs and some

scales they are used exclusively. For example, the main accompaniment

to “Superstition” by Stevie Wonder is played on only the black keys of

the keyboard. The twelve tones taken together, plus their repeating

cousins one or more octaves apart, are the basic building blocks for

melody, for all the songs in our culture. Every song you know, from

“Deck the Halls” to “Hotel California,” from “Ba Ba Black Sheep” to the

theme from Sex and the City, is made up from a combination of these

twelve tones and their octaves.

To add to the confusion, musicians also use the terms sharp and flat

to indicate if someone is playing out of tune; if the musician plays the

tone a bit too high (but not so high as to make the next note in the scale)

we say that the tone being played is sharp, and if the musician plays the

tone too low we say that the tone is flat. Of course, a musician can be

only slightly off and nobody would notice. But when the musician is off

by a relatively large amount—say one quarter to one half the distance be-

tween the note she was trying to play and the next one—most of us can

usually detect this and it sounds off. This is especially apparent when

there is more than one instrument playing, and the out-of-tune tone we

are hearing clashes with in-tune tones being played simultaneously by

other musicians.

The names of pitches are associated with particular frequency values.

Our current system is called A440 because the note we call A that is in

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the middle of the piano keyboard has been fixed to have a frequency of

440 Hz. This is entirely arbitrary. We could fix A at any frequency, such as

439, 444, 424, or 314.159; different standards were used in the time of

Mozart than today. Some people claim that the precise frequencies affect

the overall sound of a musical piece and the sound of instruments. Led

Zeppelin often tuned their instruments away from the modern A440 stan-

dard to give their music an uncommon sound, and perhaps to link it with

the European children’s folk songs that inspired many of their composi-

tions. Many purists insist on hearing baroque music on period instru-

ments, both because the instruments have a different sound and because

they are designed to play the music in its original tuning standard, some-

thing that purists deem important.

We can fix pitches anywhere we want because what defines music is

a set of pitch relations. The specific frequencies for notes may be arbi-

trary, but the distance from one frequency to the next—and hence from

one note to the next in our musical system—isn’t at all arbitrary. Each

note in our musical system is equally spaced to our ears (but not neces-

sarily to the ears of other species). Although there is not an equal change

in cycles per second (Hz) as we climb from one note to the next, the dis-

tance between each note and the next sounds equal. How can this be?

The frequency of each note in our system is approximately 6 percent

more than the one before it. Our auditory system is sensitive both to rel-

ative changes and to proportional changes in sound. Thus, each increase

in frequency of 6 percent gives us the impression that we have increased

pitch by the same amount as we did last time.

The idea of proportional change is intuitive if you think about

weights. If you’re at a gym and you want to increase your weight lifting

of the barbells from 5 pounds to 50 pounds, adding 5 pounds each week

is not going to change the amount of weight you’re lifting in an equal

way. After a week of lifting 5 pounds, when you move to 10 you are dou-

bling the weight; the next week when you move to 15 you are adding 1.5

times as much weight as you had before. An equal spacing—to give your

muscles a similar increase of weight each week—would be to add a con-

stant percentage of the previous week’s weight each time you increase.

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For example, you might decide to add 50 percent each week, and so you

would then go from 5 pounds to 7.5, then to 11.25, then to 16.83, and so

on. The auditory system works the same way, and that is why our scale

is based on a proportion: Every tone is 6 percent higher than the previ-

ous one, and when we increase each step by 6 percent twelve times, we

end up having doubled our original frequency (the actual proportion is

the twelfth root of two = 1.059463 . . . ).

The twelve notes in our musical system are called the chromatic

scale. Any scale is simply a set of musical pitches that have been chosen

to be distinguishable from each other and to be used as the basis for con-

structing melodies.

In Western music we rarely use all the notes of chromatic scale in

composition; instead, we use a subset of seven (or less often, five) of

those twelve tones. Each of these subsets is itself a scale, and the type of

scale we use has a large impact on the overall sound of a melody, and its

emotional qualities. The most common subset of seven tones used in

Western music is called the major scale, or Ionian mode (reflecting its

ancient Greek origins). Like all scales, it can start on any of the twelve

notes, and what defines the major scale is the specific pattern or distance

relationship between each note and its successive note. In any major

scale, the pattern of intervals—pitch distances between successive keys—

is: whole step, whole step, half step, whole step, whole step, whole step,

half step.

Starting on C, the major scale notes are C - D - E - F - G - A - B - C, all

white notes on the piano keyboard. All other major scales require one or

more black notes to maintain the required whole step/half step pattern.

The starting pitch is also called the root of the scale.

The particular placement of the two half steps in the sequence of the

major is crucial; it is not only what defines the major scale and distin-

guishes it from other scales, but it is an important ingredient in musical

expectations. Experiments have shown that young children, as well as

adults, are better able to learn and memorize melodies that are drawn

from scales that contain unequal distances such as this. The presence of

the two half steps, and their particular positions, orient the experienced,

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acculturated listener to where we are in the scale. We are all experts in

knowing, when we hear a B in the key of C—that is, when the tones are

being drawn primary from the C major scale—that it is the seventh note

(or “degree”) of that scale, and that it is only a half step below the root,

even though most of us can’t name the notes, and may not even know

what a root or a scale degree is. We have assimilated the structure of this

and other scales through a lifetime of listening and passive (rather than

theoretically driven) exposure to the music. This knowledge is not in-

nate, but is gained through experience. By a similar token, we don’t need

to know anything about cosmology to have learned that the sun comes

up every morning and goes down at night—we have learned this se-

quence of events through largely passive exposure.

Different patterns of whole steps and half steps give rise to alterna-

tive scales, the most common of which (in our culture) is the minor

scale. There is one minor scale that, like the C major scale, uses only the

white notes of the piano keyboard: the A minor scale. The pitches for

that scale are A - B - C - D - E - F - G - A. (Because it uses the same set of

pitches, but in a different order, A minor is said to be the “relative minor

of the C major scale.”) The pattern of whole steps and half steps is

different from that of the major scale: whole–half–whole–whole–half–

whole–whole. Notice that the placement of the half steps is very differ-

ent than in the major scale; in the major scale, there is a half step just

before the root that “leads” to the root, and another half step just before

the fourth scale degree. In the minor scale, the half steps are before the

third scale degree and before the sixth. There is still a momentum when

we’re in this scale to return to the root, but the chords that create this

momentum have a clearly different sound and emotional trajectory.

Now you might well ask: If these two scales use exactly the same set

of pitches, how do I know which one I’m in? If a musician is playing the

white keys, how do I know if he is playing the A minor scale or the C ma-

jor scale? The answer is that—entirely without our conscious aware-

ness—our brains are keeping track of how many times particular notes

are sounded, where they appear in terms of strong versus weak beats,

and how long they last. A computational process in the brain makes an

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inference about the key we’re in based on these properties. This is an-

other example of something that most of us can do even without musical

training, and without what psychologists call declarative knowledge—

the ability to talk about it; but in spite of our lack of formal musical edu-

cation, we know what the composer intended to establish as the tonal

center, or key, of the piece, and we recognize when he brings us back

home to the tonic, or when he fails to do so. The simplest way to estab-

lish a key, then, is to play the root of the key many times, play it loud, and

play it long. And even if a composer thinks he is writing in C major, if he

has the musicians play the note A over and over again, play it loud and

play it long; if the composer starts the piece on an A and ends the piece

on an A, and moreover, if he avoids the use of C, the audience, musi-

cians, and music theorists are most probably going to decide that the

piece is in A minor, even if this was not his intent. In musical keys as in

speeding tickets, it is the observed action, not the intention, that counts.

For reasons that are largely cultural, we tend to associate major

scales with happy or triumphant emotions, and minor scales with sad or

defeated emotions. Some studies have suggested that the associations

might be innate, but the fact that these are not culturally universal indi-

cates that, at the very least, any innate tendency can be overcome by

exposure to specific cultural associations. Western music theory recog-

nizes three minor scales and each has a slightly different flavor. Blues

music generally uses a five note (pentatonic) scale that is a subset of the

minor scale, and Chinese music uses a different pentatonic scale. When

Tchaikovsky wants us to think of Arab or Chinese culture in the Nut-

cracker ballet, he chooses scales that are typical to their music, and

within just a few notes we are transported to the Orient. When Billie Hol-

iday wants to make a standard tune bluesy, she invokes the blues scale

and sings notes from a scale that we are not accustomed to hearing in

standard classical music.

Composers know these associations and use them intentionally. Our

brains know them, too, through a lifetime of exposure to musical idioms,

patterns, scales, lyrics, and the associations between them. Each time

we hear a musical pattern that is new to our ears, our brains try to make

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an association through whatever visual, auditory and other sensory cues

accompany it; we try to contextualize the new sounds, and eventually,

we create these memory links between a particular set of notes and a

particular place, time, or set of events. No one who has seen Hitchcock’s

Psycho can hear Bernard Hermann’s screeching violins without thinking

of the shower scene; anyone who has ever seen a Warner Bros. “Merrie

Melody” cartoon will think of a character sneakily climbing stairs when-

ever they hear plucked violins playing an ascending major scale. The

associations are so powerful—and the scales distinguishable enough—

that only a few notes are needed: The first three notes of David Bowie’s

“China Girl” or Mussorgsky’s “Great Gate of Kiev” (from Pictures at an

Exhibition) instantly convey a rich and foreign (to us) musical context.

Nearly all this variation in context and sound comes from different

ways of dividing up the octave and, in virtually every case we know of,

dividing it up into no more than twelve tones. Although it has been

claimed that Indian and Arab-Persian music use “microtuning”—scales

with intervals much smaller than a semitone—close analysis reveals that

their scales also rely on twelve or fewer tones and the others are simply

expressive variations, glissandos (continuous glides from one tone to

another), and momentary passing tones, similar to the American blues

tradition of sliding into a note for emotional purposes.

In any scale, a hierarchy of importance exists among scale tones;

some are more stable, structurally significant, or final sounding than oth-

ers, causing us to feel varying amounts of tension and resolution. In the

major scale, the most stable tone is the first degree, also called the tonic.

In other words, all other tones in the scale seem to point toward the

tonic, but they point with varying momentum. The tone that points most

strongly to the tonic is the seventh scale degree, B in a C major scale.

The tone that points least strongly to the tonic is the fifth scale degree, G

in the C major scale, and it points least strongly because it is perceived

as relatively stable; this is just another way of saying that we don’t feel

uneasy—unresolved—if a song ends on the fifth scale degree. Music the-

ory specifies this tonal hierarchy. Carol Krumhansl and her colleagues

performed a series of studies establishing that ordinary listeners have

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incorporated the principles of this hierarchy in their brains, through pas-

sive exposure to music and cultural norms. By asking people to rate how

well different tones seemed to fit with a scale she would play them, she

recovered from their subjective judgments the theoretical hierarchy.

A chord is simply a group of three or more notes played at the same

time. They are generally drawn from one of the commonly used scales,

and the three notes are chosen so that they convey information about

the scale they were taken from. A typical chord is built by playing the

first, third, and fifth notes of a scale together. Because the sequence of

whole steps and half steps is different for minor and major scales, the in-

terval sizes are different for chords taken in this way from the two dif-

ferent scales. If we build a chord starting on C and use the tones from the

C major scale, we use C, E, and G. If instead we use the C minor scale,

the first, third, and fifth notes are C, E-flat, and G. This difference in the

third degree, between E and E-flat, turns the chord itself from a major

chord into a minor chord. All of us, even without musical training, can

tell the difference between these two even if we don’t have the terminol-

ogy to name them; we hear the major chord as sounding happy and the

minor chord as sounding sad, or reflective, or even exotic. The most ba-

sic rock and country music songs use only major chords: “Johnny B.

Goode,” “Blowin’ in the Wind,” “Honky Tonk Women,” and “Mammas

Don’t Let Your Babies Grow Up to Be Cowboys,” for example.

Minor chords add complexity; in “Light My Fire” by the Doors, the

verses are played in minor chords (“You know that it would be un-

true . . .”) and then the chorus is played in major chords (“Come on baby,

light my fire”). In “Jolene,” Dolly Parton mixes minor and major chords

to give a melancholy sound. Pink Floyd’s “Sheep” (from the album Ani-

mals) uses only minor chords.

Like single notes in the scale, chords also fall along a hierarchy of sta-

bility, depending on context. Certain chord progressions are part of

every musical tradition, and even by the age of five, most children have

internalized rules about what chord progressions are legal, or typical of

their culture’s music; they can readily detect deviations from the stan-

dard sequences just as easily as we can detect when an English sentence

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is malformed, such as this one: “The pizza was too hot to sleep.” For

brains to accomplish this, networks of neurons must form abstract rep-

resentations of musical structure, and musical rules, something that they

do automatically and without our conscious awareness. Our brains are

maximally receptive—almost spongelike—when we’re young, hungrily

soaking up any and all sounds they can and incorporating them into the

very structure of our neural wiring. As we age, these neural circuits are

somewhat less pliable, and so it becomes more difficult to incorporate,

at a deep neural level, new musical systems, or even new linguistic

systems.

Now the story about pitch becomes a bit more complicated, and it’s all

the fault of physics. But this complication gives rise to the rich spectrum

of sounds we hear in different instruments. All natural objects in the

world have several modes of vibration. A piano string actually vibrates at

several different rates at once. The same thing is true of bells that we hit

with a hammer, drums that we hit with our hands, or flutes that we blow

air into: The air molecules vibrate at several rates simultaneously, not

just a single rate.

An analogy is the several types of motion of the earth that are simul-

taneously occurring. We know that the earth spins on its axis once every

twenty-four hours, that it travels around the sun once every 365.25 days,

and that the entire solar system is spinning along with the Milky Way

galaxy. Several types of motion, all occurring at once. Another analogy is

the many kinds of vibration that we often feel when riding a train. Imag-

ine that you’re sitting on a train in an outdoor station, with the engine off.

It’s windy, and you feel the car rock back and forth just a little bit. It does

so with a regularity that you can time with your handy stopwatch, and

you feel the train moving back and forth about twice a second. Next, the

engineer starts the engine, and you feel a different kind of vibration

through your seat (due to the oscillations of the motor—pistons and

crankshafts turning around at a certain speed). When the train starts

moving, you experience a third sensation, the bump the wheels make

every time they go over a track joint. Altogether, you will feel several dif-

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ferent kinds of vibrations, all of them likely to be at different rates, or fre-

quencies. When the train is moving, you are no doubt aware that there is

vibration. But it is very difficult, if not impossible, for you to determine

how many vibrations there are and what their rates are. Using specialized

measuring instruments, however, one might be able to figure this out.

When a sound is generated on a piano, flute, or any other instru-

ment—including percussion instruments like drums and cowbells—it

produces many modes of vibration occurring simultaneously. When you

listen to a single note played on an instrument, you’re actually hearing

many, many pitches at once, not a single pitch. Most of us are not aware

of this consciously, although some people can train themselves to hear

this. The one with the slowest vibration rate—the one lowest in pitch—

is referred to as the fundamental frequency, and the others are collec-

tively called overtones.

To recap, it is a property of objects in the world that they generally vi-

brate at several different frequencies at once. Surprisingly, these other

frequencies are often mathematically related to each other in a very sim-

ple way: as integer multiples of one another. So if you pluck a string and

its slowest vibration frequency is one hundred times per second, the

other vibration frequencies will be 2 x 100 (200 Hz), 3 x 100 Hz (300 Hz),

etc. If you blow into a flute or recorder and cause vibrations at 310 Hz,

additional vibrations will be occurring at twice, three times, four times,

etc., this rate: 620 Hz, 930 Hz, 1240 Hz, etc. When an instrument creates

energy at frequencies that are integer multiples such as this, we say that

the sound is harmonic, and we refer to the pattern of energy at different

frequencies as the overtone series. There is evidence that the brain re-

sponds to such harmonic sounds with synchronous neural firings—the

neurons in auditory cortex responding to each of the components of the

sound synchronize their firing rates with one another, creating a neural

basis for the coherence of these sounds.

The brain is so attuned to the overtone series that if we encounter a

sound that has all of the components except the fundamental, the brain

fills it in for us in a phenomenon called restoration of the missing fun-

damental. A sound composed of energy at 100 Hz, 200 Hz, 300 Hz, 400

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Hz, and 500 Hz is perceived as having a pitch of 100 Hz, its fundamental

frequency. But if we artificially create a sound with energy at 200 Hz, 300

Hz, 400 Hz, and 500 Hz (leaving off the fundamental), we still perceive it

as having a pitch of 100 Hz. We don’t perceive it as having a pitch of 200

Hz, because our brain “knows” that a normal, harmonic sound with a

pitch of 200 Hz would have an overtone series of 200 Hz, 400 Hz, 600 Hz,

800 Hz, etc. We can also fool the brain by playing sequences that deviate

from the overtone series such as this: 100 Hz, 210 Hz, 302 Hz, 405 Hz, etc.

In cases like these, the perceived pitch shifts away from 100 Hz in a com-

promise between what is presented and what a normal harmonic series

would imply.

When I was in graduate school, my advisor, Mike Posner, told me

about the work of a graduate student in biology, Petr Janata. Although he

hadn’t been raised in San Francisco like me, Petr had long bushy hair

that he wore in a ponytail, played jazz and rock piano, and dressed in tie-

dye: a true kindred spirit. Peter placed electrodes in the inferior collicu-

lus of the barn owl, part of its auditory system. Then, he played the owls

a version of Strauss’s “The Blue Danube Waltz” made up of tones from

which the fundamental frequency had been removed. Petr hypothesized

that if the missing fundamental is restored at early levels of auditory pro-

cessing, neurons in the owl’s inferior colliculus should fire at the rate of

the missing fundamental. This was exactly what he found. And because

the electrodes put out a small electrical signal with each firing—and be-

cause the firing rate is the same as a frequency of firing—Petr sent the

output of these electrodes to a small amplifier, and played back the

sound of the owl’s neurons through a loudspeaker. What he heard was

astonishing; the melody of “The Blue Danube Waltz” sang clearly from

the loudspeakers: ba da da da da, deet deet, deet deet. We were hearing

the firing rates of the neurons and they were identical to the frequency

of the missing fundamental. The overtone series had an instantiation not

just in the early levels of auditory processing, but in a completely differ-

ent species.

One could imagine an alien species that does not have ears, or that

doesn’t have the same internal experience of hearing that we do. But it

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would be difficult to imagine an advanced species that had no ability

whatsoever to sense vibrating objects. Where there is atmosphere there

are molecules that vibrate in response to movement. And knowing

whether something is generating noise or moving toward us or away

from us, even when we can’t see it (because it is dark, our eyes aren’t at-

tending to it, or we’re asleep) has a great survival value.

Because most physical objects cause molecules to vibrate in several

modes at once, and because for many, many objects the modes bear sim-

ple integer relations to one another, the overtone series is a fact-of-the-

world that we expect to find everywhere we look: in North America, in

Fiji, on Mars, and on the planets orbiting Antares. Any organism that

evolved in a world with vibrating objects is likely—given enough evolu-

tionary time—to have evolved a processing unit in the brain that incor-

porated these regularities of its world. Because pitch is a fundamental

cue to an object’s identity, we would expect to find tonotopic mappings

as we do in human auditory cortex, and synchronous neural firings for

tones that bear octave and other harmonic relations to one another; this

would help the brain (alien or terrestrial) to figure out that all these

tones probably originated from the same object.

The overtones are often referred to by numbers: The first overtone is

the first vibration frequency above the fundamental, the second over-

tone is the second vibration frequency above the fundamental, etc. Be-

cause physicists like to make the world confusing for the rest of us, there

is a parallel system of terminology called harmonics, and I think it was

designed to make undergraduates go crazy. In the lingo of harmonics,

the first harmonic is the fundamental frequency, the second harmonic is

equal to the first overtone, and so on. Not all instruments vibrate in

modes that are so neatly defined. Sometimes, as with the piano (because

it is a percussive instrument), the overtones can be close, but not exact,

multiples of the fundamental frequency, and this contributes to their

characteristic sound. Percussion instruments, chimes, and other objects—

depending on composition and shape—often have overtones that are

clearly not integer multiples of the fundamental, and these are called

partials or inharmonic overtones. Generally, instruments with inhar-

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monic overtones lack the clear sense of pitch that we associate with har-

monic instruments, and the cortical basis for this may relate to a lack of

synchronous neural firing. But they still do have a sense of pitch, and we

hear this most clearly when we can play inharmonic notes in succession.

Although you may not be able to hum along with the sound of a single

note played on a woodblock or a chime, we can play a recognizable

melody on a set of woodblocks or chimes because our brain focuses on

the changes in the overtones from one to another. This is essentially

what is happening when we hear people playing a song on their cheeks.

A flute, a violin, a trumpet, and a piano can all play the same tone—

that is, you can write a note on a musical score and each instrument will

play a tone with an identical fundamental frequency, and we will (tend

to) hear an identical pitch. But these instruments all sound very different

from one another.

This difference is timbre (pronounced TAM-ber), and it is the most

important and ecologically relevant feature of auditory events. The tim-

bre of a sound is the principal feature that distinguishes the growl of a

lion from the purr of a cat, the crack of thunder from the crash of ocean

waves, the voice of a friend from that of a bill collector one is trying to

dodge. Timbral discrimination is so acute in humans that most of us can

recognize hundreds of different voices. We can even tell whether some-

one close to us—our mother, our spouse—is happy or sad, healthy or

coming down with a cold, based on the timbre of that voice.

Timbre is a consequence of the overtones. Different materials have

different densities. A piece of metal will tend to sink to the bottom of a

pond; an identically sized and shaped piece of wood will float. Partly due

to density, and partly due to size and shape, different objects also make

different noises when you strike them with your hand, or gently tap them

with a hammer. Imagine the sound that you’d hear if you tap a hammer

(gently, please!) against a guitar—a hollow, wooden plunk sound. Or if

you tap a piece of metal, like a saxophone—a tinny plink. When you tap

these objects, the energy from the hammer causes the molecules within

them to vibrate, to dance at several different frequencies, frequencies

determined by the material the object is made out of, its size, and its

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shape. If the object is vibrating at, say, 100 Hz, 200 Hz, 300 Hz, 400 Hz,

etc., the intensity of vibration doesn’t have to be the same for each of

these harmonics, and in fact, typically, it is not.

When you hear a saxophone playing a tone with a fundamental fre-

quency of 220 Hz, you are actually hearing many tones, not just one. The

other tones you hear are integer multiples of the fundamental: 440, 660,

880, 1200, 1420, 1640, etc. These different tones—the overtones—have

different intensities, and so we hear them as having different loudnesses.

The particular pattern of loudnesses for these tones is distinctive of the

saxophone, and they are what give rise to its unique tonal color, its

unique sound—its timbre. A violin playing the same written note (220

Hz) will have overtones at the same frequencies, but the pattern of how

loud each one is with respect to the others will be different. Indeed, for

each instrument, there exists a unique pattern of overtones. For one in-

strument, the second overtone might be louder than in another, while the

fifth overtone might be softer. Virtually all of the tonal variation we

hear—the quality that gives a trumpet its trumpetiness and that gives a

piano its pianoness—comes from the unique way in which the loud-

nesses of the overtones are distributed.

Each instrument has its own overtone profile, which is like a finger-

print. It is a complicated pattern that we can use to identify the instru-

ment. Clarinets, for example, are characterized by having relatively high

amounts of energy in the odd harmonics—three times, five times, and

seven times the multiples of the fundamental frequency, etc. (This is a

consequence of their being a tube that is closed at one end and open at

the other.) Trumpets are characterized by having relatively even

amounts of energy in both the odd and the even harmonics (like the clar-

inet, the trumpet is also closed at one end and open at the other, but the

mouthpiece and bell are designed to smooth out the harmonic series). A

violin that is bowed in the center will yield mostly odd harmonics and ac-

cordingly can sound similar to a clarinet. But bowing one third of the

way down the instrument emphasizes the third harmonic and its multi-

ples: the sixth, the ninth, the twelfth, etc.

All trumpets have a timbral fingerprint, and it is readily distinguish-

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able from the timbral fingerprint for a violin, piano, or even the human

voice. To the trained ear, and to most musicians, there even exist differ-

ences among trumpets—all trumpets don’t sound alike, nor do all pianos

or all accordions. (Well, to me all accordions sound alike, and the sweet-

est, most enjoyable sound I can imagine is the sound they would make

burning in a giant bonfire.) What distinguishes one particular piano from

another is that their overtone profiles will differ slightly from each other,

but not, of course, as much as they will differ from the profile for a harp-

sichord, organ, or tuba. Master musicians can hear the difference be-

tween a Stradivarius violin and a Guarneri within one or two notes. I can

hear the difference between my 1956 Martin 000-18 acoustic guitar, my

1973 Martin D-18, and my 1996 Collings D2H very clearly; they sound like

different instruments, even though they are all acoustic guitars; I would

never confuse one with another. That is timbre.

Natural instruments—that is, acoustic instruments made out of real-

world materials such as metal and wood—tend to produce energy at sev-

eral frequencies at once because of the way the internal structure of

their molecules vibrates. Suppose that I invent an instrument that, unlike

any natural instruments we know of, produces energy at one, and only

one, frequency. Let’s call this hypothetical instrument a generator (be-

cause it can generate tones of specific frequencies). If I line up a bunch

of generators, I could set each one of them to play a specific frequency

corresponding to the overtone series for a particular instrument playing

a particular tone. I could have a bank of these generators making sounds

at 110, 220, 330, 440, 550, and 660 Hz, which would give the listener the

impression of a 110 Hz tone played by a musical instrument. Further-

more, I could control the amplitude of each of my generators and make

each of the tones play at a particular loudness, corresponding to the

overtone profile of a natural musical instrument. If I did that, the result-

ing bank of generators would approximate the sound of a clarinet, or

flute, or any other instrument I was trying to emulate.

Additive synthesis such as the above approach achieves a synthetic

version of a musical-instrument timbre by adding together elemental

sonic components of the sound. Many pipe organs, such as those found

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in churches, have a feature that will let you play around with this. On

most pipe organs you press a key (or a pedal), which sends a blast of air

through a metal pipe. The organ is constructed of hundreds of pipes of

different sizes, and each one produces a different pitch, corresponding

to its size, when air is shot through it; you can think of them as mechan-

ical flutes, in which the air is supplied by an electric motor rather than by

a person blowing. The sound that we associate with a church organ—its

particular timbre—is a function of there being energy at several different

frequencies at once, just as with other instruments. Each pipe of the or-

gan produces an overtone series, and when you press a key on the organ

keyboard, a column of air is blasted through more than one pipe at a

time, giving a very rich spectrum of sounds. These supplementary pipes,

in addition to the one that vibrates at the fundamental frequency of the

tone you’re trying to play, either produce tones that are integer multiples

of the fundamental frequency, or are closely related to it mathematically

and harmonically.

The organ player typically has control over which of these supple-

mentary pipes he wants to blow air through by pulling and pushing

levers, or drawbars, that direct the flow of air. Knowing that clarinets

have a lot of energy in the odd harmonics of the overtone series, a clever

organ player could simulate the sound of a clarinet by manipulating

drawbars in such a way as to re-create the overtone series of that instru-

ment. A little bit of 220 Hz here, a dash of 330 Hz, a dollop of 440 Hz, a

heaping helping of 550 Hz, and voilà!—you’ve cooked yourself up a rea-

sonable facsimile of an instrument.

Starting in the late 1950s, scientists began experimenting with build-

ing such synthesis capabilities into smaller, more compact electronic de-

vices, creating a family of new musical instruments known collectively

as synthesizers. By the 1960s, synthesizers could be heard on records by

the Beatles (on “Here Comes the Sun” and “Maxwell’s Silver Hammer”)

and Walter/Wendy Carlos (Switched-On Bach), followed by groups who

sculpted their sound around the synthesizer, such as Pink Floyd and

Emerson, Lake and Palmer.

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Many of these synthesizers used additive synthesis as I’ve described

it here, and later ones used more complex algorithms such as wave guide

synthesis (invented by Julius Smith at Stanford) and FM synthesis (in-

vented by John Chowning at Stanford). But merely copying the overtone

profile, while it can create a sound reminiscent of the actual instrument,

yields a rather pale copy. There is more to timbre than just the overtone

series. Researchers still argue about what this “more” is, but it is gener-

ally accepted that, in addition to the overtone profile, timbre is defined

by two other attributes that give rise to a perceptual difference from one

instrument to another: attack and flux.

Stanford University sits on a bucolic stretch of land just south of San

Francisco and east of the Pacific Ocean. Rolling hills covered with pas-

tureland lie to the west, and the fertile Central Valley of California is just

an hour or so to the east, home of a large proportion of the world’s

raisins, cotton, oranges, and almonds. To the south, near the town of

Gilroy, are vast fields of garlic. Also to the south is Castroville, known as

the “artichoke capitol of the world.” (I once suggested to the Castroville

Chamber of Commerce that they change capitol to heart. The response

was not enthusiastic.)

Stanford has become something of a second home for computer sci-

entists and engineers who love music. John Chowning, who was well

known as an avant-garde composer, has had a professorship in the mu-

sic department there since the 1970s, and was among a group of pio-

neering composers at the time who were using the computer to create,

store, and reproduce sounds in their compositions. Chowning later be-

came the founding director of the Center for Computer Research in Mu-

sic and Acoustics at Stanford, known as CCRMA (pronounced CAR-ma;

insiders joke that the first c is silent). Chowning is warm and friendly.

When I was an undergraduate at Stanford, he would put his hand on my

shoulder and ask what I was working on. You got the feeling talking to a

student was for him an opportunity to learn something. In the early

1970s, while fiddling with the computer and with sine waves—the sorts

of artificial sounds that are made by computers and used as the building

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blocks of additive synthesis—Chowning noticed that changing the fre-

quency of these waves as they were playing created sounds that were

musical. By controlling these parameters just so, he was able to simulate

the sounds of a number of musical instruments. This new technique be-

came known as frequency modulation synthesis, or FM synthesis, and

became embedded first in the Yamaha DX9 and DX7 line of synthesizers,

which revolutionized the music industry from the moment of their intro-

duction in 1983. FM synthesis democratized music synthesis. Before FM,

synthesizers were expensive, clunky, and hard to control. Creating new

sounds took a great deal of time, experimentation, and know-how. But

with FM, any musician could obtain a convincing instrumental sound

at the touch of a button. Songwriters and composers who could not af-

ford to hire a horn section or an orchestra could now play around with

these textures and sounds. Composers and orchestrators could test out

arrangements before taking the time of an entire orchestra to see what

worked and what didn’t. New Wave bands like the Cars and the Pre-

tenders, as well as mainstream artists like Stevie Wonder, Hall and

Oates, and Phil Collins, started to use FM synthesis widely in their

recordings. A lot of what we think of as “the eighties sound” in popular

music owes its distinctiveness to the particular sound of FM synthesis.

With the popularization of FM came a steady stream of royalty income

that allowed Chowning to build up CCRMA, attracting graduate students

and top-flight faculty members. Among the first of many famous elec-

tronic music/music-psychology celebrities to come to CCRMA were

John R. Pierce and Max Mathews. Pierce had been the vice president of

research at the Bell Telephone Laboratories in New Jersey, and super-

vised the team of engineers who built and patented the transistor—and it

was Pierce who named the new device (TRANSfer resISTOR). In his dis-

tinguished career, he also is credited with inventing the traveling wave

vacuum tube, and launching the first telecommunications satellite, Tel-

star. He was also a respected science fiction writer under the pseudonym

J. J. Coupling. Pierce created a rare environment in any industry or re-

search lab, one in which the scientists felt empowered to do their best

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and in which creativity was highly valued. At the time, the Bell Telephone

Company/AT&T had a complete monopoly on telephone service in the

U.S. and a large cash reserve. Their laboratory was something of a play-

ground for the very best and brightest inventors, engineers, and scientists

in America. In the Bell Labs “sandbox,” Pierce allowed his people to be

creative without worrying about the bottom line or the applicability of

their ideas to commerce. Pierce understood that the only way true inno-

vation can occur is when people don’t have to censor themselves and can

let their ideas run free. Although only a small proportion of those ideas

may be practical, and a smaller proportion still would become products,

those that did would be innovative, unique, and potentially very prof-

itable. Out of this environment came a number of innovations including

lasers, digital computers, and the Unix operating system.

I first met Pierce in 1990 when he was already eighty and was giving

lectures on psychoacoustics at CCRMA. Several years later, after I had

earned my Ph.D. and moved back to Stanford, we became friends and

would go out to dinner every Wednesday night and discuss research. He

once asked me to explain rock and roll music to him, something he had

never paid any attention to and didn’t understand. He knew about my

previous career in the music business, and he asked if I could come over

for dinner one night and play six songs that captured all that was impor-

tant to know about rock and roll. Six songs to capture all of rock and

roll? I wasn’t sure I could come up with six songs to capture the Beatles,

let alone all of rock and roll. The night before he called to tell me that he

had heard Elvis Presley, so I didn’t need to cover that.

Here’s what I brought to dinner:

1) “Long Tall Sally,” Little Richard

2) “Roll Over Beethoven,” the Beatles

3) “All Along the Watchtower,” Jimi Hendrix

4) “Wonderful Tonight,” Eric Clapton

5) “Little Red Corvette,” Prince

6) “Anarchy in the U.K.,” the Sex Pistols

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A couple of the choices combined great songwriters with different

performers. All are great songs, but even now I’d like to make some ad-

justments. Pierce listened and kept asking who these people were, what

instruments he was hearing, and how they came to sound the way they

did. Mostly, he said that he liked the timbres of the music. The songs

themselves and the rhythms didn’t interest him that much, but he found

the timbres to be remarkable—new, unfamiliar, and exciting. The fluid

romanticism of Clapton’s guitar solo in “Wonderful Tonight,” combined

with the soft, pillowy drums. The sheer power and density of the Sex Pis-

tols’ brick-wall-of-guitars-and-bass-and-drums. The sound of a distorted

electric guitar wasn’t all that was new to Pierce. The ways in which in-

struments were combined to create a unified whole—bass, drums, elec-

tric and acoustic guitars, and voice—that was something he had never

heard before. Timbre was what defined rock for Pierce. And it was a rev-

elation to both of us.

The pitches that we use in music—the scales—have remained essen-

tially unchanged since the time of the Greeks, with the exception of the

development—really a refinement—of the equal tempered scale during

the time of Bach. Rock and roll may be the final step in a millennium-long

musical revolution that gave perfect fourths and fifths a prominence in

music that had historically been been given only to the octave. During

this time, Western music was largely dominated by pitch. For the past

two hundred years or so, timbre has become increasingly important. A

standard component of music across all genres is to restate a melody us-

ing different instruments—from Beethoven’s Fifth and Ravel’s “Bolero”

to the Beatles’ “Michelle” and George Strait’s “All My Ex’s Live in Texas.”

New musical instruments have been invented so that composers might

have a larger palette of timbral colors from which to draw. When a coun-

try or popular singer stops singing and another instrument takes up the

melody—even without changing it in any way—we find pleasurable the

repetition of the same melody with a different timbre.

The avant-garde composer Pierre Schaeffer (pronounced Sheh-FEHR,

using your best imitation of a French accent) performed some crucial

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experiments in the 1950s that demonstrated an important attribute of

timbre in his famous “cut bell” experiments. Schaeffer recorded a num-

ber of orchestral instruments on tape. Then, using a razor blade, he cut

the beginnings off of these sounds. This very first part of a musical in-

strument sound is called the attack; this is the sound of the initial hit,

strum, bowing, or blowing that causes the instrument to make sound.

The gesture our body makes in order to create sound from an instru-

ment has an important influence on the sound the instrument makes. But

most of that dies away after the first few seconds. Nearly all of the ges-

tures we make to produce a sound are impulsive—they involve short,

punctuated bursts of activity. In percussion instruments, the musician

typically does not remain in contact with the instrument after this initial

burst. In wind instruments and bowed instruments, on the other hand,

the musician continues to be in contact with the instrument after the ini-

tial impulsive contact—the moment when the air burst first leaves her

mouth or the bow first contacts the string; the continued blowing and

bowing has a smooth, continuous, and less impulsive quality.

The introduction of energy to an instrument—the attack phase—

usually creates energy at many different frequencies that are not related

to one another by simple integer multiples. In other words, for the brief

period after we strike, blow into, pluck, or otherwise cause an instru-

ment to start making sound, the impact itself has a rather noisy quality

that is not especially musical—more like the sound of a hammer hitting

a piece of wood, say, than like a hammer hitting a bell or a piano string,

or like the sound of wind rushing through a tube. Following the attack is

a more stable phase in which the musical tone takes on the orderly pat-

tern of overtone frequencies as the metal or wood (or other material)

that the instrument is made out of starts to resonate. This middle part of

a musical tone is referred to as the steady state—in most instances the

overtone profile is relatively stable while the sound emanates from the

instrument during this time.

After Schaeffer edited out the attack of orchestral instrument record-

ings, he played back the tape and found that it was nearly impossible for

most people to identify the instrument that was playing. Without the at-

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tack, pianos and bells sounded remarkably unlike pianos and bells, and

remarkably similar to one another. If you splice the attack of one instru-

ment onto the steady state, or body, from another, you get varied results:

In some cases, you hear an ambiguous hybrid instrument that sounds

more like the instrument that the attack came from than the one the

steady state came from. Michelle Castellengo and others have discovered

that you can create entirely new instruments this way; for example, splic-

ing a violin bow sound onto a flute tone creates a sound that strongly re-

sembles a hurdy-gurdy street organ. These experiments showed the

importance of the attack.

The third dimension of timbre—flux—refers to how the sound

changes after it has started playing. A cymbal or gong has a lot of flux—

its sound changes dramatically over the time course of its sound—while

a trumpet has less flux—its tone is more stable as it evolves. Also, in-

struments don’t sound the same across their range. That is, the timbre of

an instrument sounds different when playing high and low notes. When

Sting reaches up toward the top of his vocal range in “Roxanne” (by The

Police), his straining, reedy voice conveys a type of emotion that he can’t

achieve in the lower parts of his register, such as we hear on the opening

verse of “Every Breath You Take,” a more deliberate, longing sound. The

high part of Sting’s register pleads with us urgently as his vocal cords

strain, the low part suggests a dull aching that we feel has been going on

for a long time, but has not yet reached the breaking point.

Timbre is more than the different sounds that instruments make.

Composers use timbre as a compositional tool; they choose musical in-

struments—and combinations of musical instruments—to express par-

ticular emotions, and to convey a sense of atmosphere or mood. There is

the almost comical timbre of the bassoon in Tchaikovsky’s Nutcracker

Suite as it opens the “Chinese Dance,” and the sensuousness of Stan

Getz’s saxophone on “Here’s That Rainy Day.” Substitute a piano for the

electric guitars in the Rolling Stones’ “Satisfaction” and you’d have an

entirely different animal. Ravel used timbre as a compositional device in

Bolero, repeating the main theme over and over again with different tim-

bres; he did this after he suffered brain damage that impaired his ability

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to hear pitch. When we think of Jimi Hendrix, it is the timbre of his elec-

tric guitars and his voice that we are likely to recall the most vividly.

Composers such as Scriabin and Ravel talk about their works as

sound paintings, in which the notes and melodies are the equivalent of

shape and form, and the timbre is equivalent to the use of color and

shading. Several popular songwriters—Stevie Wonder, Paul Simon, and

Lindsey Buckingham—have described their compositions as sound

paintings, with timbre playing a role equivalent to the one that color does

in visual art, separating melodic shapes from one another. But one of the

things that makes music different from painting is that it is dynamic,

changing across time, and what moves the music forward are rhythm

and meter. Rhythm and meter are the engine driving virtually all music,

and it is likely that they were the very first elements used by our ances-

tors to make protomusics, a tradition we still hear today in tribal drum-

ming, and in the rituals of various preindustrial cultures. While I believe

timbre is now at the center of our appreciation of music, rhythm has held

supreme power over listeners for much longer.

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