The top row shows the fundamental frequency, the one you hear as the pitch — say it’s a violin string playing A 440. The second row shows the first harmonic, the string vibrating in halves, producing A 880. The harmonic is quieter than the fundamental, so you aren’t necessarily conscious of it, but you can isolate it by lightly touching the string at its halfway point while playing. The other rows show other harmonics, vibrations of the string in integer ratios, each producing a pitch that’s an integer multiple of the fundamental frequency. The second harmonic is E 1320; the third is A 1760; the fourth is C# 2200.

In an ideal string, the harmonics would continue to get infinitely higher, beyond the range of your hearing. As the harmonics get higher, they also get quieter and subtler. Still, they all have an impact on the overall sound of the instrument. All musical instruments have overtones: winds, the human throat, speaker cones, even well-tuned drumheads.

Read more about how harmonics form the basis of western music theory.

Original post on Quora

The picture shows a stylized nucleus with red protons and blue neutrons, surrounded by three grey electrons. It’s an attractive and iconic image. It makes a nice logo. Unfortunately, it’s also totally wrong. There’s an extent to which subatomic particles are like little marbles, but it’s a limited extent. Electrons do move around the nucleus, but they don’t do it in elliptical paths as if they’re little moons orbiting a planet. The true nature of electrons in atoms is way weirder and cooler.

Pictures are a terrible way to understand the nature of quantum particles. Music theory is much better.

Quantum particles are waves

The problem with textbook images like the one above is that they mislead you into thinking of particles as “things.” Particles aren’t things. They pop in and out of being in a rapid, flickery way that’s more like the way we think of energy. What we call “particles” are really just knots or bundles of energy fields.

Protons and electrons pull on each other the way refrigerators and magnets do. If electrons really were like little moons orbiting a planet, it seems like they could orbit at any distance, and could easily fall into the nucleus to collide with the protons. And yet, this never happens. Electrons always organize themselves into very specific spatial arrangements around the nucleus. This fact was totally mysterious until scientists started conceiving of electrons as probability waves in an energy field.

You can get a good idea of how particles really behave by looking at television static, which consists of huge numbers of electrons being fired at the screen at random. Now try to imagine “static” surrounding the nucleus of an atom, and you’ll get a much better picture of what’s going on than you get from imagining moons orbiting a planet.

When electrons are in orbit around an atom or molecule, their pattern of static isn’t random the way it is in TV static. When electrons orbit atoms, their energy fields are organized into patterns of overlapping ripples. You can explore these patterns with Paul Falstad’s interactive visualizations of the subatomic world — scroll down to the Quantum Mechanics sections for his simulated hydrogen atom. The colorful blobs show the probability of electrons being found in a particular place.

So what does this have to do with music theory? The electron field’s vibrations around an atom behave like harmonic oscillators. Electrons have harmonics, just like guitar strings do. Electron harmonics are three-dimensional instead of the one-dimensional harmonics of strings, but the underlying math is the same. These harmonics determine the arrangement and interactions of the electron wave, the same way that harmonics of a string form the basis of chords and scales. The electron field’s harmonics are called orbitals.

The physical world is made of electron harmonics

This screenshot of Falstad’s quantum harmonic oscillator applet shows the first harmonic of the electron field around an H2 molecule, two hydrogen atoms, each with one proton and one electron. This is the electron equivalent of the twelfth fret harmonic on a guitar string.

The blue blob represents the position of one electron, and the red blob is the other. At higher energy levels, the orbitals take on more complex shapes. There’s a direct analogy here to the more complex musical intervals that come from the higher harmonics in a guitar string.

You can think of the orbitals as a structure of cubbyholes, each of which can be occupied by one electron. The cubbyholes come in pairs, and electrons “prefer” to live in filled pairs of cubbyholes. All of the structure of objects and chemistry in the world arises from the way that atoms’ outer orbitals interact. If an atom’s outermost cubbyholes are unfilled, electrons from other atoms with unfilled orbitals can fill them, locking the atoms together into molecules. All solid and liquid materials are held together by this sharing of electrons between orbitals.

Here’s the molecular structure of ice, as rendered by Masakazu Matsumoto. The red balls are oxygen atoms. The blue ones are hydrogen atoms. The yellow rods represent the bonds caused by electrons shared between the oxygen and hydrogen atoms’ outermost orbitals.

The “sixness” of ice’s structure emerges from the way that hydrogen and oxygen orbitals combine to make open slots in groups of six. You can see the “sixness” repeated up at the macroscopic scale in the shape of snowflakes and frost.

If you raise the ice’s temperature to the melting point, what you’re really doing is shooting photons at the ice, knocking the electrons out of their orbitals so they can skip more freely from atom to atom. The atoms still stick together, but not as tightly, and not in so rigid an arrangement:

If you zap even more photons into the water, you can sever the bonds between the molecules completely, freeing them to bounce around independently in the state we perceive as steam. If you zap even more photons at the steam, you can rip the molecules apart and tear the electrons from the nuclei to form plasma. Even more energy will rip the nuclei into protons and neutrons, and ridiculously more energy will rip the protons and neutrons into their constituent up and down quarks. The quarks, protons, neutrons, nuclei, atoms and molecules are all vibrating energy fields with waveforms and harmonics of their own.

Whenever I’m bored, I like to try to imagine everything around me, all the matter and energy, as resonating energy fields, cohering the way pitches cohere into chords. Who says science isn’t fun?

Teaching science with music

Albert Einstein told interviewers that he often “thought in terms of musical architectures.” Einstein was an enthusiastic amateur violinist, and an early architect of quantum mechanics. These two facts are probably related.

Did Einstein make an explicit connection between musical harmonics and quantum harmonics? Maybe we’ll never know, but the connection exists, and future scientists can benefit from it. The concept of electron orbitals is really hard. When I was in high school, my (excellent) chemistry teacher told us not to even bother trying to visualize the true nature of electrons. She was right to not try to condescend to us or mislead us, but she gave up too easily. True, she didn’t have cool interactive computer visualizations, but the school did have a great music department. If I ever get a chance to teach chemistry, first I’m going to make sure the kids get some hands-on experience with harmonics. I’ll have them experience the way that it takes more energy to produce higher harmonics, and the way those higher harmonics produce more complex musical sounds. Then we’ll go back to chemistry class and I’ll bet the kids will have an easier time.

To understand tuning, it helps to start with the concept of the octave. Two pitches are an octave apart if their frequencies have the ratio 2:1. Standard concert A has a frequency of 440 Hz. When you play concert A on the guitar, the string vibrates to and fro 440 times every second. If you double the frequency to 880 Hz, you get an A that’s one octave higher. If you halve the frequency to 220 Hz, you get an A that’s an octave lower. The ear hears all these different pitches as being the “same” note. (Technically, they’re the same pitch class.) This ability we have to hear frequencies related by powers of two as being the “same” is known in music theory terms as octave equivalency. This ability isn’t specific to humans. Rhesus monkeys hear octaves as being equivalent too.

Octaves emerge naturally out of the overtone series. The first harmonic of a vibrating string is an octave above the fundamental. The third harmonic is two octaves above. The seventh harmonic is three octaves above, and the fifteenth harmonic is four octaves above.

After the octave, the next musical interval you get from the natural overtone series is the fifth (it’s the third harmonic.) Two pitches are a fifth apart if the ratio between their frequencies is 3:2. The note a fifth above concert A (440 Hz) is E (660 Hz.)

Fifths are a very significant interval in western music theory. If you keep going up by fifths, you visit every note in the chromatic scale (every key on the piano) until you eventually wind up back on the note where you began. So if you start on A, then go up to E, then B, then F#, and so on, eventually you’ll wind up on the A seven octaves higher from where you started. This concept is known as the circle of fifths, though it would be more accurate to call it the spiral of fifths, since you’re getting higher and higher in pitch.

The circle of fifths is foundational to western music theory. It makes it possible to transpose music effortlessly from one key to another. The circle gives rise to all sorts of useful and interesting symmetries, too, like its close relationship to the circle of semitones.

But there’s a big problem with the circle of fifths. If you use the 3/2 ratio you get from the natural overtone series, the circle doesn’t actually close. Recall that to go up by a fifth, you multiply the frequency by 3/2. To keep going up by fifths, you keep multiplying by 3/2. To go all the way around the circle of fifths from A to A, you multiply by 3/2 twelve times:

(3/2)^12 = 129.746337890625

Going around the circle of fifths twelve times is the same as going up seven octaves. To go up an octave, you multiply by two, so to go up seven octaves, you multiply by two seven times:

 2^7 = 128

Going from A to A by fifths means multiplying the frequency by 129.746337890625, but going by octaves means multiplying by 128. The discrepancy between the two multiples is known in music theory terms as the Pythagorean comma, and it has caused musicians a lot of gray hair over the past few hundred years. It would be nice if a tuning system based on fifths agreed with a system based on octaves. It would make it a lot easier to hop from one key to another without having to retune your instruments. But that is sadly not possible.

The history of western tuning systems is the story of musicians trying to resolve the contradiction between the desire to have pure-sounding overtone-based intervals and a closed circle of fifths. European musicians of the 1700s tried all kinds of compromises. You could have some of the keys sound perfectly in tune, and have others be out of tune. You could have eleven decent-sounding keys and one awful one. You could use perfect fifths and smooth out the Pythagorean comma with out-of-tune thirds. You could have pianos with many extra keys for all the subtly different versions of each note.

I don’t advise getting too bogged down in the minutiae of all these different systems. The bottom line is that the western world eventually settled on its present consensus solution, which is to just make all the intervals other than octaves a little bit wrong. This system is called equal temperament. It’s considered a “modern” idea, but it dates back at least as far as Galileo’s father Vincenzo Galilei.

In equal temperament, all intervals are built by adding semitones together, and all semitones are defined as a ratio of one to the twelfth root of two. Twelve half steps gets you the perfect octave, because multiplying by the twelfth root of two twelve times equals two. An equal-tempered fifth is seven semitones –  you multiply the frequency by 2^(7/12). This comes to about 1.4983, which isn’t quite the 3/2 from the overtone system that your ear would like, but it’s close enough to not be offensively awful-sounding. The other equal-tempered intervals are similarly “wrong,” but by similarly bearable small amounts. Every key is identical and the circle of fifths closes, so everybody is more or less happy. If you get an electronic guitar tuner, it’ll be based on equal temperament.

Some musicians lament the loss of pure fifths. One bassist I know claims that all those out-of-tune fifths are gradually making western listeners crazy, which is why we’ve had so many enormous and horrible wars in the past couple of centuries. This idea sounds silly to me, but it’s true that pure fifths are easier on the brain. On instruments where the tuning is flexible, like winds and violin, the most skilled musicians tend to seek out pure intervals by ear, adjusting their intonation slightly depending on the key. Good singers do this too. Electronic instruments are a lot easier to retune than acoustic ones, and it’s sometimes possible to program in whatever tuning system suits you. Auto-tune lets you choose any historical or microtonal tuning system you want right off a menu.

So what if you’re just trying to get your guitar in tune? You need to make peace with not being able to do it perfectly. Use an electronic tuner to get the individual strings to their correct equal-tempered pitches and deal with the fifths sounding a little wrong, or tune with harmonics and have the low register not quite match the high register. In practice, most guitarists just fudge a little bit one way or the other, and guitars rarely stay tuned the way you want them to anyway. As always, let your ear be your guide.

The electric and acoustic guitar are superficially similar, but they produce sound in totally different ways. Acoustic guitars make sound from vibrations of the body, driven by the vibrating bridge, which is in turn driven by the vibrating strings. The player controls the body’s vibrations by plucking and strumming the strings. All the power of the vibrations has to come from the player’s hands.

Electric guitars generate a little sound from their bodies directly, but it’s almost inaudible. The sound you’re hearing mostly comes from the speaker cone in the amplifier, driven by current from the wall. This current is controlled by a much weaker current originating in the guitar’s magnetic pickups. As the metal strings vibrate, they agitate the pickups’ electromagnetic field, sending a fluctuating current down the cable and into the amp circuitry. Good amps respond dramatically to very subtle touches on the electric guitar’s strings that would be inaudible on an acoustic instrument.

Jimi Hendrix was one of the first guitarists to think of his instrument as a way to modulate an electrical signal first and foremost. He didn’t just pluck and strum the strings; he scraped them and swatted them and played with their tension. And he produced his most distinctive sounds by letting the amp itself vibrate his guitar’s pickups. All of these techniques are at work in his iconic performance of “The Star-Spangled Banner” at Woodstock:

Jimi’s guitar is feeding back and heavily distorted. He also throws in a little wah-wah pedal. The time is free, or as the classical musicians say, rubato – drums don’t always need to keep a steady beat. Jimi interlaces the melody notes with inharmonic screams and yowls, produced by scraping the pick against the string’s winding. He throws in a few unresolved tritones at 1:35 and some terrifying divebombing sounds at 2:00. At around 2:30 he quotes part of “Taps.” This performance has been criticized as anti-American, but Jimi said in interviews that he considered himself to be patriotic.

The first generations of electric guitarists considered feedback to be bad, a technical mishap to be avoided. Jimi discovered ways to use it as a musical expression in its own right. If the amp is loud enough, its sound can physically shake a guitar’s pickups enough to produce a current. That current gets sent to the amp, which then vibrates the pickups harder, which sends even more current to the amp, which produces even more sound. This feedback loop builds rapidly, getting louder and louder. Every beginner electric guitarist discovers feedback accidentally by leaning their guitar against their amp without turning the volume down. Feedback can also be seeded spontaneously by the slight hum produced by any electrical system that uses alternating current, or by radio waves. Cheap, poorly shielded pickups and cables make great radio antennas. I used to live on Roosevelt Island, right across the East River from a Con Ed power plant with a whole bunch of big transformers. If I didn’t face due north while playing electric guitar, I picked up all kinds of radio signals and other electromagnetic noise. It was nice for experimental music, but not so great for producing a clean sound.

Feedback is more likely, and a lot louder, when the guitar is overdriven, its signal boosted and compressed to bring out and sustain overtones that are normally inaudible. Feedback has a mystical quality, an evolving life of its own. It’s unpredictable and hard to control exactly. It would be pointless to try to score a feedback composition because there are too many variables at work; it’s an intrinsically improvisational medium. The results can be annoying or boring, or they can be transcendant. You can experience the visual equivalent by pointing a video camera at the monitor showing its own output.

Jimi was also a pioneer in his exploration of the electric guitar’s microtonal possibilities. The conventional way to control a guitar string’s pitch is to press it against the frets, changing its length. You can also bend the strings, changing both their length and tension, for more nuanced pitch intervals. Jimi’s guitars have an additional pitch control, the whammy bar, which lifts the bridge, allowing very precise control of all six strings’ tension simultaneously. The whammy bar lets you play arresting microtonal chords effortlessly. It also quickly pulls the strings out of tune, which is why in the video Jimi is continually adjusting the tuning pegs whenever his left hand is free.

The video cuts out before this point, but at Woodstock Jimi segued from “The Star Spangled Banner” into “Purple Haze.” The song is based around a distinctive chord that has come to be nicknamed the Hendrix chord.

This chord is easy to play – any beginner could learn it – but intellectually it’s extremely intense. It contains every possible interval in the western tuning system (or implies them, I count the inversions too.)

Jimi didn’t invent the Hendrix chord. It had been a distinctive device in blues and jazz since before he was born. But where Duke Ellington and Thelonious Monk used the Hendrix chord for accents and embellishments, Hendrix pushed it front and center, using it as a cornerstone for songs like Purple Haze and Foxy Lady (in different keys than the one written here.)

The electric guitar doesn’t just offer a lot of the tonal and harmonic freedom. It also leaves the player’s mouth and feet free for more expression. You can use your feet to dance, or to control stomp boxes and expression pedals. Your voice is free for singing and talking. The electric guitar is some seriously advanced interface design.

Here’s a remix/cover of “Purple Haze” by my band Revival Revival, combining Jimi with Missy Elliot, M.I.A. and Miles Davis. Enjoy:

mp3 download, ipod format download

]]> http://www.ethanhein.com/wp/2009/jimi-hendrix-electronic-musician/feed/ 5 http://www.ethanhein.com/wp/2009/wow-chicka-wah-wah/ http://www.ethanhein.com/wp/2009/wow-chicka-wah-wah/#comments Sun, 28 Jun 2009 02:33:12 +0000 Ethan http://www.ethanhein.com/wp/?p=968 Say “oooh” as in “noodle.” Then say “aaah” as in “park.” When you say “oooh” your mouth is more closed, with less resonating space and a smaller opening. This configuration blocks the higher overtones of your voice. When you say “aaah” your jaw and lips open, creating more resonating space and letting more high overtones through. Now glide from one to the other. The resulting “ooohaaaah” is the sound the wah-wah pedal is named for. By selectively filtering an electronic instrument’s overtones, the pedal can make it sound more vocal. It’s only two vowel sounds out of the dozens your mouth is capable of producing, but it’s a start toward making a more human tone.

Here’s a documentary about the wah:

Cry Baby: The Pedal That Rocks The World from Joey Tosi on Vimeo.

Combined with a guitar, the wah can do more than vowel glides. When you mute the strings and strum through a wah, you get a percussive sound ranging from “chicka chicka” to “chucka chucka.” By filtering the overtones differently, you can make other vocal sounds too. I have a digital effects unit that can make the guitar say the word “yeah” pretty convincingly. These kinds of effects give a guitarist the emotional immediacy of the voice combined with the guitar’s wide range of pitches and richness of harmonic possibility.

The guitar isn’t the only instrument you can use with a wah, and it wasn’t the first. The pedal was invented somewhat by accident when the Thomas Organ Company was developing a tone modifier for amplifiers. The first instrument they tried with it was an amplified saxophone, and the company thought they might market it for wind instruments in big bands, as an electronic version of the Harmon mute. A guitarist who worked for the company named Del Casher heard the possibilities of the new tone modifier, and he was the first person to make a recording of it in 1966.

Frank Zappa was an early adopter, and he introduced it to Jimi Hendrix, who would be the first to break it into mass consciousness with “Voodoo Child (Slight Return).” Jimi also introduced the percussive “chicka chicka” on “Little Miss Lover.” Jimi’s solos on “All Along The Watchtower” is another distinctive early adventure with wah. Plenty of other hippie rockers followed suit. George Harrison has a song called “Wah-Wah” on All Things Must Pass, named both for the pedal and for the Beatles’ whining during their final sessions together. Eric Clapton uses wah with Cream on “Tales of Brave Ulysses” and “White Room”.

Pop culturally, wah is most associated with seventies funk and soul, like on “Theme From Shaft” by Isaac Hayes, with Charles Pitts on guitar. Curtis Mayfield also had a distinctive and much-imitated wah style. From blaxploitation soundtracks it was a short jump to the porn movies that imitated them, which is why funky wah guitar is an effective comedy shorthand for getting busy. But wah doesn’t have to be seductive. Eddie Hazel of Funkadelic used it for a dark, spacey cry on “Maggot Brain.” Click here to listen to some standard wah techniques on electric guitar. The wah pedal sounds especially good on E9, the mother of all funk chords.

Hard rock and metal guitarists have found a vocabulary for wah drawing more on Hendrix and Zappa than on funk. Zappa used it less like a speech effect and more like a simple adjustable filter. He would leave it partially open to filter the high frequencies over the course of an entire song. Distortion exaggerates out the guitar’s upper harmonics and other partials, and the wah makes a great envelope controller. Jimmy Page used it on Led Zeppelin’s “Dazed and Confused,” “Whole Lotta Love,” “No Quarter” and “Custard Pie”. Slash used it with Guns N’ Roses, and Kirk Hammett leans heavily on it with Metallica.

Bassists sometimes use the wah too, especially in the funk and soul world. Michael Henderson played with one on Miles Davis’s album On the Corner. Other wah-loving bassists include Metallica’s Cliff Burton and Black Sabbath’s Geezer Butler.

Electric pianos and harpsichords operate in very much the same way as electric guitars, so it was only a matter of time before keyboard players started investigating guitar effects. Clavinet with wah sounds so much like guitar that it’s hard to tell them apart. Garth Hudson plays some pretty groovy clav with The Band on “Up On Cripple Creek”, but nothing is as funky as Stevie Wonder on “Superstition,” “Higher Ground” and his other seventies classics. Electric piano also sounds great through wah, again because of its guitar-like tone when played through an amp with distortion. Richard Wright uses it on Pink Floyd’s “Money”, and it’s on tons of Miles Davis electric recordings, especially the ones with Keith Jarrett and Chick Corea.

Any instrument that’s amplified can be played through a wah. Miles Davis got a devastating trumpet tone with wah on Live-Evil and his other darker funk records. A few saxophone players have experimented with it too, as the pedal’s original inventors intended. David Sanborn played with one on the David Bowie album Young Americans, and Dana Colley used it with Morphine.

Violin sounds great with wah. The leading practitioners are Jean-Luc Ponty in the Mahavishnu Orchestra and Boyd Tinsley in the Dave Matthews Band. Pink Floyd even tried some wah on an acoustic piano in their song “Echoes”, which also includes wah guitar made to sound like crying birds. I myself have found that wah sounds terrific on mandolin. I’ve also tried it on harmonica, but there it’s redundant since you can do the wah effect so easily with your mouth.

Wah is just one flavor of the envelope filtering you can do with a synthesizer. A lot of the craft of electronic music comes down to creative rhythmic use of the filter. A standard technique is to get a repetitive loop happening and then sloooowwwly open and close the filter over the course of a phrase or section. Since a sequencer or computer can play the actual synthesizer notes, it frees up the musician’s hands for complex multi-parameter filter control using knobs or touchscreens. We’re only at the beginning of our collective exploration of the artificial vowel glide in music.

First of all, to get your feet under you, here’s a video by the delightful Vi Hart explaining the basic physics of sound:

Music theory and wave mechanics

I’m going to talk you through the relationship between wave mechanics and music theory using the low E string on the guitar. If you have one handy, grab it and follow along. The experiment is easier on electric guitar with the amp turned up, but it works fine on acoustic as long as the room is quiet. If you don’t have a stringed instrument available, there are a bunch of Youtube videos you can watch, including this one, this one, and this one.

Vibrational modes

When you see a cartoon of a plucked string, it shows vibration lines spanning the full length of the string, implying that the middle of the string is swinging to and fro. Real strings vibrate this way, but they also vibrate in more complex ways too. Strings can vibrate in halves, with one half bowing this way while the other bows that way. Strings can also vibrate in thirds, the middle third bowed this way while the outer thirds bow that way, and vice versa. They simultaneously vibrate in fourths, fifths, sixths, sevenths and so on, in smaller and smaller parts, in theory going all the way to infinity.

When you pluck a guitar string, its movements are a combination of all these patterns of vibration combined together.

Resonant frequencies

Guitar strings vibrate really fast, so it’s hard to see all the different vibrational patterns firsthand. Jumpropes work better. Imagine that you and I are holding a big long jumprope. I’m holding one end still, and you’re waving your end up and down. If you wave the rope up and down at a certain frequency, the jumprope’s resonant frequency, you can make it vibrate along its entire length. If you wave it up and down twice that fast, you can make it vibrate in halves. Waving it three times faster makes the rope vibrate in thirds. Waving four times faster makes it vibrate in fourths. The pattern continues indefinitely, with more and more effort required on your part to make the rope vibrate in more and more sections.

If you could organize a whole bunch of people to carefully all shake the rope at once, each person shaking at a different multiple of the rope’s resonant frequency, the rich blend of movements would produce a perfect slow-mo replica of a guitar string’s vibrational pattern. This would be hard to do in real life, so you can use computer animation to assist your visualizing. See also this amazing stroboscopic video of an upright bassist.

Harmonics

The different multiples of the jumprope’s resonant frequency are called harmonics. If the word reminds you of harmony, it should. The rate of shaking that produces the basic cartoon-style full-length vibration is the fundamental. When each harmonic is a whole-number multiple of the fundamental frequency, and the vibration is fast enough to agitate the air audibly, you hear a pleasantly musical sound.

The harmonics don’t have to be perfectly aligned to whole-number ratios. If they’re random, the sound you hear is abrasive, or dull, or harsh, or strange. If you blend together the simple whole-number multiples of the resonant frequency with more complex or random multiples, you get sounds that are musical but otherworldly, like bells, gongs and the synthesizers in hip-hop and techno.

Tonal instruments like the guitar has been designed to maximize the rational, whole-number harmonics and to minimize the random and irrational ones. When you use distortion on an electric guitar, it compresses the sound and makes it exceptionally easy to hear the harmonics. Jimi Hendrix was a pioneer of the use of electric guitar harmonics.

The harmonics of the guitar’s E string

When you pluck the low E string on a guitar, the loudest sound you hear is the fundamental tone as the string vibrates along its entire length. If it’s in tune, the E string’s middle crosses its relaxed position about eighty-two and a half times each second. This is the pitch known to western music theory as E2. It’s the second-lowest E on a piano.

The guitar’s E string is also vibrating in halves, each half crossing the relaxed position a hundred sixty-five times each second, twice the fundamental frequency, to produce the note E3. You can hear this harmonic more clearly if you silence the fundamental by deadening the string lightly with a fingertip right above the twelfth fret, the string’s halfway point, as you pluck it.

Your ear-brain system is very adept at sussing out when one frequency is twice another frequency, even if they’re being produced simultaneously by the same vibrating string. We experience this pattern-detection ability as our sense of harmony. Nearly everyone can identify E2 and E3 as being “same” pitch, even though one is “higher” than the other. The western musical name for a two to one ratio of frequencies is an octave. What music theory calls “octave equivalence” is your experience of frequencies related by powers of two as being, in a sense, “the same.” The ability to detect octaves appears to be a human universal, and apes and monkeys can detect octaves too.

There are a lot of octaves of E hidden in the string’s quieter sub-vibrations. You can make the string vibrate in quarters by plucking it while touching it lightly above the fifth fret. As the string vibrates, each quarter crosses the relaxed position three hundred thirty times per second, to produce the note E4. As the string vibrates in eighths, each section produces a very quiet E5. As the string vibrates in sixteenths, each section produces an even quieter E6. If you plucked a perfect string on a perfect guitar, you’d hear every E up past the limits of your pitch perception, each one produced by a power-of-two multiple of the fundamental E2 frequency. These harmonics are reinforced by sympathetic vibrations from the guitar’s other E string, whose fundamental is E4. As the high E mutually agitates with the low E, it contributes its own multiples of eighty-two and a half to the mix.

The major triad emerges from the overtone series

All those ghostly bright E’s hovering above the fundamental frequency are only the beginning of the complexity hidden in an ordinary guitar string. To hear the string vibrating in thirds, touch it lightly above the seventh fret while plucking it. As the E string vibrates, each third crosses the relaxed position two hundred forty-seven times each second. This frequency is the pitch B3. The ratio between B3 and E3 is known to western music theory as the perfect fifth. The ratio of B3 to E4 is a perfect fourth. Like the octave, these ratios are heard across nearly every world culture. It’s no accident that the guitar has a string tuned to B3. Its sympathetic vibrations as you play the E strings are part of the instrument’s distinctive sound.

As the E string vibrates in fifths, each fifth crosses the relaxed position four hundred twelve times per second, producing the note G#4. You can hear this harmonic by lightly touching the string between the third and fourth frets and plucking hard. The ratio of frequencies between G#4 and E4 is a major third. After octaves, fifths and fourths, major thirds are the next most common interval in western music, and in many other musical cultures as well.

The ratio of G#4 to B4 is a minor third, the frequently-heard “sad” counterpart to the “happy” major third. Not every culture ascribes these emotional qualities to these ratios in every context, but most of the time, most western listeners experience major thirds as happy and minor thirds as sad.

Higher harmonics, more complex intervals

As you continue along the harmonic series, the ratios get more complex and the sounds get more mysterious. As the string vibrates in sevenths, it produces a very high-pitched sound close to a D. The interval between E and D is a flat or minor seventh. As the string vibrates in ninths, you get a note very close to F#, a natural second above E. As the string vibrates in elevenths, you get something close to Bb or A#, a tritone above E. Tritones are a key ingredient in blues, jazz, rock and their musical descendants.

The chord hidden in a single note

With all of its harmonics, the note E on a guitar is more like a richly complex chord. You’re not just hearing E in several different octaves. You’re hearing a quiet B, an even quieter G#, a faint D and a fainter F# and a barely perceptible A#. You’re hearing the chord that jazz musicians call E9#11, whose pitches comprise the acoustic scale. Jazz musicians call it the lydian dominant mode;  western classical nicknames it the Bartók scale.

And this is all from a single note. When you play two notes at a time, or three or four, you get even more complex harmonic interaction from all the overtones.

Every vibrating physical system can have harmonics

The cool thing about all of this harmonic business is that it isn’t specific to guitar strings or air or eardrums. Anything that vibrates steadily can produce harmonics: your throat, drum heads, gongs, the mouthpieces and bodies of wind instruments, organ pipes, bones, hollow logs, shopping carts, garbage cans, ladders, water glasses, crystals, even single molecules and atoms.

Most rigid material objects have a resonant frequency. You can shatter a wine glass by singing its resonant frequency.

Bridges have resonant frequencies too. High school science teachers love to show films of the Tacoma Narrows Bridge as the wind shakes it at its resonant frequency until it collapses.

Quantum particles have harmonics too

If you want to understand the fundamental structure of matter, you need to know about harmonics. All those pictures you see of marble-like electrons orbiting a nucleus like moons orbiting a little planet are totally wrong. In reality, electrons are organized in atoms by the harmonics of the electron field. Here’s a blog post explaining this concept in detail. Who says music is frivolous?