What is the harmonic series?

The set of pitches a vibrating string or air column produces at once: the fundamental frequency f and its whole-number multiples 2f, 3f, 4f and so on. The multiples are the harmonics (or overtones), and their relative loudness is what gives an instrument its timbre.

Is a musical note a single frequency?

No. A note is a stack of frequencies sounding together: the fundamental, which is the pitch you name, and its harmonics at two, three, four times that frequency, each usually quieter than the last. A single frequency on its own is a sine wave — the sound of a hearing test or a tuning fork — and no instrument produces one.

The character of a sound — what tells a piano from a guitar from a voice on the same note. It is the recipe of harmonics: which ones an instrument sounds, how strongly, and how each one fades. A clarinet sounds mostly the odd harmonics; a string plucked by the bridge sounds the high ones more strongly than one plucked over the neck, and on any plucked string the high harmonics die away first. Change the recipe and the pitch stays, but the instrument changes.

Why does an octave sound like the same note?

Because two notes an octave apart are the first two harmonics of one string, in the ratio 2:1: the upper one vibrates exactly twice per cycle of the lower, so their sum repeats every single cycle. Nothing fits together more simply, and the ear treats them as one note in two registers.

Why is a perfect fifth consonant?

Because it is harmonics 2 and 3 of the same note, the ratio 3:2, and the two waves together repeat every two cycles of the lower one. After the octave it is the simplest relationship two different notes can have, which is why it sounds stable and open.

Why does a minor second sound dissonant?

Two notes a semitone apart are harmonics 15 and 16 of a common fundamental, 16:15, and their combined wave only repeats every fifteen cycles. In between, the two nearly equal waves drift in and out of step, which the ear hears as beating and roughness.

Is the harmonic series the same as equal temperament?

No. The series gives exact ratios — 3:2, 5:4 — while equal temperament divides the octave into twelve equal semitones so every key is playable. A tempered fifth is 2 cents narrow and a tempered major third 14 cents wide; close enough that the consonance ranking is unchanged, but the ratios are approximations.

Why is the tritone dissonant?

Less than its reputation suggests, and it depends on the music around it. Six semitones is exactly half an octave, so the tritone is its own inversion and neither note points at the other as home, and it sits one semitone from the fifth on one side and the fourth on the other. Its smoothest form in the harmonic series is 7:5 — harmonics 5 and 7, the third against the seventh of a dominant chord — repeating every five cycles, rougher than a third but far smoother than a minor second. In a dominant seventh it is the tension that makes a resolution; in a blues or a whole-tone passage it is simply a colour.

Why does a note have a pitch at all if it is many frequencies?

Because the frequencies are whole-number multiples of one another, so their combined wave repeats — once per cycle of the fundamental, however many harmonics are stacked on it. Pitch is the rate at which that whole pattern comes round again, and your ear needs to hear it repeat several times, a few dozen milliseconds, before it settles on one.

Why Some Notes Sound Good Together: The Harmonic Series, Interval by Interval

One string, its overtones, and the sums you can see — why a fifth is smooth, a semitone grinds, and a chord feels like arrival.

Why do some notes sound good together and others clash? The answer is inside every single note. Pluck a string, and what you hear as one note is several — a fundamental and its harmonics — and that stack is the reference point for what sounds good.

It matters because the pattern found within a single note seems to be the pattern we follow when we put many notes together.

The figure follows the text: each section lights the harmonics it is about, the links in the text set it too, and the ones marked ♪ sound it.

One frequency, and no harmonics

A note is one frequency — supposedly

We talk about pitch as if it were a single number. A is 440 Hz; the C the figure is drawn on is 131. Tune to it, sing it, name it — one note, one frequency. But a single frequency is a very particular sound, and you have almost certainly never heard an instrument make it. Tap to hear one ♪: the first harmonic alone, one sine wave, struck and left to ring, with nothing to add it to. A tuning fork, a hearing test, a soft synth bass at best. That is what "one frequency" sounds like — pure, thin, and not like a note on anything you play.

One note is many

Now pluck a string. It vibrates along its whole length at that frequency — and at the same time in halves, in thirds, in quarters, and each of those motions is a pitch of its own, two, three, four times the fundamental. The figure draws one row per motion and, under the rule, their sum: the single wave they make between them, which is the only thing that ever reaches your ear. Watch the bottom row as the rows above it multiply; scroll, and each step is plucked.

Add the second harmonic, at twice the frequency and half the strength, and the sum at the bottom is no longer a sine: a taller crest and a shallower trough, once per cycle. Some edge comes into the pluck.

The third, at three times the frequency and a third of the strength, sharpens the crest and puts a ripple in the trough. Listen to the start of the note: the high harmonic is there in the first instant and gone a moment later, while the fundamental rings on. That is what a string does — its high harmonics die first, and every note is brightest at the pluck.

The fourth. Each harmonic is quieter than the one before, and each one you add makes the wave leaner: a slow rise and a sharp drop.

The first eight, and it is a string. The sum is nearly a saw-tooth — a ramp and a fall, once per cycle of the fundamental — which is the shape a string takes when it is plucked hard, close to the bridge; and the sound is the sound of that: a bright snap that settles, in about a second, to the fundamental you would name. The figure draws the first instant, every harmonic at the series' own strength; your ear hears the whole second, the top of the stack falling away.

That is what a note is. When we say "the C at 131 Hz" we mean the fundamental, the lowest of the stack and the one that lasts — the pitch your ear assigns the whole bundle to. The rest of the stack is always there, and the pitch is not what it is for.

The bottom row is what you hear

Those rows are a way of drawing. Your ear is not given eight signals to sort out: air carries one pressure wave, your eardrum moves one way at a time, and what arrives is the row at the bottom — the lit rows added together, a single wiggle. Every claim on this page is a claim about that one wave.

And the thing to see about it is that it repeats. One shape, over and over, once per cycle of the fundamental, no matter how many harmonics are stacked into it — because each of them completes a whole number of turns in that time, which is exactly what being a whole-number multiple means. The shaded band marks one repeat; the count at the end of each row is how many turns that harmonic takes inside it.

That repetition is the pitch. Eight frequencies go in, one pattern comes out 131 times a second, and you hear one note. Pitch is not a frequency your ear fishes out of the bundle; it is the rate at which the whole bundle comes round again.

Which makes the band a clock. One cycle of this C lasts about eight thousandths of a second, and your ear does not read a pitch off an instant — it needs the pattern to come round several times before it settles on it, a few dozen milliseconds here. That is why a click has no pitch at all, and why the very start of a note, before anything has repeated, is the part that sounds like the instrument rather than like a note.

Two notes played together are the same question asked of a longer pattern: the sum still repeats, but after how many cycles — and can your ear settle on it while the note lasts? That is the whole of what follows.

The recipe is the instrument

What the rest of the stack is for is timbre: the character of a sound, the thing that tells a piano from a guitar from a voice singing the same note. Every instrument sounds every harmonic in its own proportions — its own recipe — and change the recipe and you change the instrument, without touching the pitch. Three recipes, on the same C:

Only the odd harmonics, blown. A pipe closed at one end — a clarinet, a pan pipe, a bottle blown across the top — cannot sound the even ones at all, and a blown note swells in and holds for as long as there is breath rather than dying away. The sum squares off, and the sound goes hollow and woody. Same fundamental, same pitch; a different instrument.

A string plucked over the neck. Only the odd harmonics again, and faint ones — the middle of the string is where the even harmonics stand still, so a pluck there cannot start them, and it barely starts the high ones. The sum is nearly a sine; the pluck is soft and round. Same string, same note.

The same string plucked by the bridge. Every harmonic starts, in the series' own proportions — the recipe you heard built up above — and the pluck twangs. Nothing about the string has changed but where the finger landed, and the recipe — the timbre — is a different one.

A piano's hammer, a bow, a reed, the shape of your mouth on a vowel — each is a way of setting a recipe, and your ear reads it in the first fraction of a second. (These are toys: an ideal string, its harmonics dying at a made-up rate. A real note's recipe also shifts as it decays, and a real body adds resonances of its own.) So a note is a fundamental and a recipe; the fundamental is the pitch, the recipe is the timbre. The recipe is the instrument's. The ladder is not: f, 2f, 3f, 4f — the same rungs, in the same places, on every note of every instrument, and the rest of this page is about the ladder alone, every harmonic at the series' own strength.

The notes inside a note

Name the rungs and the ladder has a second meaning. The first eight harmonics of a C are C, C, G, C, E, G, B♭, C — the note itself in three octaves, its fifth, its major third, its seventh. Nobody chose those notes to go with C; they are C, heard closely. And they are the notes that sound at home against it — the first thing you learn to hear over a root, the chord a song comes back to. That is the claim of this page, and the sections below are the evidence: every interval we use is the distance between two rungs of the ladder that every note carries, and the higher up the ladder a pair sits, the rougher it sounds. The figure numbers the rungs rather than naming them, on purpose: the lesson is not which notes are in there but how early each interval turns up.

The octave: 2:1

Harmonics 1 and 2 are an octave apart — the second vibrates exactly twice for every cycle of the first — and the band across the strips shows the sum repeating every single cycle of the lower note: at this C, eight milliseconds — the shortest pattern two different notes can make, settled before you could call it a sound. No two different notes fit together more simply than that. It is why notes an octave apart are so consonant that we hear them as the same note, give them the same name, and, in NoSheet, the same color: the whole instrument is twelve notes over and over.

The fifth: 3:2

Harmonics 2 and 3 are a perfect fifth. Three cycles of the upper note against two of the lower, and the sum repeats every two cycles of the lower one — down at the C we started from, fifteen thousandths of a second, so the pattern has come round a dozen times before the note is properly under way and your ear has nothing left to work out. That is the sound of a power chord, of a violin's or a mandolin's open strings, of a bass player's first two notes. After the octave it is the most consonant interval there is, and the first thing the app teaches you to hear.

The fourth and the thirds

Keep climbing and each next pair of neighbors is the next interval down the list. Harmonics 3 and 4 are a perfect fourth, 4:3, repeating every three cycles. Harmonics 4 and 5 are a major third, 5:4, every four. Harmonics 5 and 6 are a minor third, 6:5, every five. Each is a little more complex than the last, and each sounds a little less like one note and a little more like two.

Now light 4, 5 and 6 together. That is a major triad — root, third and fifth — and it is sitting inside the harmonic series, ready made. The sum repeats every four cycles, as simply as a major third does on its own. This is why the major triad is the most consonant chord there is, why it feels like arrival, and why NoSheet builds its harmony games on it. The minor triad is in the series too, but higher up — 10:12:15, repeating every ten cycles — which is exactly how much darker it sounds.

Why some notes sound good together, and some don't

The rule is the clock from the bottom row, and the band is drawing it. Two harmonics m and n of one string, sounded together, make a wave that repeats every n cycles of the lower one. The fifth, 3:2, repeats every two. The major third, 5:4, every four. The major second, 9:8, every eight. The minor second, 16:15, every fifteen.

That count is a measure of how much there is to take in. A two-cycle pattern is a small thing to hold — hear it once and you have heard all of it — while a fifteen-cycle one is fifteen times as much wave before anything comes round again, and until it does your ear has nothing to recognise.

Put those in a register you would actually play in — the lower note down at this C, one cycle of it eight milliseconds — and the fifth's pattern is fifteen thousandths of a second long while the semitone's runs past a tenth of a second: longer than the window your ear would have liked to settle in, so for as long as the note lasts there is no one pattern to settle on. What it tracks instead is two nearly-equal waves sliding in and out of step, swelling and fading — the slow wah of two pitches almost the same, which is what roughness is.

Consonance is nothing more mysterious than how soon the wave repeats, and the length of the band across these pages is the ranking: the shorter, the sooner it is one sound. It is not quite the ranking a musician would give — the major sixth, 5:3, repeats every three cycles, sooner than either third, and the numbers are right: the sixth is the simpler ratio. The thirds matter more to music because they decide major and minor, not because they are smoother.

A stack ranks itself the same way. The major seventh, 15:8, repeats every eight cycles; a dominant seventh chord, 4:5:6:7, every four — which is why it is tense and still sounds like a chord.

Why the 7th harmonic isn't on a piano

Light harmonics 4 and 7 and the sum's row says ♭7, the brace 7:4, and then −31¢: a third of a semitone flat of the minor seventh on your instrument. Harmonic 7 is a real note, the "harmonic seventh", and barbershop quartets tune to it; but it is not one of the twelve, and a piano cannot play it. Harmonic 11 is worse, 49¢ off, nearly a quarter-tone between a fourth and a tritone. The twelve notes we use are the harmonics that happened to fall close to a neat grid, and these are the ones that did not. A cents figure on the brace says how far the ratio sits from the nearest interval on the instrument — which is the next section's subject from the other side.

Equal temperament: close enough

The ratios on this page are exact, and your guitar does not play them. Its frets divide the octave into twelve equal semitones so that every key is the same shape — equal temperament — and in exchange every interval but the octave is slightly off its ratio. The fifth is 2¢ narrow, which nobody hears. The major third is 14¢ wide, which you can hear as a slow beat if you hold a chord and listen for it. The ranking survives, and not by luck. Your ear does not measure a ratio, it recognises one — hearing a third that is 14¢ wide as a major third slightly out of tune is the same act as hearing a slightly flat singer as still singing the note. And the simple ratios are the ones with room to be wrong: a third can be a sixth of a semitone off and still arrive as a third, which is exactly what your guitar is counting on. That is the whole trick of temperament, and it is why the numbers on this page survive being approximated. The figure sounds the pure ratios so you can hear what the instrument is approximating; the instrument pages sound the instrument.

Why the tritone is dissonant — and when it isn't

Six semitones is half an octave, and the tritone is the one interval that is its own upside down: turn it over — put the lower note an octave up — and you get a tritone again. Every other interval comes as a pair with its inversion, a fifth with a fourth, a major third with a minor sixth, and that lopsidedness is part of what makes one of the two notes sound like home and the other like the note leaning on it. The tritone cuts the octave in half and names no home.

The ratio your ear rounds it to is 7:5, harmonics 5 and 7 — the third against the seventh of a dominant chord — repeating every five cycles: rougher than a third, far smoother than a second, and the sound of a barbershop seventh. So "the most dissonant interval" is not what the clock says about it. What is unusual is where it sits: one semitone from the fifth on one side and one from the fourth on the other, a hair away from the two steadiest intervals there are and not quite either.

That is why its dissonance is contextual in a way a minor second's is not. A semitone grinds wherever you put it; the tritone takes its character from what surrounds it. In a dominant seventh chord it is the tension that makes the resolution feel like one, and the ear hears it leaning toward the notes a semitone away. In a blues it sits there all night and nobody flinches. In a whole-tone or diminished setting, where the notes are spaced evenly and none of them is a home, it stops being a pull at all. The tritone is not the harshest interval in the octave — the minor second and the major seventh are rougher by the clock. It is the most ambiguous one, and the ambiguity is the job.

What NoSheet does with this

Not everything in music is in the series. It ranks the intervals and hands over the triad ready made; it does not say which twelve notes to keep — the seventh harmonic did not make it — or why the thirds matter more to us than the smoother sixth. The series is the reference point; music makes its choices against it. Scales are those choices made in fifths, five for a pentatonic and seven for a scale, and that is the circle of fifths.

The app teaches intervals in the order the series gives them. Root and fifth first, because 3:2 is the interval your ear already knows how to hold on to; then the major triad, 4:5:6, the first chord the series makes; then the pentatonics and the modes. Every game plays a phrase and waits for you to play it back, so what you are practicing is not the ratios but the thing they explain: which notes sound like home against a root, and how far from home the others are.

Hear the series once, and the numbers stop being theory. A fifth is two cycles. A semitone is fifteen. That is why some notes sound good together: you can hear the difference in the first second, and now you know what you are hearing.

Questions

Why do some notes sound good together?: Because their frequencies are in a simple ratio — an octave is 2:1, a fifth 3:2 — so their sound waves line up again after a cycle or two. The more cycles it takes for the combined wave to repeat, the rougher the pair sounds: a major third takes four, a minor second fifteen. Those ratios are the harmonics that every single note already contains.
What is the harmonic series?: The set of pitches a vibrating string or air column produces at once: the fundamental frequency f and its whole-number multiples 2f, 3f, 4f and so on. The multiples are the harmonics (or overtones), and their relative loudness is what gives an instrument its timbre.
Is a musical note a single frequency?: No. A note is a stack of frequencies sounding together: the fundamental, which is the pitch you name, and its harmonics at two, three, four times that frequency, each usually quieter than the last. A single frequency on its own is a sine wave — the sound of a hearing test or a tuning fork — and no instrument produces one.
What is timbre?: The character of a sound — what tells a piano from a guitar from a voice on the same note. It is the recipe of harmonics: which ones an instrument sounds, how strongly, and how each one fades. A clarinet sounds mostly the odd harmonics; a string plucked by the bridge sounds the high ones more strongly than one plucked over the neck, and on any plucked string the high harmonics die away first. Change the recipe and the pitch stays, but the instrument changes.
Why does an octave sound like the same note?: Because two notes an octave apart are the first two harmonics of one string, in the ratio 2:1: the upper one vibrates exactly twice per cycle of the lower, so their sum repeats every single cycle. Nothing fits together more simply, and the ear treats them as one note in two registers.
Why is a perfect fifth consonant?: Because it is harmonics 2 and 3 of the same note, the ratio 3:2, and the two waves together repeat every two cycles of the lower one. After the octave it is the simplest relationship two different notes can have, which is why it sounds stable and open.
Why does a minor second sound dissonant?: Two notes a semitone apart are harmonics 15 and 16 of a common fundamental, 16:15, and their combined wave only repeats every fifteen cycles. In between, the two nearly equal waves drift in and out of step, which the ear hears as beating and roughness.
Is the harmonic series the same as equal temperament?: No. The series gives exact ratios — 3:2, 5:4 — while equal temperament divides the octave into twelve equal semitones so every key is playable. A tempered fifth is 2 cents narrow and a tempered major third 14 cents wide; close enough that the consonance ranking is unchanged, but the ratios are approximations.
Why is the tritone dissonant?: Less than its reputation suggests, and it depends on the music around it. Six semitones is exactly half an octave, so the tritone is its own inversion and neither note points at the other as home, and it sits one semitone from the fifth on one side and the fourth on the other. Its smoothest form in the harmonic series is 7:5 — harmonics 5 and 7, the third against the seventh of a dominant chord — repeating every five cycles, rougher than a third but far smoother than a minor second. In a dominant seventh it is the tension that makes a resolution; in a blues or a whole-tone passage it is simply a colour.
Why does a note have a pitch at all if it is many frequencies?: Because the frequencies are whole-number multiples of one another, so their combined wave repeats — once per cycle of the fundamental, however many harmonics are stacked on it. Pitch is the rate at which that whole pattern comes round again, and your ear needs to hear it repeat several times, a few dozen milliseconds, before it settles on one.