Pluck a guitar string and you hear one note. But there is a trick you can do with your own hands: touch the string lightly above the twelfth fret and pluck, and the same string suddenly sings an octave higher, thin and bell-like. Touch it above the seventh fret and you get a note an octave and a fifth up. The string was never making one tone. It was making a whole stack of them at once, and your ear has been adding them into a single "note" since you were born.
That stack is why a violin does not sound like a flute when both play the same A, and it is also the root of one of music's oldest headaches: the stacks do not line up neatly. Tune a piano so that every fifth is perfect and the octaves come out wrong; tune it so the octaves are perfect and you must nudge every fifth. Acoustics, tuning and timbre is the course where musicians meet the physics under their instrument. This piece plays real sounds in your browser; keep your volume low, tap Play to start and Stop to end. Every sound control also shows its numbers, so it works with the sound off.
A string, an air column or a vocal fold vibrates in many patterns at once. The slowest pattern sets the pitch you name; the faster ones vibrate at whole-number multiples of it. If the fundamental is 220 Hz, the others sit at 440, 660, 880, 1,100 and so on. These are the harmonics or partials, and together they make the harmonic series.
The series is a map of music's intervals. Partial 2 is an octave above the fundamental (1,200 cents). Partial 3 is an octave plus a fifth (1,901.96 cents, a hair above the equal-tempered 1,900). Partial 5 is two octaves plus a major third, 13.7 cents flatter than a piano's. Partial 7 is 31 cents flatter than the nearest piano note, which is why a "natural seventh" on a brass instrument or a string harmonic sounds slightly odd to Western ears. Helmholtz's On the Sensations of Tone (1863) built a whole theory of consonance on these overlaps.
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The shapes are the textbook ones: sawtooth 1/n, square odd n at 1/n, triangle odd n at 1/n². Waveform shows one cycle. Phone speakers barely reproduce frequencies below about 200 Hz, so try the fundamental near 330 Hz or above if you hear little.
Try it. The sine is the only shape with no overtones: it sounds like a tuning fork or a pure whistle. Switch to the sawtooth and add partials one by one: the sound goes from a flute-like tone to a buzzy, brassy one, because the strength of the high partials is what we hear as brightness. Now press Mute partial 1. The fundamental vanishes from the chart, the waveform barely changes its repeat length, and the pitch you hear stays the same. Your brain infers the pitch from the spacing of the partials, not from the lowest one. That "missing fundamental" is why a phone speaker that cannot make 100 Hz can still play you a bass line.
The partials are not arbitrary; the shape of the vibrating thing chooses them. A string held at both ends can wave in 1, 2, 3 … loops, so it makes every multiple. A pipe open at both ends (a flute) does the same. A pipe closed at one end (a clarinet's cylindrical bore behaves much like this) can only fit an odd number of quarter-waves, so it makes 1, 3, 5, 7 … times the fundamental and no even ones. That missing set of even partials is a large part of the clarinet's hollow sound.
Worked numbers: a guitar's low E string is 0.648 m long and sounds 82.41 Hz, so its waves travel along it at 2 × 0.648 × 82.41 ≈ 107 m/s. An open pipe 0.5 m long sounds 343 / (2 × 0.5) = 343 Hz; closed at one end, the same length sounds half that, 171.5 Hz, an octave lower.
Solid bars and bells break the rule. A free bar, such as a xylophone, vibraphone or a metallophone from a gamelan, has partials at 1, 2.76, 5.40 and 8.93 times the fundamental (from the beam equation), nowhere near whole numbers. We hear that as a clang with a pitch that is harder to pin down. Instrument makers fight it: marimba and vibraphone bars are carved thin in the middle to pull the first overtones toward neat multiples. Even a piano string is a little stiff, so its partials run sharp: partial n is at n·f1·√(1 + B·n²). That is why piano tuners "stretch" the octaves.
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Stiffness slider is B × 10−4 (so 4 means B = 0.0004, an illustrative value; real strings vary). The bar uses the free-free beam ratios 1 : 2.76 : 5.40 : 8.93. Ticks mark exact whole-number multiples. Pipe lengths ignore end corrections.
Choose the closed pipe and the sticks jump to 1, 3, 5, 7. Choose the free bar and none of them sit on a tick. Then choose the string, raise stiffness to 40 and watch the top partial slide to the right. At the default B = 0.0004 on a 110 Hz string, partial 8 comes out 21.9 cents sharp (889–891 Hz rather than 880).
Play 440 Hz and 444 Hz together. The two waves drift in and out of step: when their peaks line up the sound is loud, when peak meets trough it is quiet. The loudness swells and fades at a rate equal to the difference of the frequencies, here 4 times a second. These are beats, and piano tuners count them to tune.
Widen the gap and something else happens. Past roughly 15–20 beats a second the swells blur into a rough, fluttering buzz; widen it further and the ear can finally separate two pitches, and the roughness fades. In the mid-1960s Plomp and Levelt showed that the ear behaves as a bank of overlapping filters, and two tones that land in the same filter beat against each other and sound rough. That one idea, plus the harmonic series, explains why some intervals sound smooth: they are the ones whose partials either coincide exactly or stay out of each other's filters.
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Pure sine tones. The chart shows the loudness envelope over one second, not the individual cycles (440 of them would be a solid block). The boundaries between "beats", "rough" and "two tones" depend on pitch and listener; the numbers in the readout are rough guides.
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A model, not a measurement: pairwise roughness after Plomp and Levelt, in the parameterisation published by William Sethares, summed over every pair of partials. Middle C = 261.63 Hz. Partial strengths fall by 12% per step (harmonic) or are 1, .88, .77, .68 (bar-like).
With six harmonics, the dips in the curve fall at 316, 386, 498, 702 and 884 cents: the minor third (6:5), major third (5:4), fourth (4:3), fifth (3:2) and major sixth (5:3). The "consonant" intervals are simply where the partials line up. Now switch to the bar-like timbre: the dips move to 616, 870, 1,008 and 1,166 cents, and the fifth is no longer special. Sethares argued that an ensemble whose instruments have inharmonic partials will sound best in scales matched to those partials rather than to the harmonic series. That is an argument about why scales differ across the world, not a proof, but the curve it rests on is computed live above.
Here is the arithmetic problem under every keyboard. A pure fifth is the ratio 3:2, 701.955 cents. Stack twelve of them from C and you should come back to a C, seven octaves up. You do not.
The leftover, 23.46 cents, is the Pythagorean comma, about a quarter of a semitone. There is a second mismatch. Four pure fifths up and two octaves down gives a major third of 407.82 cents, but the pure major third from the harmonic series (5:4) is 386.31. The 21.51-cent gap between them is the syntonic comma (81/80). So you cannot have pure fifths and pure thirds and a keyboard that closes. Every tuning system chooses what to give up.
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Each step moves round the octave circle by one fifth. The red arc is the gap after twelve steps. Beat counts compare partial 5 of middle C with partial 4 of the note four fifths up. Quarter-comma fifth = 1200·log₂(5^¼) = 696.578 cents.
Drag to 700 and the gap closes exactly: that is equal temperament, which shares the comma equally among all twelve fifths, 1.955 cents each. Zhu Zaiyu in China published exact figures for it in 1584, and Simon Stevin in the Low Countries worked it out at about the same time (his manuscript was only printed in 1884). Drag to 696.578 and the four-fifths third becomes the pure 386.31: that is quarter-comma meantone, which squeezes each fifth by a quarter of the syntonic comma to make thirds sweet. The price is the 41-cent gap you see at the end.
Four classical answers, each of which can be written as twelve numbers in cents above C:
Bach's Well-Tempered Clavier (Book I, 1722) is often described as a hymn to equal temperament. It was not necessarily: "well-tempered" meant a tuning in which every key was usable, and there were many such tunings. Which one Bach used is still debated.
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Tables: Pythagorean and meantone use a chain of twelve fifths from E♭ to G♯ (the wolf falls between G♯ and E♭); just uses the standard symmetric 5-limit chromatic scale (16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8). Roughness uses the same model as above, six harmonics, root at C-based pitch; the index is scaled so the equal-tempered C major triad is 100. The A♭ chord uses G♯ and the chain's wolf fifth as spelled.
Play the home-key triad in each system and look at the numbers: just intonation and meantone are smoother than equal. Then pick the far chord. In meantone the third of F♯ major is 427.37 cents, a full 41 cents wider than pure, and the A♭ triad contains the 737.64-cent wolf fifth. Old keyboards had "key colours" for this reason: each key sounded different, and composers chose keys for it. Equal temperament won in the 1800s because it let music modulate anywhere, and it won at a cost that the ear learned to ignore.
Western keyboard tuning is one answer to one set of questions. Other traditions asked different ones, and their tunings are not "imperfect" versions of ours.
Javanese and Balinese gamelan use two main tuning families, sléndro (five notes to the octave, with large steps) and pélog (seven notes, with uneven steps, usually played as five-note subsets). The crucial fact is that tuning differs from one ensemble to the next: the instruments of a gamelan are made and tuned together as a set, and the ethnomusicologist Colin McPhee remarked that there might be "as many scales as there are gamelans" (some teachers dispute how far that goes). The textbook figure for sléndro is "five roughly equal steps", about 240 cents each, but treat it as a description, not a standard. In Balinese gamelan, instruments are built in pairs tuned slightly apart on purpose, one a little high and one a little low, so every note shimmers. The beat is called ombak, "wave", and the aim is a steady beat rate across all registers.
Maqam (Arabic), makam (Turkish) and dastgah (Persian) organise melody into modes whose steps include sizes between our semitone and whole tone. The common shorthand of "quarter tones" is a notation convention: written music rounds every note to a 24-per-octave grid, but the pitches players use vary with the mode, the region, the period and the performer, and theorists disagree about the sizes. Turkish theory in the Arel–Ezgi–Uzdilek system divides the octave into 53 equal commas, with a whole tone of 9 commas. In that system the third of makam rast is 17 commas, 384.9 cents, close to a just major third, while in 24-division Arabic notation the rast third is written at 350 cents. These are two descriptions of related practices, not one right answer.
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Rast on C, 53-comma steps 9-8-5-9-9-4-9 (the Acem form), cents = commas × 1200/53. The sléndro-like scale is the idealised five equal steps, not a measured ensemble; no pélog is shown because there is no single pélog to show. Grey ticks are equal-tempered semitones.
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The beat rates here are illustrative; each ensemble and tradition sets its own. The point is the arithmetic.
The ombak slider hides a neat consequence. To keep a steady beat of, say, 6 per second, the two instruments of a pair must be 46.6 cents apart at 220 Hz, 23.5 cents apart at 440 Hz and 11.8 cents apart at 880 Hz. A constant beat rate means a pair spacing that shrinks as the notes rise, which a tuner has to build into the metal.
So why is a violin not a flute? Partly the partials: a flute's spectrum is nearly pure, a violin's is rich. But two more things count. First, the spectral envelope: a body of wood or a column of air resonates more at some frequencies than others and shapes how strong each partial is, regardless of what the pitch is. Second, the attack: the first fraction of a second, with its noise and quick changes, is a large part of how we recognise an instrument. Record a piano, cut off the start and play only the steady middle, and many listeners find it hard to say what it was.
The cleanest way to hear the envelope is your own voice. Vocal folds make a buzzing sawtooth-like source; the throat and mouth are a filter with resonances called formants. Change the shape of your mouth and the formants move; the pitch does not. The first two formants, F1 and F2, mostly decide which vowel you hear. Round textbook averages for adult men's vowels (after Peterson and Barney, 1952) are shown below as presets. Women, children and other languages differ.
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A simple source-filter sketch: harmonics n·f₀ weighted by 1/n and by two resonance curves with bandwidths of 80 and 120 Hz. Presets (F1/F2 in Hz): ee 270/2290, eh 530/1840, ah 730/1090, aw 570/840, oo 300/870. It's a sketch of the idea, not a speech synthesiser.
Press ee, then oo: the harmonics are at the same places, but the bumps in the envelope move and the vowel changes. Now raise the pitch to 400 Hz. The harmonics spread so far apart that the envelope has few of them to pick out, and vowels get harder to tell apart. This is why high sopranos are hard to understand and why the same effect is used by a trained singer's "singer's formant" near 3,000 Hz, a cluster of energy that helps a voice carry over an orchestra. The instrument and the voice are the same story: a source, then a body that filters it.
A note is a chord of partials, a scale is a deal between pure ratios and a keyboard that must close, and a tone colour is the shape of a spectrum. Once you know that, you hear why intervals are smooth, why every tuning is a compromise, and why the same pitch can belong to a flute, a violin or a vowel.
Try this: this week, find a stringed instrument or a piano and play the same note on two instruments, or sing "ee" then "oo" on one pitch. Hum a steady note and slowly change your mouth shape, and listen to the colour move while the pitch does not. If you have a piano, hold a low key silently and strike the key an octave and a fifth above sharply, then release the upper one: you may hear the low string ring along.
Sources. Harmonic series, standing waves and Fourier analysis are standard physics (Rossing, Moore and Wheeler). The Pythagorean comma (531441/524288 ≈ 23.46 cents) and syntonic comma (81/80 ≈ 21.51 cents) were computed and checked in node. Equal-temperament history (Zhu Zaiyu 1581/1584; Stevin) and the Well-Tempered Clavier caveat: encyclopaedia summaries; the exact tuning Bach used is disputed. Gamelan tuning variation and ombak, the Arel–Ezgi–Uzdilek 53-comma system and the 24-division maqam notation: encyclopaedia summaries of ethnomusicological sources, with the McPhee remark as reported there. Roughness model: Plomp and Levelt (mid-1960s) as parameterised by Sethares. Vowel formants: Peterson and Barney (1952) averages as commonly tabulated, rounded. Piano inharmonicity and free-bar ratios from the standard beam equation. All interactive numbers were computed in node.