A string does everything at once

Pluck a string fixed at both ends and it vibrates as a whole, at a frequency set by its length, tension and mass.

It also vibrates in halves, at twice that frequency. And in thirds, at three times. And in quarters, fifths, sixths, and so on upward, all at the same time, all in the same piece of wire.

That is the harmonic series, and it is not a musical idea. It is a consequence of the boundary conditions: a string clamped at both ends can only support standing waves with a node at each end, which means only whole numbers of half-wavelengths fit, which means only whole-number multiples of the fundamental frequency are available.

Everything else on this site is, one way or another, downstream of it.

What the series contains #

Read the intervals between the first few partials and a familiar object assembles itself.

Partial 1 to 2 is an octave. 2 to 3 is a perfect fifth. 3 to 4 is a perfect fourth. 4 to 5 is a major third, 5 to 6 a minor third.

Partials 4, 5 and 6 together are a major triad. That is the single most important sentence in this essay: the major triad is not a cultural artefact bolted onto acoustics, it is a contiguous subset of what every string and every pipe produces on its own. Every note anybody plays on a harmonic instrument contains a faint major triad inside itself.

The order of arrival matters too. The simplest intervals appear first and lowest, where the partials are strongest, and they are exactly the intervals that every tuning system is built out of and that every tradition treats as consonant. That is not a coincidence — consonance is partials coinciding, and the partials that coincide are these.

Why the numbers are whole #

The whole-number relationship deserves a sentence of mechanism, because it is what makes the series harmonic rather than arbitrary.

A wave on a string travels at a speed $v$ fixed by tension and mass per unit length. A standing wave requires the string’s length $L$ to be a whole number of half-wavelengths: $L = n\lambda/2$. So $\lambda = 2L/n$, and since $f = v/\lambda$:

$$f_n = \frac{nv}{2L}.$$

The frequencies are $n$ times the fundamental, for integer $n$, and the integers come from counting half-wavelengths rather than from anything about music. An air column in a pipe open at both ends gives the same result for the same reason. A pipe closed at one end supports only odd $n$, which is why a clarinet has essentially no even partials and overblows to a twelfth instead of an octave.

The seventh partial, which is not on the keyboard #

Follow the series far enough and it stops agreeing with the notes anybody plays.

The seventh partial is at 7 times the fundamental. Fold it into an octave and it is 7/4 of the fundamental — 969 cents, which is 31 cents flat of the equal-tempered minor seventh. That is nearly a third of a semitone, and it is audible immediately.

The eleventh partial is 551 cents above its octave, halfway between a perfect fourth and a tritone, and matches nothing at all. The thirteenth is similarly stranded.

This is worth stating carefully, because the usual framing has it backwards. The keyboard is out of tune with the string, not the other way round. The seventh partial is exactly where the physics puts it; the minor seventh on a piano is a compromise arrived at by a completely different process, and the two disagree.

The 7:4 interval — the harmonic seventh — is genuinely smoother than the tempered one. Barbershop quartets sing it, and the characteristic ringing quality of a barbershop seventh chord is exactly this: the singers have tuned to the partial rather than to the key.

Where the series comes apart #

Real strings are not ideal, and the deviation is measurable and consequential.

An ideal string has no stiffness — it resists being stretched and not being bent. A real string resists bending, and the stiffness adds a restoring force that grows with the curvature of the mode. Higher modes curve more, so they are pushed sharp.

The result is inharmonicity: partial $n$ sits not at $nf_1$ but at approximately

$$f_n = n f_1 \sqrt{1 + Bn^2},$$

where $B$ is a stiffness coefficient. For a good long piano string $B$ is small and the effect is a few cents by the tenth partial. For a short thick one — the bottom of a small upright — it is large, and the tenth partial can be over half a semitone sharp.

That has a direct practical consequence: pianos are tuned with stretched octaves. A tuner setting the octave above a bass note matches the upper note’s fundamental to the lower note’s second partial, which is already sharp — so the octave comes out wide. Accumulated across the keyboard, the top of a piano is around 30 cents sharp of the bottom’s nominal pitch and the bottom is around 30 cents flat. That curve was measured by Railsback in 1938, it is present in every well-tuned piano, and a piano tuned to mathematically exact octaves sounds wrong.

Instruments that are not harmonic at all #

Strings and pipes are one-dimensional. Plates, membranes and bells are not, and their modes are not whole-number multiples of anything.

A circular drumhead’s modes stand in ratios like 1 : 1.59 : 2.14 : 2.30 : 2.65 — solutions of a Bessel function, and nothing resembling integers. A struck bar gives 1 : 2.76 : 5.40. A bell gives another set again.

The perceptual consequence is that these instruments have a weak or ambiguous sense of pitch. A timpano has a definite pitch because its loading and the enclosed air pull several of its modes into near-harmonic ratios — the design does the work the geometry does not. A cymbal has essentially none.

It also means beat-based tuning does not apply to them. There is no pair of partials that should coincide, so there is nothing for beatless to mean, and tuning an inharmonic instrument is a different craft with different criteria.

Gamelan tuning is the largest working example. The instruments are metallophones and gongs with inharmonic spectra, the tuning systems fit those spectra, and no theory based on the harmonic series predicts them.

What a listener hears instead of the series #

The series is present in the signal, and it is emphatically not what a listener hears.

A note with partials at 200, 400, 600 and 800 hertz is heard as one sound with a pitch of 200 hertz and a quality. The partials are not heard as separate notes, though they are separately present and can be picked out with practice or with the right resonator.

Even more strikingly, remove the 200 hertz component entirely and the pitch does not change. A tone with partials at 400, 600 and 800 is still heard at 200 — the missing fundamental — because the auditory system infers a pitch from the pattern of spacings rather than reading it off the lowest component. This is why a small radio speaker that cannot reproduce 100 hertz still conveys a bass line.

So the series determines pitch and timbre, and does so through an inference rather than a measurement. The ear is not a microphone, and the harmonic series is the clearest case: what is in the air and what is heard are different objects.

Where the energy goes #

The frequencies are set by the geometry. How strong each partial is depends entirely on how the string is excited, and that is where a player’s control lives.

A mode can only be excited if the string is displaced at a point where that mode moves. Pluck a string exactly at its midpoint and every even-numbered mode has a node there — so none of them is excited at all, and the resulting tone is odd-partials-only and hollow, like a clarinet.

Pluck near the bridge and the displacement is close to a node of nothing, so every mode is excited, and the tone is bright and thin. Guitarists move their picking hand between the two positions constantly, and they are selecting a spectrum.

The same principle governs the piano’s hammer position. Piano hammers strike at roughly one-seventh to one-ninth of the string’s length, and the choice is deliberate: striking at exactly one-seventh would place a node of the seventh partial at the strike point and suppress it. The seventh partial is 31 cents flat of anything on the keyboard, so suppressing it removes a note that would clash with the tempered scale. Nineteenth-century piano makers arrived at that empirically and it is now standard.

Touching a node #

The clearest demonstration that the modes are real and separable takes one finger.

Touch a string lightly at its midpoint and pluck. The fundamental requires motion at the midpoint, so the touch kills it; the second partial has a node there and is unaffected. What sounds is the octave, pure and clear, and it keeps ringing after the finger is removed.

Touch at a third of the length and the third partial survives — an octave and a fifth above the fundamental. At a quarter, two octaves. This is the harmonics technique on every string instrument, it is available on any stretched string including a guitar in a shop, and it is direct physical evidence that all these modes are present simultaneously in an ordinary plucked note.

It also explains the natural harmonic’s tuning quirk. The third harmonic is a pure 3:2 above the octave, not a tempered one, so a harmonic on a violin’s A string is two cents sharp of the stopped note at the same nominal pitch. String players know this and adjust; it is one of the places where the tempered scale and the physical string disagree audibly and in ordinary playing.

Whose music, and when #

The mathematics of the vibrating string was worked out in the eighteenth century by d’Alembert, Euler and Daniel Bernoulli, and the fact that a string vibrates in several modes at once was Bernoulli’s contribution and was controversial.

The musical facts long predate the mathematics. Dividing a string in simple ratios and hearing the results is Pythagorean, and the natural trumpet and horn — instruments with no valves, playing only the harmonic series of a single tube — made the series audible to every European listener for centuries before anyone could write down why.

Those instruments are a good demonstration of the series’ quirks. A natural horn can play partials 2 through 16, which gives a usable scale in the upper range and an unusable one lower down, and includes the flat seventh partial and the stranded eleventh. Composers wrote around them; players learned to bend the awkward ones with the hand in the bell; and the eleventh partial in eighteenth-century horn writing is often notated as an F that the player is expected to correct.

Closed pipes, and the clarinet’s twelfth #

The string is the standard example and the pipe is the one that shows the boundary conditions doing real work.

A pipe open at both ends behaves like a string: pressure nodes at both ends, all whole-number multiples available, overblows to the octave. A flute does this.

A pipe closed at one end has a pressure antinode at the closed end and a node at the open one. Only odd multiples fit, so the second partial does not exist, and the first available overblow is the third partial — a twelfth rather than an octave.

That single fact reshapes the clarinet. A woodwind’s fingering system repeats at the overblow interval, so a flute needs fingerings for twelve semitones before the pattern repeats and a clarinet needs nineteen. The clarinet’s notoriously complicated fingering, its awkward throat register, and the break that every beginner struggles with are all consequences of a boundary condition at one end of a tube.

A closed pipe also sounds an octave lower than an open one of the same length, which is why a clarinet is longer-sounding than a flute of similar size and why stopped organ pipes are half the length of open ones at the same pitch.

Where the model stops #

The ideal string. No stiffness, no damping, fixed ends, small amplitude. Real strings violate all four, and the inharmonicity above is only the first correction.

Amplitudes are not predicted. The frequencies come from the boundary conditions; the relative strengths come from how the string is excited. Plucking a string at its midpoint suppresses all even partials; plucking near the bridge produces a bright spectrum with strong upper partials. Every figure here draws a plausible amplitude set rather than a derived one.

Steady state. The series describes a sustained vibration. The first fifty milliseconds of a real note are a mess of transients that carry much of the instrument’s identity and are not in the picture.

One string. A piano note has three, slightly detuned; a guitar has sympathetic resonance from the others; every instrument has a body with its own modes. The single-string picture is a component of a real note rather than a description of one.

The series as a scale, and the instruments that had no choice #

Before valves, a brass instrument could play the harmonic series of its tube and nothing else. That constraint produced a repertoire and a set of habits worth knowing about.

The natural trumpet’s usable range is partials 2 to 16 or so. Low down, the partials are far apart — an octave, then a fifth, then a fourth — so only fanfares and arpeggios are available. High up, around partials 8 to 16, they are a step apart and a diatonic melody becomes possible. That is the clarino register, it is where Baroque trumpet writing lives, and it is why those parts are uniformly and terrifyingly high.

The series’ quirks are audible in the repertoire. The eleventh partial sits halfway between F and F-sharp and is neither; the thirteenth is similarly stranded. Composers avoided them, wrote around them, or expected the player to bend them with lip or hand. Handel’s trumpet parts step carefully around exactly these notes.

When valves arrived in the nineteenth century they gave the player several tubes and therefore several series, and the awkward partials stopped mattering. What was lost was the characteristic sound of a player negotiating a constraint, which is why period-instrument performance of Baroque trumpet parts sounds different in a way that is not only about timbre.

The ladder from here #

Later rungs: standing waves derived from the wave equation. Modes of pipes, open and closed. Inharmonicity and the Railsback curve. The missing fundamental and pitch inference. Formants, and why a vowel is a vowel. Plates, membranes and bells. Gamelan tuning and its instruments. Excitation, and how plucking position sets the spectrum. Sympathetic resonance. And the natural horn, which is an instrument that plays nothing but this series and had a repertoire written for it.

Mersenne published the laws relating a string’s frequency to its length, tension and mass in 1636, before Newton was born and before anybody had a wave equation to derive them from. He got them right by measurement, using strings long enough that he could count the vibrations by eye.