Only some notes fit, and that is where discreteness comes from

A guitar string is a continuous object. There is nothing discrete about it anywhere — not the material, not the tension, not the space it occupies. Pluck it and it produces a definite pitch, and if the string is shortened by a fret it produces another definite pitch, and between them there is no note the string is willing to sustain.

Something continuous is producing something that comes in a list. Two centuries before anybody had a reason to expect that, the vibrating string was already doing it, and the explanation is entirely mechanical.

Two travelling waves, going opposite ways #

A standing wave is not a third kind of wave. It is two travelling waves added together, moving in opposite directions.

Send a wave down a string with a fixed end and it comes back. The reflection travels the other way through the same string, and by superposition the string’s actual shape is the sum. That sum has a property neither component has: certain points never move at all, because at those points the two waves are always exactly opposed.

Those points are the nodes, and what is worth noticing about them is that the cancellation there is permanent. Two waves half a cycle apart cancel wherever they meet; what the geometry of a reflection adds is that the same two are held opposed at the same places at every instant. So nothing is stationary about the constituents — only the map of where they agree and disagree stays put, and it is that map a listener hears as a note.

Those permanently still points are the nodes, and the points that swing furthest are the antinodes. Between two consecutive nodes lies half a wavelength, which is the fact that does all the work below.

The reflection also arrives inverted, and that inversion is the physics of the boundary rather than a convention. A fixed end cannot move; the only way for a wave to arrive there and for the end to stay put is for the reflected wave to cancel the incoming one exactly at that point. So the reflection is upside down, and the fixed end is guaranteed to be a node.

Counting what fits #

Now the constraint. Both ends are clamped, so both ends must be nodes, and nodes are half a wavelength apart. The length of the string must therefore be a whole number of half-wavelengths:

$$L = \frac{n\lambda}{2}, \qquad n = 1, 2, 3, \ldots$$

Wavelength is fixed by that condition, the wave speed is fixed by the string’s tension and density, and $v = f\lambda$ then fixes the frequency:

$$f_n = \frac{n v}{2L} = \frac{n}{2L}\sqrt{\frac{T}{\mu}}.$$

The integer arrived by counting. Nothing was quantised by decree; a whole number appeared because a whole number of half-wavelengths is what fits between two nodes, and half-fitting is not an option.

That last figure makes the general principle visible. Change the boundary conditions and the list changes. A pipe open at both ends has antinodes at both ends and gets the full harmonic series; a pipe closed at one end has a node there and an antinode at the other, and only the odd multiples fit. A clarinet is that pipe. It overblows to a twelfth — three times the frequency — rather than to an octave, and every clarinettist knows this as an awkward fingering fact before knowing it as a consequence of one end being stopped.

The structure — a differential equation plus a boundary condition, yielding a discrete set of allowed solutions — is exactly the structure that produces atomic energy levels. Historically it went the other way round: the mathematics of standing waves was fully developed in the eighteenth century, and quantum mechanics inherited it whole, which is why the equations for a particle in a box are the equations for a string on a violin with the symbols renamed.

Why a violin does not sound like a flute #

A plucked string does not vibrate in one mode. It vibrates in a great many at once, and the mixture is what an instrument’s character is made of.

The initial shape of a plucked string is a triangle — a kink where the finger was. That is not any single mode, but by superposition it can be written as a sum of modes, and the amounts of each are fixed by where the string was plucked. Every mode then oscillates at its own frequency, and the string’s subsequent motion is the sum.

The perceived pitch comes from the fundamental. The timbre comes from the recipe: how much second harmonic, how much third, and how the mixture changes as the note decays. A flute is nearly a pure fundamental. A violin has a rich series of strong upper harmonics. A struck bell is not harmonic at all — its modes are not whole-number multiples of anything, which is why a bell has a shimmer instead of a pitch, and why bell founders spend their effort tuning individual modes into agreeable ratios by grinding metal from the inside.

Plucking position selects the recipe directly, and by a rule that follows straight from the figure. A mode with a node at the plucking point cannot be excited, because the pluck displaces a point that mode requires to stay still. Plucking a guitar string exactly halfway along suppresses every even harmonic and gives a hollow, flute-like tone; plucking near the bridge, far from every low-order node, excites the high harmonics strongly and gives the bright, nasal sound. Nothing about the string changed. Only which modes were invited.

Two waves, again #

It is worth checking that the standing wave really is two travelling waves added rather than a new object, because the claim is doing structural work.

The same standing pattern appears whenever travelling constituents are held in a fixed relationship. Two sources interfering produce dark and bright lines that stay where they are while the waves themselves keep moving — a standing pattern with travelling parts, which is exactly the standing wave’s situation with the geometry rearranged. In both cases what is stationary is the map of agreement, not the medium.

In both cases two waves overlap, in both cases the sum has places that never move, and in both cases nothing about the individual waves has changed. The difference is only in the arrangement: two sources side by side give a fixed pattern in space, and a wave meeting its own reflection gives a fixed pattern along a line.

That equivalence has a practical payoff. Anything that reflects a wave will produce standing waves, whether or not it was meant to — in a radio feed line, an ultrasound probe, an exhaust pipe, a room with parallel walls. Much of acoustic and electrical engineering consists of arranging for the reflection not to happen, which is what impedance matching is for, and the standing wave ratio is the standard measure of how badly it has been arranged.

What the picture leaves out #

The diagram is a catalogue of shapes, not a description of any real vibration. Three things it cannot show are exactly the three things that decide how an instrument sounds.

It cannot show amplitude. All modes are drawn the same size; in a real note they are not, and the ratios are the timbre.

It cannot show decay. Higher modes lose energy faster, because they bend the string more sharply and internal friction scales with curvature. So a plucked note is bright at the start and mellows as it rings — the recipe is not fixed, it evolves, and the evolution is a large part of what makes a real instrument sound alive and a naive synthesis sound dead.

It cannot show coupling. A string alone moves almost no air and would be nearly inaudible. What is heard is the body of the instrument, driven through the bridge, with its own set of resonances that emphasise some frequencies and swallow others. The measured spectrum of a violin is the string’s harmonic series filtered by the body’s response, and it is the body that a maker’s reputation rests on.

What the discreteness costs #

A list of allowed frequencies is a wonderful thing to have and an expensive thing to own, because the list is short and fixed, and anything not on it is unavailable.

A string of a given length, tension and thickness can produce its fundamental and its harmonics and nothing in between. Every other pitch has to be obtained by changing one of the three quantities in $f_1 = (1/2L)\sqrt{T/\mu}$, and the whole visible design of stringed instruments is the machinery for doing so. Frets and fingerboards change $L$. Tuning pegs change $T$. Multiple strings of different gauges change $\mu$. A guitar carries six strings because one string with a usable fingerboard cannot cover the range, and it carries twenty frets because the alternative — a continuous slide — gives up the reliability that made the discreteness attractive in the first place.

The cost is easiest to count on a piano, which does not fret at all and therefore needs one string set per note. Its range runs from 27.5 Hz to 4,186 Hz, a factor of 152. Length alone cannot deliver that: the longest string is about two metres and the shortest about five centimetres, which is a factor of forty, and the bottom of the instrument would need a string twelve metres long to make up the difference. So the remaining factor is bought by loading the bass strings with copper winding to raise $\mu$, and by tension. About 230 strings are required, the frame carries something like twenty tonnes of tension, and the frame has to be cast iron because nothing else will hold it. That mass of ironwork is the physical price of a quantisation condition.

Brass instruments pay the same bill differently and more visibly. A bugle has no valves, so it can play only its harmonic series — which is why bugle calls are made of the handful of notes that series contains, and why the gaps at the bottom of it are audible as the shape of every military call ever written. Valves were added in the 1810s for no other reason than to switch in extra tubing and shift the whole series, and the trombone’s slide is the one instrument that solved the problem by refusing the discreteness altogether.

What a sharp note costs in time #

There is a second charge, and it is the deeper one: a frequency is only as well defined as the length of time it is allowed to run.

The mode frequencies above were derived for a string vibrating forever. A note that starts and stops is not a single frequency — it is a packet, and building a packet requires a spread of components. The relationship is quantitative: a tone lasting a time $\Delta t$ has a frequency spread of roughly $\Delta f \approx 1/\Delta t$, and no amount of care in the construction reduces it.

So a note held for a second has a pitch defined to about a hertz, which is far finer than the ear needs. A note lasting twenty milliseconds has a spread of about fifty hertz, and at the bottom of the piano that is more than the interval between adjacent keys — the pitch is genuinely not there to be heard, rather than being heard imperfectly. This is why very short sounds are clicks rather than notes, why a bass drum has a thud and not a pitch, and why bass notes in music are given longer to speak than treble ones by nearly every composer who has ever written any.

It is also the classical original of a much more famous statement. Time and frequency are related by a transform, so a quantity narrow in one is broad in the other, and the product has a floor. Replace frequency by energy and the sentence becomes the energy–time uncertainty relation, with the same proof and the same content. The reason an excited atomic state with a short lifetime has a broad spectral line is the reason a short note has no pitch, and nothing quantum is required for either — only the fact that a finite pulse is a mixture.

Choosing a mode by touching a node #

The node argument has a use that takes about five seconds to test on any stringed instrument, and it is the cleanest confirmation available that the modes are really there and really independent.

Touch a guitar string very lightly at its midpoint — do not press it to the fret — and pluck. The result is a clear note an octave above the open string, and it goes on ringing after the finger is lifted.

What the finger did was impose a node. Every mode that requires the midpoint to move is damped out immediately; every mode that already has a node there is untouched, and those are exactly the even harmonics, whose lowest member is the octave. Touch at a third of the length and the surviving modes are the multiples of three, giving an octave and a fifth. The available notes are a list, the list is read straight off the figure at the top of this page, and the technique is old enough to have its own name in three languages.

The list will not close #

There is a consequence of the whole-number ratios that shaped several centuries of music, and it is a piece of arithmetic rather than a matter of taste.

The string produces a harmonic series, so the intervals that sound consonant are simple ratios: the octave is 2:1, because it is the second harmonic, and the fifth is 3:2, because it is the third harmonic against the second. Build a scale by stacking pure fifths and it ought to come back round: twelve fifths should be seven octaves.

They are not equal. Twelve fifths is $(3/2)^{12} = 129.746$ and seven octaves is $2^7 = 128$, and the ratio between them — the Pythagorean comma — is about a quarter of a semitone. Powers of three are never powers of two, so no amount of care closes the circle; it is the same impossibility as finding a common measure for the side and diagonal of a square, and Pythagoras’s followers are said to have taken that one badly as well.

An instrument that can bend its pitch does not care, and a singer or a violinist adjusts each interval to whatever the harmony wants. An instrument with a fixed list of notes has to choose. Equal temperament spends the discrepancy evenly: every fifth is flattened by two cents, which is inaudible, and the major third is consequently sharp by fourteen cents, which is not, and which is why a piano chord and a barbershop chord are audibly different objects.

So the keyboard’s twelve notes are not a natural list at all. They are a compromise between the list the string offers and a defect in the arithmetic — a quantisation condition and a closure condition that cannot both be satisfied.

Where the model stops #

The ideal string is a genuinely idealised object, and the deviations are audible rather than theoretical.

Perfect flexibility is the big assumption. A real string resists being bent, and that stiffness adds a restoring force that grows with curvature — so higher modes, being more curved, are pulled up in frequency. The overtones of a real piano string are therefore slightly sharp of exact multiples, by an amount that grows with mode number. This is called inharmonicity, and it is why pianos are tuned with stretched octaves: the top of the instrument is tuned sharp and the bottom flat, to match the overtones the strings actually produce rather than the ones the theory predicts. A mathematically perfect piano tuning sounds wrong.

Fixed ends. The ends are not perfectly rigid — if they were, no energy could reach the body and no sound would come out. The instrument works precisely because the boundary condition is slightly wrong. Perfect confinement and perfect isolation are the same thing, which is why an ideal resonator would be inaudible and every real one is a compromise between holding energy and releasing it.

Small amplitude. A hard pluck stretches the string and raises its tension, which raises the pitch. The effect is audible on a heavily plucked bass string, which sounds sharp for the first instant. Superposition is failing there, mildly — and it fails in exactly the way a pendulum’s period starts depending on amplitude, because both are the same linear approximation running out.

Constant tension and density. Real strings are wound, worn, and unevenly stretched, and the modes of a non-uniform string are not sinusoidal at all.

Standing waves that are not on strings #

The counting argument needs no string. It needs a wave, a region, and a boundary condition.

Sound in a room produces standing waves at wavelengths matching the room’s dimensions, which is why some bass notes boom and others vanish depending on where the listener sits. A microwave oven sets up a standing wave in a metal box — the walls are conductors, so the field must vanish at them, which is the boundary condition a clamped string obeys — and the hot and cold spots are its antinodes and nodes. That is the origin of the turntable, and it makes it possible to measure the speed of light with an oven, a chocolate bar and the manufacturer’s stated frequency.

Light between two mirrors gives the modes of a laser cavity, and the requirement that a whole number of wavelengths fit the round trip is what makes a laser’s output so nearly monochromatic — the mirrors doing their job by total internal reflection in some designs and by coating in others. Electrons confined to an atom give energy levels. Even the thermal motion in a gas was first counted this way, by asking how many standing waves fit in a box — an argument that worked beautifully for sound and failed catastrophically for light, and the failure is where quantum theory began.

The ladder from here #

Later rungs: the derivation of the modes from the wave equation and separation of variables. Pipes, and the end correction that makes them acoustically longer than they are. Two-dimensional modes on a drumhead, where the frequencies stop being whole-number multiples and the nodes become curves. Chladni figures, which make those nodal curves visible in sand. Inharmonicity and piano tuning. Resonance, and why a driven system responds enormously at a mode frequency. Room acoustics and modal density. And the transition to the continuum, where a long enough string has modes so closely spaced that the discreteness stops mattering.

Pythagoras is said to have found the whole-number ratios in vibrating strings and concluded that number governed the universe. The conclusion was overreaching and the observation was exactly right.