Plucked String Modal Synthesis (2D): Harmonics, Inharmonicity, Pickup Position and Spectrum

Presets

Live readouts

Graphs

Pickup signal over the first three periods (real time)

Spectrum of the pickup signal: amplitude of each partial in dB

Stiffness: how far each partial lies above n times the fundamental, in cents

Decay time T60 of each partial (damping grows with frequency)

Pluck

Numerical checks

running...

How to use

The string vibrates in slow motion above the fretboard: the orange line is the real shape; the coloured traces are the three strongest modes that add up to it. Drag the string up or down and release to pluck it at that point (the sound plays, so check your volume). Presets set up a guitar, a stiff piano string, a harp string, a pluck at a node, one pure mode, and a rubber band. The lower graphs show what a pickup at the marked position would deliver.

The science

The ideal string. A string of length L, tension T and mass per length mu obeys the wave equation u_tt = c^2 u_xx, c = sqrt(T / mu) with fixed ends. Its normal modes are standing waves sin(n pi x / L) with frequencies f_n = n c / (2 L) = n f1: exactly harmonic. The page takes f1, L and mu as inputs and derives the tension, T = mu (2 L f1)^2.

A plucked triangle. Pulling the string to height h at position x0 = beta L and releasing it gives the initial shape of two straight segments. Projecting it onto the modes gives a_n = 2 h sin(n pi beta) / ( n^2 pi^2 beta (1 - beta) ) so the harmonics fall like 1/n^2 and every harmonic with n beta an integer is missing (pluck at a fifth of the length and the 5th harmonic vanishes). A soft finger rounds the corner; that multiplies a_n by exp(-(n pi s)^2 / 2), a Gaussian smoothing of width s L.

The motion. u(x,t) = sum_n a_n sin(n pi x / L) e^(-sigma_n t) [ cos(w_n' t) + (sigma_n / w_n') sin(w_n' t) ] The energy of the pluck is T h^2 L / (2 x0 (L - x0)), shared among the modes in proportion to w_n^2 a_n^2.

Stiffness and damping. A real string resists bending, which stretches the partials: w_n = n w1 sqrt(1 + B n^2) with the inharmonicity B = pi^2 E I / (T L^2) (about 1e-5 for a guitar string, 1e-4 to 1e-3 for piano strings). Damping is modelled as sigma_n = sigma0 + sigma2 w_n^2: air drag affects all partials alike, internal friction hits the high ones harder, which is why a plucked note loses its brightness first. The T60 of partial n is 6.91 / sigma_n.

The pickup. A magnetic pickup responds to the string velocity at its position x_p, a piezo to the displacement. Partial n is weighted by sin(n pi x_p) (and by w_n for velocity): a pickup near the bridge hears the high partials, one at the middle misses the even ones.

Honest simplifications. (1) One polarisation only; real strings also move sideways and exchange energy through the bridge, giving the double decay of a guitar. (2) The bridge and body are rigid and the string is not coupled to a soundboard, so there is no body resonance. (3) Damping is linear and the amplitude small, so there is no tension modulation (the pitch glide of a hard pluck). (4) The audio is the pickup signal of this model, not a recording. (5) The picture is slowed down by the factor you choose; the sound is real time.

Checks run in the page. An independent finite-difference wave solver against the modal sum; the Parseval energy of 3000 modes against the elastic energy of the triangle; the frequency measured from zero crossings; the vanishing fifth harmonic; the spectral peak of partial 10 of a stiff string, measured in the rendered waveform, against the stiff-string formula; and the e^-1 decay of a single mode.