The Attack Changes the Tuning

Leave the tuning peg alone. Leave the fret alone. Change only how far you pull the string before letting go.

The obvious expectation is a louder note, perhaps a more aggressive color. But a sufficiently large pluck can change something less obvious: the frequencies of the string's vibration. They can begin above their settled values and slide downward as the note decays.

I wanted to understand how the same string, spanning the same two endpoints, could briefly behave as though its tuning had changed. The answer starts with a distinction easy to miss while looking down a guitar neck: the distance between the ends is not the length of the path between them.

The extra road between two fixed places

Pull a string sideways and its path becomes longer. In the simplest picture, it is two straight segments meeting at your picking hand. The endpoints have not moved, but the string has stretched.

That extra stretch adds tension. Greater tension makes transverse waves travel faster, which tends to raise the string's modal frequencies—the frequencies associated with its different patterns of vibration. A sufficiently large pluck therefore changes the conditions governing the note, not merely the amount of motion under those conditions. As vibration loses energy, the short-time average of that extra tension relaxes and the frequencies approach their settled values. [1, §3.3; 2, §2]

“Longer string means lower note” is the wrong shortcut here. That familiar rule describes changing the speaking length while other quantities are held fixed. Sideways stretching also changes tension. Lee and colleagues explicitly discuss the competing effects of increased tension and increased path length; in their approximation the tension effect wins. [1]

The geometry supplies a useful hint about why a hard attack need not be just a scaled-up soft one. For a shallow triangle, the extra path grows approximately with the square of the sideways displacement.

I checked an ideal example: endpoints 65 centimetres apart, with the pluck one fifth of that distance from an end. A one-millimetre displacement adds about 4.8 micrometres of path. Two millimetres adds about 19.2 micrometres. Twice the displacement, nearly four times the extra path.

Those are geometry calculations, not measurements of a guitar or predictions of its instantaneous pitch. Turning extra path into a frequency trajectory also requires the string's elasticity, initial tension and dynamics. But the calculation exposes the important departure: the string's own motion can alter its restoring conditions. The material need not have an exotic nonlinear elastic law; the geometry already supplies a nonlinearity. [2, §§1–2]

A note with moving components

In a 2009 study, Nelson Lee, Julius O. Smith III, Jonathan Abel and David Berners analyzed recorded guitar tones one partial at a time. They isolated a component with a bandpass filter, then used short-time Fourier analysis and interpolation to estimate how its frequency changed. [1, §2]

Their illustrated high-e recording has a second partial near 660 hertz. It falls by nearly one hertz over roughly a second and a half. That is a small fractional shift—about 0.15 percent, or roughly 2.6 cents. A semitone spans 100 cents. This conversion only gives the scale of the paper's approximate description; I have not recovered exact endpoints from its recording.

It is also the second partial, not a one-hertz fall of the fundamental.

The authors tracked the first fifteen partials and normalized each by its own settled frequency. Their similar trajectories support a compact model: each partial has an initial fractional frequency excess that relaxes approximately exponentially. The illustrated fit has a time constant of about 0.305 seconds. That does not mean the glide suddenly ends then; an exponential keeps approaching its baseline over several time constants. [1, §§2–3]

A common fractional shift is not a common number of hertz. If one partial is twice as high as another, the same percentage change moves it twice as far in hertz. The components can travel together proportionally without collapsing into one frequency.

The paper also examines a softer high-e pluck. Its smaller glide can be approximated with the same time constant and a different initial excess. That is a useful comparison, not a law assigning one decay rate to every guitar, string and touch. The authors' aim is to reproduce the trajectory for sound synthesis, not to recover a complete set of independently measured string properties. [1, §§1, 3.2]

A larger glide—in a different kind of evidence

A separate paper from the same conference makes the mechanism more explicit. Balázs Bank models a low-E string with a nominal fundamental of 82.4 hertz and a five-millimetre triangular displacement. His example reports a frequency increase of about 3.6 percent, roughly half a semitone. [2, §2/Fig. 1]

That example is a simulation, not a recording of someone striking a low E. It is not a contradiction of Lee's much smaller recorded shift: different string, different excitation, different evidence.

Bank separates rapidly oscillating tension from a slowly changing, short-time-average increase. In the tension-modulation approximation, that average contribution is related to the string's vibrational energy—the extra energy in the moving string, not all the elastic energy already stored in a silent, tuned one. As the vibration dissipates, this contribution dwindles. [2, §§2, 4]

The distinction also explains why a single exponential can be useful without being compulsory. If one mode's displacement envelope decays exponentially, its squared-amplitude contribution has half the amplitude envelope's time constant. But different modes can lose energy at different rates. Their combined contribution need not be one perfect exponential; Bank explicitly notes that qualification. [2, §5.2]

These are different research contributions, not two independent recording experiments confirming the same number. They also draw on shared earlier nonlinear-string theory. A successful compact model is valuable without being the complete story of every string, bridge and instrument body.

Brightness is a different axis

In the small-amplitude explanation of plucking position, moving the hand changes the mixture of modes while their frequencies remain fixed. That is one way a gesture changes tone color.

Here the frequencies themselves move. Imagine a spectrum drawn as peaks on a graph. Changing their relative heights is different from moving their horizontal positions. A real attack can do both.

This is the distinction I wanted: the picking hand can choose how strongly different motions begin, but it can also push the string beyond the approximation that gives those motions fixed frequencies. “Harder means louder” leaves something out. “Harder means brighter” still leaves something out.

Neither paper, as studied here, tells me exactly what a listener would report from a particular guitar. Tracking a physical partial is not the same as measuring perceived pitch. I read both complete papers, including their methods and captions, but did not listen to their examples, inspect raw recordings or run a new experiment. The small high-e shift is not an audibility threshold, and a sharp attack is not by itself a diagnosis of an instrument fault.

The next question is enticingly practical: in a controlled listening comparison, what happens if one sound's frequencies settle while another's frequencies stay fixed and only its spectral balance changes? Would listeners reliably separate pitch motion from a changing color?

For now, the mechanical lesson is enough to change the way I frame the note. The fret and tuning peg specify important conditions. The attack can briefly revise them.

Sources

[1] Nelson Lee, Julius O. Smith III, Jonathan Abel and David Berners, “Pitch Glide Analysis and Synthesis from Recorded Tones”, DAFx-09, Como, September 1–4, 2009. Especially §§2–3 and Figs. 1–7 for recorded-tone analysis and fitting; §3.3 for its physical approximation.

[2] Balázs Bank, “Energy-Based Synthesis of Tension Modulation in Strings”, DAFx-09, Como, September 1–4, 2009. Especially §§1–2/Table 1 for the regime and geometry, Fig. 1 for the simulated low-E example, §4 for energy and average tension, and §5.2 for the exponential approximation. Rendered figure images and linked sound examples were not inspected here; figure descriptions above use the papers' text and captions.