The Physics of a Plucked String

Every pitched note you hear from an oud, a tanbur, or a setar begins with the same physics: a stretched string, displaced and released, vibrating between two fixed points. The pitch you hear, the overtones that colour it, and the way the sound evolves over time all follow from a small set of physical principles. Understanding these principles is not just academic — it explains why frets are spaced the way they are, why thick bass strings behave differently from thin treble strings, and why plucking near the bridge sounds nothing like plucking over the soundhole.

What determines pitch

Established The fundamental frequency of a vibrating string — the pitch you hear — is governed by three properties: the vibrating length of the string, the tension applied to it, and the string's linear mass density (its mass per unit length). The relationship, known as Mersenne's law, is:

f = (1 / 2L) × √(T / μ)

In plain language: f is the frequency in hertz, L is the vibrating length (from nut to bridge, or from a fretted position to the bridge), T is the tension in newtons, and μ (mu) is the linear mass density in kilograms per metre.

Established This formula tells you three things a string player already knows intuitively. Shortening the string (pressing it against the fingerboard or a fret) raises the pitch. Tightening the tuning peg raises the pitch. And a thicker, heavier string at the same length and tension vibrates more slowly — which is why bass courses on the oud use wound strings with greater mass, not just longer or slacker ones.

The harmonic series

Established A vibrating string does not produce a single pure tone. It vibrates simultaneously at the fundamental frequency and at integer multiples of that frequency — 2f, 3f, 4f, and so on. These are called harmonics (or overtones, or partials). The second harmonic is twice the fundamental frequency, which is exactly one octave higher. The third harmonic is three times the fundamental — an octave and a perfect fifth above the original pitch.

Established These integer ratios are not arbitrary. They arise because the string can sustain standing waves only when a whole number of half-wavelengths fits between the two fixed endpoints. The first mode (the fundamental) has one half-wavelength spanning the full string length. The second mode has two half-wavelengths, with a node — a point of zero displacement — at the midpoint. The third mode has three half-wavelengths, with nodes at one-third and two-thirds of the string length. You can hear these modes individually by lightly touching a string at a node point and plucking: the finger damps every mode that does not have a node at that point, isolating the harmonic.

Established The harmonic series is where "natural" musical intervals come from. The octave is the ratio 2:1. The perfect fifth is 3:2. The perfect fourth is 4:3. These simple whole-number ratios were recognized by Pythagorean theorists and form the basis of Pythagorean tuning — the foundation on which both makam and dastgah interval theory ultimately rest. The connection between a vibrating string and a tuning system is direct and physical: the intervals that sound most consonant are the ones that correspond to the lowest, simplest ratios in the harmonic series.

Why frets get closer together

Established If you look at the neck of a tanbur or a fretted baglama, the frets near the nut (the top of the neck) are widely spaced, and they get progressively closer together as you move toward the bridge. This is not a quirk of construction — it is a direct consequence of how frequency relates to string length.

Established Pitch perception is logarithmic: an octave always corresponds to a doubling of frequency, regardless of starting pitch. Moving from 220 Hz to 440 Hz is one octave; moving from 440 Hz to 880 Hz is another. Because frequency is inversely proportional to string length, halving the vibrating length raises the pitch by one octave. Each successive equal interval requires a smaller absolute change in length. In 12-tone equal temperament, each semitone multiplies the frequency by 2^(1/12), approximately 1.0595. In the 53-comma system used in Turkish makam theory, each Holdrian comma multiplies the frequency by 2^(1/53), approximately 1.0131. In both cases, the fret positions follow a geometric (not arithmetic) progression — each fret is a fixed ratio of the remaining string length from the previous one, which means the physical gaps shrink as you ascend.

Inharmonicity: when real strings depart from the ideal

Established The formula above and the harmonic series it predicts assume a perfectly flexible string — one with zero stiffness. Real strings are not perfectly flexible. They resist bending, and this bending stiffness causes the higher partials to vibrate at frequencies slightly higher than the exact integer multiples predicted by the ideal model. The 2nd partial is a little sharp of 2f, the 3rd partial is a little sharper of 3f, and the deviation grows with each successive partial.

Established This effect, called inharmonicity, depends on the string's material properties and dimensions. It is proportional to the fourth power of the string diameter and inversely proportional to the tension and the square of the length. Thick, short, stiff strings exhibit the most inharmonicity; thin, long, flexible strings exhibit the least.

Practice For plucked string instruments used in makam and dastgah music, inharmonicity matters most in the bass register. The lower courses of the oud — wound nylon strings with relatively high mass and diameter — produce partials that are measurably sharper than exact harmonics. The tanbur's long neck (approximately 100 cm vibrating length for most strings) works in the opposite direction: greater length reduces inharmonicity, contributing to the instrument's clear, bell-like tone that makes precise comma-level intervals audible. Nylon and gut strings are substantially less stiff than steel, so instruments using them generally have less inharmonicity than steel-strung instruments — one reason the oud's harmonic relationships remain relatively clean despite its shorter scale length.

Plucking position and timbre

Established Where you pluck the string changes the timbre dramatically. The physics is straightforward: plucking at a point that coincides with a node of a particular harmonic cannot excite that harmonic. Plucking at the exact midpoint of the string, for example, suppresses the 2nd harmonic (and all even-numbered harmonics), producing a hollow, fluty quality dominated by odd harmonics. Plucking near the bridge excites a broad range of higher harmonics, producing a bright, incisive sound. Plucking over the soundhole or closer to the fingerboard emphasizes the fundamental and lower harmonics, producing a warmer, rounder tone.

Practice Oud players use this deliberately. A passage that calls for a warm, contemplative sound — a taksim opening, a slow seyir exploration — might be played with the risha (plectrum) positioned closer to the fingerboard or over the rosette. A rhythmic, articulated passage might use a plucking position closer to the bridge for clarity and brightness. The same string, at the same pitch, can produce distinctly different timbres depending on where the risha strikes. This is not a subtlety — it is a primary expressive tool.

Oud and guitar: why they sound different

Established Classical guitars and ouds share a similar acoustic design — plucked strings driving a thin wooden soundboard — but they sound quite different. Several physical factors account for this. The oud uses nylon strings (historically gut), which are less stiff and produce fewer prominent high-frequency overtones than steel. Nylon strings also have shorter sustain: the internal damping of the material absorbs energy faster, so notes decay more quickly, giving the oud its characteristic percussive attack followed by a rapid fade.

Established The oud's bowl-back body, assembled from thin staves of hardwood, behaves differently from the flat-back or slightly arched body of a guitar. The curved back acts as a more diffuse reflector, distributing sound less directionally. The overall effect — nylon strings, bowl-back resonance, short scale, higher internal damping — is a warm, intimate tone with a strong fundamental and relatively few sharp high-frequency components.

Practice This tonal character is not incidental to the music. The warm, rapidly decaying tone of the oud suits makam and maqam performance, where successive notes in a melodic line are meant to be distinct rather than blurred together. The shorter sustain also keeps dense ornamentation — trills, grace notes, hammer-ons — clean and articulate. A steel-strung instrument with long sustain would smear these micro-ornaments together.

Digital string synthesis: the Karplus-Strong algorithm

Established In 1983, Kevin Karplus and Alex Strong published a remarkably simple algorithm for synthesizing plucked-string sounds digitally. The method begins with a short buffer filled with random noise — representing the initial broadband energy of a pluck. The algorithm then cycles through this buffer repeatedly, replacing each sample with the average of itself and its neighbour. This is mathematically equivalent to a delay line (whose length sets the pitch) with a simple lowpass filter in the feedback loop.

Established The result sounds strikingly like a plucked string, because it reproduces the essential physical behaviour: high-frequency partials decay faster than low-frequency ones, just as they do on a real string where internal friction damps shorter wavelengths more quickly. Varying the initial noise content, the filter characteristics, and the delay length produces timbres ranging from a soft gut string to a bright metallic pluck.

This is the core technique behind the string synthesis in Meshk, the interactive practice companion to this site. When you hear a plucked oud or tanbur sound in Meshk, the audio engine is running a variant of this algorithm — a noise burst, a tuned delay line, and a filter that shapes how the tone evolves after the initial attack.