How is string tension calculated?

An ideal string vibrates at f = √(T/μ)/(2L), so the tension is T = μ(2Lf)², where L is the scale length and μ the linear density (mass per metre). For plain steel μ = ρ·πd²/4 exactly. Wound strings have a core and a wrap, so the tool treats them as a solid cylinder of the same outer diameter with an effective density back-calculated from maker tension charts, good to about 5–10%. If the packet lists a unit weight in lb/in, enter it for an exact result.

Why are frets not evenly spaced?

Each semitone multiplies the frequency by 2^(1/12), so the vibrating length must be divided by the same factor. Fret n sits at d = L(1 − 2^(−n/12)) from the nut, the spacing shrinks up the neck and the 12th fret lands exactly at half the scale. The traditional rule of 18 (take 1/17.817 of the remaining length each time) is the same formula.

How different are just intonation and equal temperament?

Just intonation uses simple ratios (major third 5/4, fifth 3/2) and sounds purer, but only in one key. Equal temperament splits the octave into 12 equal steps so every key works, at the cost of major thirds about 14 cents sharp and fifths about 2 cents flat. The cents column in the fret table shows the difference, with values over 10 cents in red.

Why does plucking at the middle sound hollow?

A plucked string starts as a triangle. Expanded as a sine series, harmonic n has amplitude proportional to sin(nπp)/n², where p is the pluck point as a fraction from the bridge. If the pluck point sits on a node of harmonic n (sin(nπp) = 0), that harmonic is not excited at all. Plucking at p = 1/2 removes every even harmonic, giving a hollow tone; plucking near the bridge adds high harmonics and sounds brighter. An electric guitar pickup filters the sound in the same way depending on where it sits.

Why are pianos tuned with stretched octaves?

Steel wire resists bending, so partial n is at f_n = n f₁ √(1 + Bn²), slightly above the whole-number multiple, with B = π³Ed⁴/(64TL²). Tuners set octaves so the partials do not beat, which makes each octave a little wider than 2:1. Over the keyboard this pushes the treble sharp and the bass flat: the Railsback stretch curve. Short, thick strings have larger B, so uprights and baby grands show it more than concert grands.

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🎸String Tension Calculator - Fret Positions, Pluck Harmonics, Inharmonicity

Pick an electric, acoustic, classical, bass, ukulele or violin set and get the tension of every string (N, kgf, lbf) from scale length, pitch, gauge and material, or work backwards to the gauge for a target tension. The fret page lists equal-tempered fret positions with just-intonation comparison. The pluck page animates the string as a sum of modes, shows its spectrum and missing harmonics and lets you hear a synthesized tone. The inharmonicity page computes the B coefficient of piano wire and how sharp each partial runs.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use String Instrument Physics: Tension & Fret Calculator

1On the tension tab choose a string-set preset, or enter note, gauge and material for each string
2Set the scale length and A4 reference to see the tension of each string and the whole set
3Switch to the fret calculator, set scale length and fret count, then copy the CSV or print the table
4On the pluck tab drag the pluck point and pickup, watch the spectrum and press play to listen
5On the inharmonicity tab pick a piano string to see sharp partials and why pianos use stretched tuning

FAQ

How is string tension calculated?: An ideal string vibrates at f = √(T/μ)/(2L), so the tension is T = μ(2Lf)², where L is the scale length and μ the linear density (mass per metre). For plain steel μ = ρ·πd²/4 exactly. Wound strings have a core and a wrap, so the tool treats them as a solid cylinder of the same outer diameter with an effective density back-calculated from maker tension charts, good to about 5–10%. If the packet lists a unit weight in lb/in, enter it for an exact result.
Why are frets not evenly spaced?: Each semitone multiplies the frequency by 2^(1/12), so the vibrating length must be divided by the same factor. Fret n sits at d = L(1 − 2^(−n/12)) from the nut, the spacing shrinks up the neck and the 12th fret lands exactly at half the scale. The traditional rule of 18 (take 1/17.817 of the remaining length each time) is the same formula.
How different are just intonation and equal temperament?: Just intonation uses simple ratios (major third 5/4, fifth 3/2) and sounds purer, but only in one key. Equal temperament splits the octave into 12 equal steps so every key works, at the cost of major thirds about 14 cents sharp and fifths about 2 cents flat. The cents column in the fret table shows the difference, with values over 10 cents in red.
Why does plucking at the middle sound hollow?: A plucked string starts as a triangle. Expanded as a sine series, harmonic n has amplitude proportional to sin(nπp)/n², where p is the pluck point as a fraction from the bridge. If the pluck point sits on a node of harmonic n (sin(nπp) = 0), that harmonic is not excited at all. Plucking at p = 1/2 removes every even harmonic, giving a hollow tone; plucking near the bridge adds high harmonics and sounds brighter. An electric guitar pickup filters the sound in the same way depending on where it sits.
Why are pianos tuned with stretched octaves?: Steel wire resists bending, so partial n is at f_n = n f₁ √(1 + Bn²), slightly above the whole-number multiple, with B = π³Ed⁴/(64TL²). Tuners set octaves so the partials do not beat, which makes each octave a little wider than 2:1. Over the keyboard this pushes the treble sharp and the bass flat: the Railsback stretch curve. Short, thick strings have larger B, so uprights and baby grands show it more than concert grands.

Features

8 common string-set presets: electric, acoustic, classical, bass, ukulele and violin
Tension in N, kgf and lbf, with an override for the maker’s unit weight
Equal-tempered fret table with spacing, just positions and cent offsets; copy as CSV or print
Animated plucked string built from modes, with missing harmonics and pickup filtering
In-browser additive synthesis to hear ideal and stiff strings
Piano-wire inharmonicity coefficient B and partial-deviation chart

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