About the String Vibration Frequency Calculator
This string vibration calculator finds the frequencies at which a stretched string fixed at both ends resonates. Enter the vibrating length, the tension and the string’s linear mass density — either directly in grams per metre or from the wire diameter and material — and it returns the wave speed on the string, the fundamental frequency, any harmonic you choose, its wavelength and the nearest note on the equal-tempered scale.
It is useful for physics students studying standing waves (Mersenne’s laws), for guitar, violin and piano builders and repairers checking a string gauge or scale length, and for anyone designing a monochord or sonometer experiment. The table lists the first eight harmonics so you can see the overtone series that gives a string its tone.
The model is an ideal flexible string: uniform, perfectly flexible and rigidly fixed at both ends. Real strings — especially thick, stiff piano strings — have slight inharmonicity, so their upper partials run a little sharp of exact whole-number multiples.
With the default inputs, the fundamental frequency (f₁) is 329.65 Hz. Change any value above to recalculate instantly.
How to use the string vibration frequency calculator
- 1Enter the vibrating length between the fixed ends.
- 2Enter the string tension in newtons.
- 3Give the linear density directly, or the wire diameter and material.
- 4Choose a harmonic number to inspect (1 for the fundamental).
- 5Read the frequency and nearest note, and check the harmonic table.
Formula and method
A transverse wave on a stretched string travels at v = √(T/μ), where T is the tension and μ the mass per unit length. With both ends fixed, only standing waves that fit a whole number of half-wavelengths into the length survive, so λₙ = 2L/n and the allowed frequencies are fₙ = n·v/(2L). These are Mersenne’s laws: pitch rises with tension (as √T), and falls with length and with mass per unit length.
For a solid round wire, μ is the material density times the cross-sectional area, ρπd²/4. Wound strings are heavier than a plain wire of the same outside diameter, so for those enter the manufacturer’s unit weight directly. The nearest note uses equal temperament with A4 = 440 Hz.
- fₙ
- Frequency of the nth harmonic (Hz)
- n
- Harmonic number (1 = fundamental)
- L
- Vibrating length (m)
- T
- Tension (N)
- μ
- Linear mass density (kg/m)
- ρ, d
- Material density (kg/m³) and wire diameter (m)
Worked examples
Guitar high E string (0.010" steel, 25.5" scale)
A 0.254 mm steel wire has μ = 7850 × π × (0.000127)² = 0.398 g/m. At 72.6 N, v = √(72.6 ÷ 0.000398) = 427 m/s, so f₁ = 427 ÷ (2 × 0.648) = 329.6 Hz — the E above middle C.
Heavy bass string, 5 g/m on a 34" scale
v = √(180 ÷ 0.005) = 189.7 m/s and f₁ = 189.7 ÷ (2 × 0.864) = 109.8 Hz, just below A2 (110 Hz). Raising the tension slightly to about 180.7 N would bring it exactly to pitch.
Third harmonic of the high E string
The third harmonic is 3 × 329.65 = 988.9 Hz with a wavelength of 2 × 0.648 ÷ 3 = 0.432 m. It sounds an octave and a fifth above the open string — the B played as a natural harmonic over the 7th fret.
Frequently asked questions
What is the formula for the frequency of a vibrating string?+
f = (n / 2L)·√(T/μ), where L is the vibrating length, T the tension, μ the mass per unit length and n the harmonic number. The fundamental (n = 1) is f₁ = √(T/μ) / 2L.
How does tension affect pitch?+
Frequency is proportional to the square root of tension. To raise a string by one octave (double the frequency) you need four times the tension; one semitone needs about 12.2% more tension.
Why do thicker strings sound lower?+
A thicker or denser string has more mass per unit length, which slows the wave on it. Since f ∝ 1/√μ, a string with four times the linear density sounds an octave lower at the same length and tension.
What are harmonics on a string?+
Harmonics are standing-wave patterns at whole-number multiples of the fundamental frequency. The second harmonic is an octave up, the third an octave plus a fifth, the fourth two octaves. Their mix gives each instrument its tone colour.
How do I find the linear density of a wound string?+
Wound strings are not solid wire, so the diameter formula underestimates their mass. Use the unit weight published by the string maker (often in lb/in; multiply by 17,858 to get g/m) and enter it directly.