Guitar String Vibration Frequency Calculator
Computes the fundamental frequency of a guitar string from vibrating length, tension and linear mass density via f = (1/2L) sqrt(T/mu).
—
Vibrating string frequency: formula and example
For an ideal vibrating string, Mersenne's law puts the fundamental at f = (1 / 2L) · √(T / μ). Here L is the speaking length (m), T the tension (N), and μ the linear mass density (kg/m). On a 0.65 m scale in standard guitar tuning you get E2 = 82.4 Hz, A2 = 110 Hz, D3 = 146.83 Hz, G3 = 196 Hz, B3 = 246.94 Hz, and E4 = 329.63 Hz. If you double the tension the pitch climbs by √2, about 7 semitones; cut the length in half and the frequency doubles, which is a full octave.
Context and applications
Luthiers lean on Mersenne to choose gauges that land on the target pitch without the string feeling too tight. Scale length plays into this. A Fender 25.5" (648 mm) string tuned to E4 needs roughly 7% more tension than a Gibson 24.75" (629 mm) string of the same gauge, which is why a Fender feels stiffer under the fingers. Nylon on a classical guitar (μ ≈ 0.0004 kg/m on the high E) runs at about half the tension of steel on an acoustic (μ ≈ 0.0007 kg/m), and that gap is what you feel as the softer touch.
FAQ
Why does my low E go sharp when I bend? A bend stretches the string, which raises T and so raises f. Since f ∝ √T, even a small change in tension is enough to hear.
Does stiffness affect the result? Mersenne assumes the string is perfectly flexible. Real ones aren't, and their bending stiffness pushes the upper partials slightly sharp. That's inharmonicity, and you notice it most on thick wound bass strings.
What μ should I use? Check the maker's specs. D'Addario and Ernie Ball list the unit weight of each string in lb/in or kg/m. A plain steel 0.010" comes out around 0.00040 kg/m, while a wound 0.046" bass string is closer to 0.0067 kg/m.
Related Tools
Belt Span Natural Frequency
Calculate the natural vibration frequency of a belt's free span, f_n = (1 ÷ (2·L))·√(T/m), from the free span length L (m, the distance between pulleys), the belt tension T (N) and the mass per unit length m (kg/m). A belt's free span, between two pulleys, behaves like a stretched STRING (like a guitar string): when disturbed, it vibrates at a natural frequency depending on its tension and mass. The HIGHER the tension, the HIGHER the frequency (tighter string, higher pitch); the higher the mass per metre, the lower the frequency. This relation is the basis of a clever, widely used method to MEASURE belt tension in the field: the SONIC (or frequency) tension meter — the technician 'plucks' the belt to make it vibrate, and a sensor (or phone app) measures the sound frequency; knowing the span length and belt mass, the tension is computed back (inverting the formula). It is far more practical and accurate than the old methods of measuring deflection under a force. Keeping the correct tension is essential: a slack belt slips (loses power, heats, wears) and an over-tight belt overloads the bearings and shortens belt life. Enter the span length, the tension and the mass per unit length.
Pizzicato String Audible Decay Calculator
Estimates audible vibration time of a plucked open string in guitar or violin from oscillator quality factor Q and fundamental frequency.
Ball Pass Frequency, Outer Race (BPFO)
Computes the ball pass frequency of the outer race (BPFO), the signature a localized defect on a bearing outer ring leaves in the vibration spectrum: BPFO = (number of elements ÷ 2) × shaft rotation frequency × (1 − element diameter ÷ pitch diameter × cosine of the contact angle). The result, in hertz, is the frequency at which a peak appears every time a ball or roller rides over the flaw; because it is not an integer multiple of shaft speed, it is distinguishable from unbalance and misalignment, which show up at 1× and 2× rotation. Since the races slip slightly, the measured frequency usually falls 1% to 2% below the theoretical one, so look for the band rather than the exact line. A 6205 deep-groove ball bearing has a BPFO near 3.6 times shaft speed, and it is always worth checking that BPFO plus the inner race frequency equals exactly the number of elements times the rotation frequency. Enter the number of rolling elements, the shaft rotation frequency, the rolling element diameter, the pitch diameter and the contact angle.
Guitar String Tension by Mass Calculator
Computes required tension in newtons on a guitar string for a given fundamental frequency from string length and linear mass density.
Larmor Frequency
Compute the Larmor angular frequency ω = γ·B (rad/s).
Jerk (Rate of Acceleration Change)
Compute the jerk, J = Δacceleration/Δtime, the rate of change of acceleration over time — the third derivative of position. High jerk causes jolts, vibration and wear; controlling it (jerk-limited or S-curve profiles) makes motion smooth, protecting mechanisms and improving the finish on CNC machines and elevators. Enter the acceleration change and the time interval.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.