Musical String Frequency & Tension Calculator
This calculator estimates the frequency of an ideal stretched string from length, tension, linear mass density, and harmonic mode. It is useful for instrument design, string-gauge comparisons, and physics demonstrations of standing waves.
The output also compares the computed frequency with nearby musical notes so you can see whether the string is close to a familiar pitch.
How to use the calculator
- Enter the vibrating string length in centimeters.
- Enter the tension in newtons.
- Enter linear mass density in grams per meter.
- Select the harmonic mode, then click Calculate frequency.
Formula used
The script converts length to meters and mass density to kilograms per meter, then applies the ideal string frequency equation:
Here n is the harmonic number, L is vibrating length, T is tension, and μ is linear mass density.
Worked example
A 65 cm string under 70 N of tension with a 0.5 g/m linear density has a fundamental near 288 Hz. Increasing tension raises pitch by the square root of the tension change; doubling tension does not double the frequency.
What each input changes
String frequency responds predictably to each physical input. Shortening the active length raises pitch because the standing wave has less distance to travel. Raising tension also raises pitch, but the effect follows a square-root relationship, so a 21 percent tension increase raises frequency by about 10 percent. Increasing linear mass density lowers pitch because a heavier string moves more slowly under the same pull.
The harmonic selector multiplies the fundamental mode. The second harmonic is twice the fundamental frequency, the third harmonic is three times the fundamental, and so on. That makes the harmonic field useful for checking overtone demonstrations as well as ordinary open-string pitch.
| Change | Frequency effect | Design implication |
|---|---|---|
| Shorter vibrating length | Higher pitch | Frets, bridges, and scale length set the playable range. |
| Higher tension | Higher pitch | Useful for fine tuning, but limited by string strength and instrument load. |
| Higher mass per meter | Lower pitch | Heavier gauges reach low notes without requiring extreme length. |
| Higher harmonic number | Proportional increase | Models ideal overtones and natural harmonics. |
Design checks before trusting a result
Check units first. Length must be the vibrating scale length, not the total string length including wrap or tailpiece sections. Linear mass density should describe the actual string material and winding, usually from a manufacturer specification or a measured string mass divided by measured length. Tension should be the static pull in newtons, not a tuner reading or weight label in pounds unless it has been converted.
For instrument setup, compare the calculated frequency to a measured pitch from a tuner. If the calculator predicts the right general range but the measured pitch is sharp in higher harmonics, the string may have enough stiffness or endpoint error to cause inharmonicity. If the measured pitch changes dramatically while playing, the real tension or scale length may be changing under load.
Common interpretation mistakes
Do not treat the closest-note result as a tuning prescription by itself. The note comparison is based on a small reference table, while real instruments may target different octaves, temperaments, or calibration standards. A frequency near A4 at 440 Hz is meaningful only if that is the intended note and the instrument is using that concert pitch reference.
Also remember that safer instrument design usually changes gauge or scale length before pushing tension to an extreme. If a desired pitch requires much more tension than similar strings on the same instrument, the better conclusion may be that the string is too light, the scale is too long, or the target pitch belongs in another octave.
Limitations
Real strings have stiffness, winding structure, imperfect endpoints, temperature effects, and inharmonicity. Use this as a first-order physics estimate, then tune with measurements when building or setting up an instrument.
When documenting a setup, save the calculated frequency alongside the measured pitch and the string specification. That makes later gauge, tension, or scale-length changes easier to compare.