Musical String Frequency & Tension Calculator

Calculate String Frequency

Physics of Vibrating Strings

What Determines String Pitch?

The frequency (pitch) of a vibrating string is determined by three fundamental physical properties: the tension (force pulling on the string), the length of the vibrating segment, and the linear mass density (mass per unit length). These parameters are combined in a single elegant formula that governs the behavior of all stringed instruments, from violins to guitars to pianos. Understanding this relationship is essential for instrument makers, musicians, and physics students exploring wave mechanics.

When a string vibrates, it does so in patterns called modes or harmonics. The lowest frequency, called the fundamental frequency or first harmonic, produces the pitch we perceive. Higher harmonics vibrate at integer multiples of this fundamental and create the rich timbre or tone color of the instrument. The relationship between string properties and frequency is linear with some parameters and inversely proportional with others, reflecting the underlying physics of wave propagation in tensioned media.

The Wave Equation for Strings

The fundamental frequency of a vibrating string is given by:

where:

• = frequency in Hz (cycles per second)
• = harmonic number (1 for fundamental, 2, 3, 4... for higher harmonics)
• = length of the vibrating string in meters
• = tension in the string in Newtons (N)
• = linear mass density (mass per unit length) in kg/m

Understanding Each Parameter

String Length (L): Longer strings vibrate more slowly, producing lower frequencies. The relationship is inversely proportional: doubling string length halves the frequency. This is why bass instruments like cellos and double basses have long strings, while small instruments like ukuleles and mandolins have short strings. Frets on guitars allow musicians to effectively shorten the vibrating length and raise the pitch.

Tension (T): Higher tension increases the frequency (higher pitch). However, the relationship is not linear—frequency is proportional to the square root of tension. This means that to double the frequency, you must increase tension by a factor of 4 (quadruple). This is why tuning pegs on string instruments work gradually: small tension changes can produce large frequency changes due to the square root relationship.

Linear Mass Density (μ): Heavier strings (higher linear mass density) vibrate more slowly and produce lower frequencies. The relationship is inverse and proportional to the square root: doubling the linear mass density reduces frequency by a factor of √2. This is why bass strings on guitars are thicker than treble strings—the increased mass per unit length lowers the frequency for the same length and tension.

Harmonic Mode (n): The fundamental frequency corresponds to n=1. Higher harmonics (n=2, 3, 4...) vibrate at integer multiples of the fundamental. A string vibrating at its second harmonic (n=2) oscillates at twice the fundamental frequency, producing a pitch one octave higher. Higher harmonics are naturally excited when a string is plucked or struck, contributing to the instrument's tone.

Worked Example: Guitar String Calculation

A standard acoustic guitar's high E-string has these specifications:

• Length: 65 cm (typical scale length)
• Tension: 80 Newtons (approximately 8 kg of force)
• Linear Mass Density: 0.7 g/m = 0.0007 kg/m
• Harmonic Mode: 1 (fundamental)

Using the formula:

f = (1 / (2 × 0.65)) × √(80 / 0.0007)
f = (1 / 1.3) × √(114,286)
f = 0.769 × 338
f ≈ 330 Hz

This corresponds to approximately E4 (329.6 Hz in standard tuning), confirming our calculation. If tension increases to 120 N (1.5× original), the frequency becomes 330 × √1.5 ≈ 404 Hz, raising the note by about 2.4 semitones.

Practical Applications in Instrument Making

Guitar/Violin Construction: Luthiers (instrument makers) select string diameters and materials to achieve desired frequencies while maintaining reasonable playing tension. A string too thick becomes difficult to press; too thin breaks easily. The optimal balance is found through the frequency formula.

Tuning Stability: Environmental changes like temperature affect both tension (strings expand/contract) and linear mass density (material properties change). Understanding the frequency formula helps musicians anticipate how instruments will drift out of tune and how to compensate.

Piano Design: Pianos use the frequency formula to calculate optimal string lengths and weights for each note. Bass notes require very long strings with heavy wire, while treble notes use short, thin strings. The crossover from wound bass strings to plain treble steel occurs where the formula predicts equal efficiency.

Harp and Zither Design: These instruments use strings at fixed lengths, so tension and material selection entirely determine the pitch. Designers must calculate precise tensions for each string to achieve the desired scale while keeping materials practical and playable.

Advanced Concepts: Damping and Inharmonicity

The basic formula assumes ideal strings with no damping or stiffness. Real strings have some rigidity (stiffness), especially metal strings, which causes a slight departure from the simple square-root-of-tension relationship. This effect, called inharmonicity, makes higher harmonics slightly sharp compared to theory. Piano technicians account for inharmonicity when stretching the tuning—deliberately making lower notes slightly sharper and higher notes slightly flatter to compensate for string stiffness.

Comparison Table: String Frequency Sensitivity

Limitations and Assumptions

• This formula assumes ideal strings with no damping or stiffness (inharmonicity is ignored).
• Real strings exhibit slight stiffness that makes higher notes slightly sharp.
• The formula applies only to strings with fixed ends (bridges) at both ends.
• Temperature and humidity variations affect both tension and material properties, changing frequency.
• String age and material fatigue can alter linear mass density slightly over time.
• For very thick strings or extreme parameters, relativistic or quantum effects (not included) become important.

Historical Context

The physics of vibrating strings was among the first wave phenomena studied mathematically. Marin Mersenne (1588–1648) discovered empirically that frequency is proportional to the square root of tension. Later, Daniel Bernoulli and Jean le Rond d'Alembert developed the wave equation that fully describes string vibration. This early success in understanding waves led to broader applications in acoustics, electromagnetism, and quantum mechanics.