Tuning fork

Motion of an A-440 tuning fork (greatly exaggerated) vibrating in its principal mode

A tuning fork is a fork-shaped acoustic resonator used in many applications to produce a fixed tone. The main reason for using the fork shape is that, unlike many other types of resonators, it produces a very pure tone, with most of the vibrational energy at the fundamental frequency. The reason for this is that the frequency of the first overtone is about 5²/2² = 25/4 = 6+1⁄4 times the fundamental (about 2+1⁄2 octaves above it).[2] By comparison, the first overtone of a vibrating string or metal bar is one octave above (twice) the fundamental, so when the string is plucked or the bar is struck, its vibrations tend to mix the fundamental and overtone frequencies. When the tuning fork is struck, little of the energy goes into the overtone modes; they also die out correspondingly faster, leaving a pure sine wave at the fundamental frequency. It is easier to tune other instruments with this pure tone.

Another reason for using the fork shape is that it can then be held at the base without damping the oscillation. That is because its principal mode of vibration is symmetric, with the two prongs always moving in opposite directions, so that at the base where the two prongs meet there is a node (point of no vibratory motion) which can therefore be handled without removing energy from the oscillation (damping). However, there is still a tiny motion induced in the handle in its longitudinal direction (thus at right angles to the oscillation of the prongs) which can be made audible using any sort of sound board. Thus by pressing the tuning fork's base against a sound board such as a wooden box, table top, or bridge of a musical instrument, this small motion, but which is at a high acoustic pressure (thus a very high acoustic impedance), is partly converted into audible sound in air which involves a much greater motion (particle velocity) at a relatively low pressure (thus low acoustic impedance).[3] The pitch of a tuning fork can also be heard directly through bone conduction, by pressing the tuning fork against the bone just behind the ear, or even by holding the stem of the fork in one's teeth, conveniently leaving both hands free.[4] Bone conduction using a tuning fork is specifically used in the Weber and Rinne tests for hearing in order to bypass the middle ear. If just held in open air, the sound of a tuning fork is very faint due to the acoustic impedance mismatch between the steel and air. Moreover, since the feeble sound waves emanating from each prong are 180° out of phase, those two opposite waves interfere, largely cancelling each other. Thus when a solid sheet is slid in between the prongs of a vibrating fork, the apparent volume actually increases, as this cancellation is reduced, just as a loudspeaker requires a baffle in order to radiate efficiently.

Commercial tuning forks are tuned to the correct pitch at the factory, and the pitch and frequency in hertz is stamped on them. They can be retuned by filing material off the prongs. Filing the ends of the prongs raises the pitch, while filing the inside of the base of the prongs lowers it.

Currently, the most common tuning fork sounds the note of A = 440 Hz, the standard concert pitch that many orchestras use. That A is the pitch of the violin's second string, the first string of the viola, and an octave above the first string of the cello. Orchestras between 1750 and 1820 mostly used A = 423.5 Hz, though there were many forks and many slightly different pitches.[5] Standard tuning forks are available that vibrate at all the pitches within the central octave of the piano, and also other pitches.

Tuning fork pitch varies slightly with temperature, due mainly to a slight decrease in the modulus of elasticity of steel with increasing temperature. A change in frequency of 48 parts per million per °F (86 ppm per °C) is typical for a steel tuning fork. The frequency decreases (becomes flat) with increasing temperature.[6] Tuning forks are manufactured to have their correct pitch at a standard temperature. The standard temperature is now 20 °C (68 °F), but 15 °C (59 °F) is an older standard. The pitch of other instruments is also subject to variation with temperature change.

The frequency of a tuning fork depends on its dimensions and what it's made from:[7]

${\displaystyle f={\frac {N}{2\pi L^{2}}}{\sqrt {\frac {EI}{\rho A}}},}$

where

f is the frequency the fork vibrates at, (SI units: 1/s)

N ≈ 3.516015 is the square of the smallest positive solution to cos(x)cosh(x) = −1,[8] which arises from the boundary conditions of the prong’s cantilevered structure.

L is the length of the prongs, (m)

E is the Young's modulus (elastic modulus or stiffness) of the material the fork is made from, (Pa or N/m² or kg/(ms²))

I is the second moment of area of the cross-section, (m⁴)

ρ is the density of the fork's material (kg/m³), and

A is the cross-sectional area of the prongs (tines) (m²).

The ratio I/A in the equation above can be rewritten as r²/4 if the prongs are cylindrical with radius r, and a²/12 if the prongs have rectangular cross-section of width a along the direction of motion.

Tuning forks have traditionally been used to tune musical instruments, though electronic tuners have largely replaced them. Forks can be driven electrically by placing electronic oscillator-driven electromagnets close to the prongs.

In clocks and watches

Quartz crystal resonator from a modern quartz watch, formed in the shape of a tuning fork. It vibrates at 32,768 Hz, in the ultrasonic range.

A Bulova Accutron watch from the 1960s, which uses a steel tuning fork (visible in center) vibrating at 360 Hz.

The quartz crystal that serves as the timekeeping element in modern quartz clocks and watches is in the form of a tiny tuning fork. It usually vibrates at a frequency of 32,768 Hz in the ultrasonic range (above the range of human hearing). It is made to vibrate by small oscillating voltages applied to metal electrodes plated on the surface of the crystal by an electronic oscillator circuit. Quartz is piezoelectric, so the voltage causes the tines to bend rapidly back and forth.

The Accutron, an electromechanical watch developed by Max Hetzel[9] and manufactured by Bulova beginning in 1960, used a 360-hertz steel tuning fork as its timekeeper, powered by electromagnets attached to a battery-powered transistor oscillator circuit. The fork provided greater accuracy than conventional balance wheel watches. The humming sound of the tuning fork was audible when the watch was held to the ear.

Medical and scientific uses

1 kHz tuning fork vacuum tube oscillator used by the U.S. National Bureau of Standards (now NIST) in 1927 as a frequency standard.

Alternatives to the common A=440 standard include philosophical or scientific pitch with standard pitch of C=512. According to Rayleigh, physicists and acoustic instrument makers used this pitch.[10] The tuning fork John Shore gave to George Frideric Handel produces C=512.[11]

Tuning forks, usually C512, are used by medical practitioners to assess a patient's hearing. This is most commonly done with two exams called the Weber test and Rinne test, respectively. Lower-pitched ones, usually at C128, are also used to check vibration sense as part of the examination of the peripheral nervous system.[12]

Orthopedic surgeons have explored using a tuning fork (lowest frequency C=128) to assess injuries where bone fracture is suspected. They hold the end of the vibrating fork on the skin above the suspected fracture, progressively closer to the suspected fracture. If there is a fracture, the periosteum of the bone vibrates and fires nociceptors (pain receptors), causing a local sharp pain. This can indicate a fracture, which the practitioner refers for medical X-ray. The sharp pain of a local sprain can give a false positive. Established practice, however, requires an X-ray regardless, because it's better than missing a real fracture while wondering if a response means a sprain. A systematic review published in 2014 in BMJ Open suggests that this technique is not reliable or accurate enough for clinical use.[13]

• Electronic tuner
• Pitch pipe
• Savart wheel
• Tonometer

Wikimedia Commons has media related to Tuning forks.

Look up tuning fork in Wiktionary, the free dictionary.

• Onlinetuningfork.com, an online tuning fork using Macromedia Flash Player.
• "Tuning Fork" . Collier's New Encyclopedia. 1921.
• "Tuning Fork" . Encyclopædia Britannica. Vol. 27 (11th ed.). 1911. p. 392.