Editing String instrument (section)

==Changing the pitch of a vibrating string==
{{main|Mersenne's laws}}

There are three ways to change the pitch of a [[vibrating string]]. String instruments are tuned by varying a string's tension because adjusting length or mass per unit length is impractical. Instruments with a [[fingerboard]] are then played by adjusting the length of the vibrating portion of the strings. The following observations all apply to a string that is infinitely flexible (a theoretical assumption, because in practical applications, strings are not infinitely flexible) strung between two fixed supports. Real strings have finite curvature at the bridge and nut, and the bridge, because of its motion, is not exactly nodes of vibration. Hence the following statements about proportionality are approximations.

===Length===
[[File:Proportional string fingering.png|thumb|200px|right|String fingering is proportional and not fixed,<ref>[[Walter Piston|Piston, Walter]] (1955). ''Orchestration'', p. 5.</ref> as on the piano]]

Pitch can be adjusted by varying the [[length]] of the string.<ref name=":0">{{Cite web|title = Oxford Music Online by subscription|url = http://www.oxfordmusiconline.com/|website = www.oxfordmusiconline.com|access-date = 2015-09-17|url-status = live|archive-url = http://archive.wikiwix.com/cache/20110224031012/http://www.oxfordmusiconline.com|archive-date = 2011-02-24}}</ref>{{Better source needed|date=June 2023|reason=That's just the home page of Oxford Music Online.}} A longer string results in a lower pitch, while a shorter string results in a higher pitch. A [[concert harp]] has pedals that cause a hard object to make contact with a string to shorten its vibrating length during a performance.<ref name="woosterped">{{cite web |last1=Wooster |first1=Patricia McNulty |title=Pedal Harp 101 |url=https://www.harpspectrum.org/pedal/wooster.shtml |website=harp spectrum.org |access-date=March 18, 2021}}</ref> The frequency is inversely proportional to the length:

:<math>
f \propto \frac{1}{l}
</math>

A string twice as long produces a tone of half the frequency (one octave lower).

===Tension===

Pitch can be adjusted by varying the [[Tension (physics)|tension]] of the string. A string with less tension (looser) results in a lower pitch, while a string with greater tension (tighter) results in a higher pitch. Pushing a pedal on a [[pedal steel guitar]] raises the pitch of certain strings by increasing tension on them (stretching) through a mechanical linkage; release of the pedal returns the pitch to the original. Knee levers on the instrument can lower a pitch by releasing (and restoring) tension in the same way.<ref>{{cite web|last=Brenner|first=Patrick|title=Early History of the Steel Guitar|url=http://steelguitaramerica.com/instruction/history/|website=steelguitaramerica.com|publisher=Patrick Brenner|access-date=March 17, 2021}}</ref> A homemade [[washtub bass]] made out of a length of rope, a broomstick and a washtub can produce different pitches by increasing the tension on the rope (producing a higher pitch) or reducing the tension (producing a lower pitch). The frequency is proportional to the square root of the tension:

:<math>
f \propto \sqrt{T}
</math>

=== Linear density ===

The pitch of a string can also be varied by changing the [[linear density]] (mass per unit length) of the string. In practical applications, such as with [[double bass]] strings or bass [[piano]] strings, extra weight is added to strings by winding them with metal. A string with a heavier metal winding produces a lower pitch than a string of equal length without a metal winding. This can be seen on a 2016-era set of gut strings for double bass. The higher-pitched G string is often made of synthetic material, or sometimes animal intestine, with no metal wrapping. To enable the low E string to produce a much lower pitch with a string of the same length, it is wrapped with many wrappings of thin metal wire. This adds to its mass without making it too stiff. The frequency is inversely proportional to the square root of the linear density:

:<math>
f \propto {1 \over \sqrt{\mu}}
</math>

Given two strings of equal length and tension, the string with higher mass per unit length produces the lower pitch.