Changing the pitch of a vibrating string
There are three ways to change the pitch of a vibrating string. String instruments are tuned by varying the strings' 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 strung between two fixed supports. Real strings have finite curvature at the bridge and nut, and the bridge, because of its motion, are not exactly nodes of vibration. Hence the following statements about proportionality are (usually rather good) approximations.
Pitch can be adjusted by varying the length of the string. A longer string will result in a lower pitch, while a shorter string will result in a higher pitch. The frequency is inversely proportional to the length. A string twice as long will produce a tone of half the frequency (one octave lower).
Pitch can be adjusted by varying the tension of the string. A string with less tension (looser) will result in a lower pitch, while a string with greater tension (tighter) will result in a higher pitch. The frequency is proportional to the square root of the tension:
The pitch of a string can also be varied by changing the linear density (mass per unit length) of the string. The frequency is inversely proportional to the square root of the linear density:
One can see the waveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of a CRT (Cathode Ray Tube) screen such as one of a television or a computer (not of an oscilloscope). This effect is called the stroboscopic effect, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the refresh rate of the screen. The same can happen with a fluorescent lamp, at a rate which is the difference between the frequency of the string and the frequency of the alternating current. (If the refresh rate of the screen equals the frequency of the string or an integer multiple thereof, the string will appear still but deformed.) In daylight, this effect does not occur and the string will appear to be still, but thicker and lighter, due to persistence of vision.
A similar but more controllable effect can be obtained using a stroboscope. This device allows the frequency of the xenon flash lamp to be exactly matched to the frequency of vibration of the string; in a darkened room, this clearly shows the waveform. Otherwise, one can use bending or, perhaps more easily, by adjusting the machine heads, to obtain the same frequency, or a multiple of, the AC frequency to achieve the same effect. For example, in the case of a guitar, the bass string pressed to the third fret gives a G at 97.999 Hz; with a slight adjustment, a frequency of 100 Hz can be obtained, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia, 50 Hz. In most countries of the Americas, where the AC frequency is 60 Hz, one can start from A# at 116.54 Hz, on the fifth string at the first fret, to obtain a frequency of 120 Hz.
Source: Wikipedia (All text is available under the terms of the GNU Free Documentation License and Creative Commons Attribution-ShareAlike License.)