Frequencies of the strings on a standardly-tuned 6-string guitar in Hz are: E2=82.4, A2=110, D3=146.8, G3=196, B3=246.92, E3=329.6. What is the fundamental frequency of these strings?
Frequencies of vibrating stretched strings are measured in Hertz (Hz). The equation describing the fundamental frequency can be written as
`f_1 = (sqrt(T/("m/L")))/(2L)`
where `T ` is the tension in the string, `m ` is the mass and `L ` is the length in cm.
The fundamental frequency `f_1 ` is the frequency of the predominant vibration of a string when plucked normally, that is, plucked as an open string with no fingers holding down the string on a fret. Higher harmonics of the strings vibration can be heard when it is plucked normally, but the fundamental frequency creates the dominant sound. The complete profile of frequencies occurring and being heard is called the 'standing wave'. Naturally, strings vibrate with wavelength twice their full length (2L) (creating the fundamental frequency), but also at successive divisions of that length (L,2L/3,L/2,3L/4 etc) (creating harmonics, or overtones). Harmonics can be upgraded to the fundamental frequency by holding the string down loosely successively halfway, 2/3 of the way, 3/4 of the way etc along, effectively shortening the length of the string from end to end (the nodes).
So the fundamental frequencies of the strings on a guitar are simply the frequencies of the predominantly heard notes when the strings are plucked normally, ie E2, A2, D3, G3, B3, E3.