FREQUENCY OF A VIBRATING STRING

The various string instruments I'm aware of (piano family, harp family, dulcimer family, fretted and unfretted strings, and wash-tub bass?)  all have strings that are anchored at both ends.  No, duh.  And on all these instruments the strings generate sounds through  transversel vibrations,  ie, perpendicular to the string.

The fundamental and strongest transverse vibrations has minimal motion of the string at the ends and maximum displacement in the middle of the string.

Formula

There is a formula to calculate the frequency of this vibration, credited to Pythagoras, the  same mathematician who brought you the right triangle formula.  It looks like this:

F = 1/2l * sqrt (T/M)

where F is the transverse vibrating frequency
L is the string length
T is the string tension
M is the cross-sectional mass of  the string

I don't know what the units were.  Basically, the formula says several things....

(1) that the frequency is inversely proportional to the length.
(2) the frequency is directly proportional to the square root of the tension
(3) the frequency is inversely proportional to the cross sectional mass of the string

Examples:

Since an octave interval is a 2:1 ratio, and a perfect interval is a 3:2 ratio, we can state several examples:

(1) to play an octave higher,  make it  half its original length but keep the same string at the same tension

(2) to play a string a perfect fifth higher, make it 2/3  its original length but keep the same string at the same tension

(3)  to make a string sound an octave lower , reduce the string tension to 1/4 of the original tension but keep the same string at the same length,

(4) to make a string sound an octave lower, switch to another string that has 4 times the cross-section mass,. but install it with the same length tension as before.

There are many more possible examples, but you get the idea.

This formula has a slight modification that seems to handle situations of different string material and the like.  I don't know about formulas for the integer multiple harmonics of a string.  I have heard, and think I've verified, that wound piano strings produce harmonics that are not exact multiples of the fundamental, due to their coiled shape.  But I'd need to look that up somewhere to verify it.

Bearing all this in mind, you can understand why a piano frame or a harps is shaped the way it is, and why hammered dulcimers are trapezoid shape.  The higher notes are at the short-string side of the instrument so that there doesn't need to be so great a difference in the string tension or mass as there would have been if all the strings were the same length.

Also, you can now see why the bridges on a hammered dulcimer are not centered.  It allows one string to play two different pitches, one on either side of the bridge.  Many dulcimers have one of their bridges that splits the string in a length ratio of 2:3, so that on each string crossing that bridge, the short side note is a perfect fifth higher than the long side note.

Steel string makers for guitar and mandolin put the diameter ratings on their strings.  So, when you are comparing string sets, remember that the smaller diameter set, installed on the same instrument, will need to be at a lower tension to produce the same pitch.

When a luthieris defines fret locations on an instrument neck,  and if those frets are all supposed to be a half-step apart, then each fret location must make the proportional change in string length to produce a half-step increase in freqneucy.  If you are working with tempered intervals, a half step ratio is 2 ** (1/12).  So, the next fret must be located to reduce the string length by that factor.  There is also string thickness to bear in mind, as well as other factors I wouldn't know about, since I don't build instruments.  But the basic concept of fret spacing comes directly from this formula.

Musical Accoustics