FREQUENCY OF A VIBRATING
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
The fundamental and strongest transverse vibrations has minimal motion
of the string at the ends and maximum displacement in the middle of the
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
(1) that the frequency is inversely proportional to the length.
(2) the frequency is directly proportional to the square root of the
(3) the frequency is inversely proportional to the cross sectional mass
of the string
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
(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.