Calculating String Tensions

In this article I will explain about the relations between the pitch of a string, the material the string is made of, the string length, the diameter of the string and the string tension. After this theoretical explanation I will give practical formulas to calculate string tension when pitch, material, length and diameter are known, and how to calculate the diameter of a string at a given tension when pitch, material and length are known. This has practical applications for choosing strings for a musical instrument. I will explain about choosing string tensions for lutes, give an example and list frequencies and densities so you can make a spreadsheet for calculating the strings for your own lutes.

Mersenne’s Law

In an effort to understand about string tension, I ended up with Mersenne. In his Harmonie Universelle of 1636 he shows by experiments how the frequency of a plucked or bowed string is dependend on its length, weight and tension. These three relations are given here.

The frequency of a string is inversely related to twice its length. That means that if all else stays the same, a string will sound an octave higher when it is halved (the frequency doubles when the string length halves). This is clearly demonstrated on a lute, where the 12th fret, giving the octave, is exactly at half the string length.

Next, the frequency is directly related to the square root of its tension. This is harder to imagine, but we can try by tuning an imaginary string so much higher that its tension is quadrupled. The resulting pitch of the string would have risen by an octave (the frequency doubles when the tension is squared). Vincenzo Galileo had proven this already around 1590 by experiment.

Finally, the frequency is inversely related to the square root of the mass-per-unit-of-length. This is a complicated way of saying that if all else stays the same, the frequency will rise when the mass of the string is decreased; a lighter string at the same tension and string length will have a higher pitch. This is easily imaginable on a lute, as higher strings are thinner than lower strings. The exact relation happens to be that a four times heavier string will give a tone one octave lower (a quadrupled mass gives a halved frequency).

The mass-per-unit-of-length is dependend on the string’s diameter and the density (relative weight) of the material used. Its defenition is the cross-section of the string multiplied by its density, in a formula this looks like:

M = π* (0.5 * d)^2 * ρ

M = mass-per-unit-of-length
d = diameter of string
ρ = density of string material

These three relations result in a formula, which is called Mersenne’s Law:

F = 1/2L * √T * 1/√(π * (0.5 * d)^2 * ρ)

F = Frequency
L = String length

Now we also know how to calculate the string tension:

T = 4 * F^2 * L^2 * π * (0.5*d)^2 * ρ

and the diameter:

d = 1/L * √T * 1/F * 1/√(π * ρ)

String tension for lutes

The choice of string tension on a lute is dependend on a number of factors, not least of which is personal preference. And also, different materials require different string tensions, and so do different playing techniques. However, a few general remarks that can serve as a guide can be made. Here follows a list of recommended string tensions for the first string of a lute, in relation to its string length:

 String length 85cm 81cm 77cm 74cm 70cm 66cm 62cm 59cm 56cm 52cm Tension of first string 42N 41N 40N 39N 38N 37N 36N 35N 34N 33N

Conventional wisdom has it that the second string should have a tension about 90% of that of the first string. The rest of the strings have a tension of about 90% of that of the second string, and octave strings have a tension of about 90% of their fundamentals. But, again, personal preferences might make you decide to change some of these tensions.

Gut

Especially the first and thinnest string on a lute, when made of gut, has a limited working range: if you give it too much tension, it will break, and if the tension is too low, it will not sound well. The thinnest gut strings made historically had a diameter of around 0.38 or 0.40mm, and on a string length of around 60cm they will sound well on a tension of about 35 to 37N. The historical advice to tune your first string to just below breaking point seems to confirm this. If you use (high twist or loaded) gut strings for your basses you may find the fundamental and octave strings touching each other after you plucked them. This is because gut strings can be very flexible. To solve this you can increase the spacing between fundamental and octave strings on the nut. It does not need to be much, so do make sure that if these bass courses run on the fingerboard you will still be able to finger them.

Nylgut, Nylon and Carbon

As nylgut and nylon strech more than gut, the final diameters of these strings under tension will be a little less than measured before stretching, so you may wish to use slightly higher tensions for these strings. Players on carbon strings tend to use an even higher tension, especially for the first two courses, as do players with nails. Some modern lutes are actually build for modern strings and players with a more modern playing technique: they sound better with nylon or carbon strings, a higher string tension and nails than with gut strings, finger tips and a lower string tension. But beware of putting too much tension on an instrument that cannot stand it: your lute may break before your strings!

Example

As an example I have calculated the strings for a 10-course g’-lute with a string length of 59cm, tuned at a pitch of a’ = 440Hz and entirely strung with gut.

Because d = 1/L * √T * 1/F * 1/√ (π * ρ), we need to know string length L in meter, string tension T in Newton (kgm/s2), frequency F in Hertz and density ρ in kg/m3.

L = 0.59m
T is given as the average advisable tension from the table.
F is given in the table.
ρ = ca 1360 kg/m3 for gut
For the sake of convenience I have written the resulting diameters in mm.

 Course 1 2 3 4 5 6   7   8   9   10 Note g' d' a f c g G f F e E d D c C Frequency in Hz 392 294 220 175 131 196 98 175 88 165 82 147 73 131 65 Tension in N 35 32 28 28 28 25 28 25 28 25 28 25 28 25 28 Diameter in mm 0.40 0.51 0.64 0.80 1.07 0.68 1.43 0.76 1.60 0.80 1.71 0.90 1.92 1.01 2.16

Once you know the formulas, it is convenient to make a little spreadsheet for your lutes. To help, I will list the frequencies of notes in equal temperament, and the density of gut, nylgut, nylon and carbon.

 Note a' g#' g' f#' f' e' d#' d' c#' c' b a# a' = 440Hz Frequency in Hz 440.0 415.3 392.0 370.0 349.2 329.6 311.1 293.7 277.1 261.6 246.9 233.1 a' = 415 Hz Frequency in Hz 415.3 392.0 370.0 349.2 329.6 311.1 293.7 277.1 261.6 246.9 233.1 220.0

For notes an octave lower, halve these frequencies and for notes an octave higher, double them.

 Material Gut Nylon Carbon Density 1360 kg/m3 1140 kg/m3 ca 1800 kg/m3

The exact densities may vary between manufacturers, as additives (e.g. glue, varnish, chemicals) will influence the density slightly, but the above figures give results matching those of the string calculator of the string supplier Kürschner. Nylgut is supposed to have practically the same density as gut, but as it stretches more under tension, most lute players like to use strings a little thicker than gut, e.g. 0.42mm nylgut in stead of 0.40mm gut. You could take that stretching into account by using a density a little below that of gut, say 1200 kg/m3.