String tension is an important issue for designers of stringed musical instruments, as all the static forces bearing on the structure of an instrument are due to string tension. The relationship of string tension, pitch, and scale length is of interest to all designers of instruments. The tension of a musical instrument string is a function of its mass (or weight) per unit of length, the vibrating length of the string, and the pitch of the note produced when the string vibrates. What may not at first appear obvious is that this relationship holds not only for strings of homogeneous construction (i.e. plain strings) but for wound strings as well. This page contains the formulas for determining string tension using English units of measurement, javascript string tension calculators, and some links to other string tension related tools and information. There is also some info for non-technical folks that may help in understanding the relationships between string tension and other quantities.

Those interested in this subject may also want to take a look at Human Perception of String Tension and Compliance in Stringed Musical Instruments, which discusses how players actually perceive string tension and compliance.

String Tension Formula

The formula for determining string tension in English units of measurement is:

where:

T = Tension, in pounds;
UW = unit weight of the string, in pounds per linear inch;
L = vibrating length of the string (for an open string, this would be the scale length) in inches;
F = frequency of the note to which you will tune the string, in Hz;

Information on unit weights of various types and sizes of strings is available from D'Addario Strings in a document entitled Everything You Ever Wanted to Know About String Tension (.pdf). Click here for a table of notes and their equivalent frequencies.

A few things should be apparent from the formula:

1. Everything else being the same, pitch increases as tension increases;

2. Everything else being the same, tension increases as scale length increases;

3. Everything else being the same, tension increases as unit weight increases, thus a heavier gage string will be under greater tension as a lighter one of the same length tuned to the same note;

Given these, it should be obvious why higher pitched instruments tend to have shorter strings and lower pitched instruments longer strings and why fretting a string (i.e. Shortening the string) raises the pitch of that string when it vibrates. It also should be obvious why the lower pitched strings of an instrument are wound - this increases the unit weight and thus lowers the pitch. Although it is certainly possible to increase the unit weight of a string simply by increasing its diameter, doing so also increases the string's side bending stiffness. Overly (side bending) stiff strings behave more like bars than strings when they vibrate. Consider what folks in the 16^th century had to do to build an instrument that covered a wide pitch range with a very limited range of available strings.. The instrument in the picture is called an orpharion. It is related to the bandora and the lute, and is tuned the same as the latter. The splayed frets (also sometimes called fanned frets) make the scale length on the bass side longer than the scale length on the treble side.

This paper is from D'Addario Strings, which surprisingly is the only string manufacturer to provide tension data for all their strings. It also contains unit weight data for all their strings, which can be used to approximate unit weights of similarly constructed strings from other manufacturers as well.

The kind folks at D’Addario are both knowledgeable and generous enough to provide us with this data. I thank them by buying their products.

Don’t want to look up the mass data?  The clever folks at Green Man Humming Productions have a great java applet that will calculate string tensions for you, using the D’Addario string tension data.

Unit Weight Formula

If you know the scale length, fequency and string tension you can solve for unit weight of the string. This is useful for example in the case where you want to figure out what string to use for, say, the low B string of a seven string guitar. The scale length is the same for the rest of the strings, and you'd want the tension to be more or less the same as the tension of those other strings, too. The frequency is that of the note the open string should sound, in this case low B at 61.735 hz. Once you determine the unit weight you can look up a string of suitable construction and that unit weight from the D'Addario Everything You Ever Wanted to Know About String Tension (.pdf) document.

The formula for determining unit weight of a string in English units of measurement is:

where:

UW = unit weight of the string, in pounds per linear inch;
T = Tension, in pounds;
L = vibrating length of the string (for an open string, this would be the scale length) in inches;
F = frequency of the note to which you will tune the string, in Hz;