An Introduction to Strings
In the first article we established the relationship between the length, pitch and density of strings as being fundamental in the understanding of how strin gs work. In this installment, we will consider how these relationships work mathematically so we can calculate string diameters and deepen our understanding of these relationships.
Since our gut string material is uniform there is only thing we can do to change the weight of the string and that is to change the diameter. The thicker the string; the more it weighs and the thinner the string; the lighter it is. This is why it is important to be able to calculate the diameter of a string based on the playing length of the string and its target pitch.
A basic formula for calculating the diameter of a gut string is shown in Figure 1. This formula was developed by the great Japanese lute player Toyohiko Satoh and first published in the Journal of the Lute Society of America. This formula uses a constant as the fixed density for gut and is, therefore, easy to understand and use for practical string calculations.
String Diameter Formula:
F x l x d = 49 x
F = frequency in Hz (Hertz)
L = string length in meters
d = diameter of string in millimeters
k = tension in Kg (kilograms)
49 = a constant representing the density of gut
440 x .50 x d = 49 x √ 5
d = __49 x _√ 5
440 x .50
d = .50
d = .50
Using this formula we can observe what happens when the parameters are changed. For instance, if the length of the string gets longer while the pitch and tension remain the same, the diameter will decrease. If the tension of the string increases while the pitch and length remain the same, the diameter of the string will increase. By playing with the formula with different values you can learn the relationships between these important concepts in string variables.
In using this formula it will be noticed that, in reality, there is usually just two of these variables that can be changed to actually alter the gauge of the string because, on most stringed instruments, the length of the strings is fixed. Once the instrument has been designed for a particular use the string length cannot usually be changed. While it is true that on some instruments with movable bridges such as violins and viols it is possible to change the position of the bridge and thereby change the string length, this length cannot usually be changed enough to significantly alter the relationships between length, pitch and tension. In addition, the pitch of the string is usually fixed, more or less. It is true that one can tune an instrument to a-440, or a-415, or a-460 and thereby change the intended pitch, technically, it is not effective to change the intended pitch on a particular instrument more than a few semitones without severely effecting the performance of the string. This why it is not possible to design a stringed instrument that will play the full musical scale and create one that will play, for instance, both a bass register and a treble register. The relationships between length, pitch and tension limit the function of a string to a fairly narrow range of possibilities. In fact, on most stringed instruments the only thing we can effectively change is the mass of the string thereby changing the tension of the string. To change the mass of a gut string we need to change the diameter of the string and this is why it is so important to be able to calculate the gauge of a given string.
In calculating the diameter of gut strings one of the first considerations is the diameter of the first, or top string. This is the highest pitch and it is usually this pitch from which all the other gauges of strings will be scaled. Therefore, it is important to get the length, tension and weight of this string correct to start with and it is most often the first string gauge that is calculated. The first string is the string that has the most tension on it so it is important to be conscious of the tensile strength of gut. The tensile strength is the maximum stress that a string can receive before breaking. This unit is usually measured in kilo-pounds per square inch. That is, if you were to cut a string and look at the end, or the the cross-section of a string, it would be a circle with a certain area, given the diameter of the string. This is where the square inch part of the measurement comes from. and the kilo-pounds is a unit of force, so tensile strength is expressed as a force per unit area.
In measuring strings, it is most, (I think), convenient to think of the diameter of the string in millimeters and the tension of the string in kilograms. The proper scientific method requires using pascals or newtons as the force and meters as the area units, but since we calculate strings in millimeters and kilograms for convenience it makes my head hurt to do the conversions. Therefore, I like to think of tensile strength in terms of kilograms per square millimeters, (Kg/mm²).
The tensile strength of gut is complicated further because, in the real world, the tensile strength varies due to the way gut is processed. Gut for small, treble strings, is usually processed to be harder and stronger than thicker gut that is processed to be softer and more flexible. Therefore, treble strings, those with diameters less than, say, 1.00mm, have a higher tensile strength than thicker strings over 1.00mm. Now, the tensile strength of gut will vary from one maker to another and even from one batch of gut to another, but, in general, treble gut will have a tensile strength of about 40 Kg/mm² and larger diameters
will have a tensile strength of about 30 Kg/mm².
The next installment will consider what the maximum length strings can have for various pitches without breaking and how strings vibrate on an instrument to produce a tone.