PLAIN STEEL STRING DIAMETER VERSUS TENSION:
Using information about plain steel strings found on the D'Addario website (
Steel Strings) the tension per cross sectional area can be calculated.
The website data gives string diameters together with tension, for various pitches and for a 25.5 inch scale length.
This uses the data given for E-1st and B-2nd guitar strings, all plain (unwound) steel.
The calculation is: Take the string diameter, halve it (to get the radius), square that and multiply by pi (3.14159) to get cross-sectional area. Divide that area into the tension. Note: The website gives the diameter in inches after the letter L, for example, a PL010 string is 0.010 inch in diameter.
Here are the results:
E-1st (329.63 Hertz)
0.008" diameter, 10.4lbs: 206,901 lbs/square inch
0.0085" diameter 11.7lbs: 206,186 lbs/square inch
0.009" diameter 13.1lbs: 205,919 lbs/square inch
0.010" diameter 16.2lbs: 206,264 lbs/square inch
0.0105" diameter 17.9lbs: 206,721 lbs/square inch
0.011" diameter 19.6lbs: 206,244 lbs/square inch
0.012" diameter 23.3lbs: 206,017 lbs/square inch
0.013" diameter 27.4lbs: 206,431 lbs/square inch
B-2nd (246.94 Hertz)
0.010" diameter 9.1lbs: 115,685 lbs/square inch
0.0105" diameter 10.0lbs: 115,487 lbs/square inch
0.011" diameter 11.0lbs: 115,749 lbs/square inch
0.013" diameter 15.4lbs: 116,023 lbs/square inch
0.0135" diameter 16.6lbs: 115,971 lbs/square inch
0.014" diameter 17.8lbs: 115,630 lbs/square inch
0.016" diameter 23.3lbs: 115,884 lbs/square inch
0.017" diameter 26.3lbs: 115,869 lbs/square inch
Notice a pattern here: For a certain note at a certain scale length, the
tension per cross-sectional area is the same <u>no matter what the gauge of the string is</u>!
The averages of the above tensions per cross-sectional area are:
For E-1st, 206,335 lbs/square inch.
For B-2nd, 115,810 lbs/square inch.
A second pattern emerges in tension per cross-sectional area versus pitch (frequency of the note). Comparing the average E-1st and B-2nd values above yields the following:
Tension per cross-sectional area ratio:
206,335/115,810 =
1.782
Frequency ratio:
329.63 Hertz/246.94 Hertz = 1.335.
But, frequency ratio squared is
(329.63/246.94)^2 =
1.782
So, Frequency ratio squared is equal to tension per cross-sectional area ratio!
The effect of that is a string tuned to a higher pitch experiences greatly increased tension, proportional to the square of the frequency. So, an upper limit to how high a note a string can play without breaking is soon reached. Now, a 12-string fretted guitar usually has a high G string. That high G is 392.00 Hertz. Using what we know, that string (on the same 25.5 inch scale length) will have a tension per cross sectional area of:
[(392.00/329.63)^2] x 206,335 lbs/square inch = 1.414 x 206,335 lbs/square inch = 291,797 lbs/square inch.
(Using D'Addario website data confirms this: a 0.008" string tuned up to G has 14.7 lbs of tension, giving 292,447 lbs/square inch.)
From experience, trying to take a guitar string up to A above that G can't be done. It will break every time. If it could be achieved, the A string would be at 367,642 lbs/square inch. So, the breaking strength of a plain steel guitar string is about 300,000 to 330,000 lbs/square inch.
A high G string is in extreme tension. A high G# string is right on the edge of breaking.
Further, it is now obvious that
the tendency of a string to break when tuned to high G or G# on a steel guitar is not dependent upon the gauge of the string. Gauge may have a secondary effect on tendency to break, such as how much effect there is on the string from flaws like nicks and bends and how carefully it is constructed for the high-note application.
<FONT SIZE=1 COLOR="#8e236b"><p align=CENTER>[This message was edited by Richard Vogh on 16 April 2001 at 04:43 PM.]</p></FONT>