mercredi 1 mai 2013

CALCULATING THE TENSION OF A STRING: THE TAYLOR EQUATION

A simple enough way to implement the famous "equation of Brook Taylor" (English mathematician and musician of the seventeenth century) :

- Measure one, or better several strings of the considered diameter ;
- Weigh them with the most accurate balance as possible: the greater the length of string, and the most precise will be the weighing, of course.
- Determine the MASS PER LENGTH UNIT (m) by dividing the value weighed in kilograms by the length in meters.

Where:
f is the frequency in Hertz of the pitch,
k denotes the harmonic wanted (here equal to 1),
L is the length of the string,
F is the tension of the string we are looking for,
m is the mass per length unit that we have calculated.

We raise it to the square to eliminate the root, we make some permutations to rewrite the equation in a more convenient form :

Consider an example: Let a nylon Tynex string tuned A4, 440 Hz, 0.4 m (40cm) long.
We take a string 1mm in diameter, which measures initially 1.33 m long and weighs 1g or 0,001 kg (be careful not to tangle with the zeros!).
The mass per length unit is m = 0.001 / 1.33 = 0.00081 Kilo /meter.

We can now replace the terms of our equation by numeral values:

F = 0.00081 X 440 X440 X 4 X 0.4 X 0.4 = 100.36 N

N  for "Newton" which is the unit of measurement of the tension. To obtain this value in kg we must divide by the coefficient 9.81. This gives us: F = 100.36 / 9.81 = 10.23 kg (At the sea level..).

If a string of the same length but 0.9 mm in diameter is placed, a tension of 8.71 kg is obtained, it may be a little dull... For a string of 1.25 mm in diameter, on the other hand, we get 17.17 kg of tension, almost double! Rude for the fingers and the soundboard ...

One note: as the tension is proportional to the square of the length, one can see why, especially in the trebble range, a difference of one or two centimeters can be problematic: the strings break in rehearsal ... and as this same tension is also proportional to the square of the frequency, the solution is, of course, often, to tune the pitch lower ...

These explanations are all my master's of science and friend Yann Baol Le Noalleg, poet, mathematician, physicist and distinguished Breton speaker...

1 commentaire:

1. pas tout à fait