# How does Tension, or length, on a string effect overtones?

In a recent experiment I saw that notes with the same pitch, played on strings of varying tension, have a different number of overtones. The pitch is kept constant by altering length. Less tension correlated with more overtones. Does this have something to do with length of the string?

• An ideal string has only mass and tension. Standing waves are possible at every integer multiple of the fundamental frequency, so you get the full harmonic series. Real strings have stiffness, which introduces a $\frac{\partial ^4 y}{\partial x^4}$ term into the equation of motion. Harmonics shift higher in frequency, no longer falling on exact integer multiples. The shorter the string, the more pronounced the stiffness effect. So I don't know about shorter strings having more overtones, but maybe they're more noticeable, because they sound discordant. Commented Jan 28, 2018 at 16:25
• Is there any way to mathematically derive the amplitudes of these overtones? Commented Jan 31, 2018 at 10:51
• If you have the initial conditions. A typical initial condition would be a plucked string—stationary triangular shape at $t=0$. Commented Feb 1, 2018 at 3:50

To build on Ben51's comments, we do indeed need to deviate from the classical wave equation. Gracia and Sanz-Perela 2016 use:

$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} - \frac{E S K^2}{\rho}\frac{\partial^4 u}{\partial x^4}$$

Where $$E$$ is Young's modulus for the string material, $$\rho$$ is it's linear density, $$S$$ is the cross-sectional area of the string, and $$K$$ is the 'radius of gyration', which they estimate as $$K =$$ R/2 for a cylindrical string of radius $$R$$.

To answer your question, the authors then go on to suggest that the frequency spectrum of a string with stiffness takes the form:

$$f_n = n \ f_o \sqrt{1 + B n^2} \ \ (n ≥ 1)$$

Where $$B$$ is an inharmonicity parameter. They suggest for piano strings it is about $$10^{-3}$$.

(Remember that for an ideal string: $$f_n = n \ f_o$$)

• +1, but you may wish to add the definition of $S$ as the cross sectional area of the string :) Commented Feb 14, 2018 at 17:19
• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. Commented Feb 14, 2018 at 17:26
• This is why concert grands are the biggest: the longer the string, the more harmonic are the overtones--the $+\Delta f$ shift from string stiffness is minimized. Per Ben51's Fourier decomp of the triangle formed from plucking: piano's have the hammer striking at 1/7 the string length, since that overtone is discordant following: C, C, G, C, E, G, B-flat
– JEB
Commented Feb 14, 2018 at 18:16