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String Core Diameters

We have initial string tensions $ T_{0n}$ , lengths $ L_n$ , and frequencies $ f_n$ which imply the initial mass densities:

$\displaystyle \mu_{0n} = (2 f_n L_n)^{-2} T_{0n},\quad n=1,\ldots,88
$

Non-wound string diameters:

$\displaystyle d_{1n} = \sqrt{\frac{4\mu_{0n}}{\pi\rho_s}},\quad n=1,\ldots,88
$

rounded to the nearest multiple of $ 0.025$ mm, resulting in new linear mass densities

$\displaystyle \mu_n = \frac{\pi}{4}\rho_s d_{1n}^2
$

and tensions

$\displaystyle T_n = (2f_nL_n)^2\mu_n
$


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``Piano-Hammer Modeling'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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