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Modeling the stiffness of the string

The stretching of the frequencies of the partials can be calculated as:

$\displaystyle f_{n}=n f_{0}\sqrt{1+Bn^{2}}

where $ B= \frac{\pi^3 E d^4}{64 l^2 T}$ =inharmonicity factor,
$ f_{0}$ =fundamental frequency of the string.

E=Young modulus of elasticity, d=diameter.

\epsfig{file=eps/freqshift.eps,width=9cm} \\
Shift of partials for a cello D string, f0=147 Hz,B=3e-4.
x-axis: partial number, y-axis: frequency (Hz)

Physical constants of a Dominant violin string, from Pickering

String E (N/m) d(m) l (m) T B
E 15.7 0.000307 0.565 72.56 5.1627e-15
A 7.470 0.000676 0.59 56.25 6.5418e-14
D 6.4 0.000795 0.567 43.72 1.5707e-12
G 6.035 0.000803 0.567 44.57 1.4963e-12

Physical constants of a gut violin string, from Schelleng.

String B      
E 1.5598e-05      
A 4.8527e-05      
D 2.4841e-04      
G 1.3e-3      

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Download stiffbowed.pdf
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``Impact of String Stiffness on Virtual Bowed Strings'', by Stefania Serafin<>, (From CCRMA DSP Seminar Presentation, Music 423).
Copyright © 2019-02-05 by Stefania Serafin<>
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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