Tension Modulation

In every freely vibrating string, the fundamental frequency declines over time as the amplitude of vibration decays. This is due to tension modulation, which is often audible at the beginning of plucked-string tones, especially for low-tension strings. It happens because higher-amplitude vibrations stretch the string to a longer average length, raising the average string tension $\Rightarrow$ faster wave propagation $\Rightarrow$ higher fundamental frequency.

The are several methods in the literature for simulating tension modulation in a digital waveguide string model [498,234,512,516,517,499,285], as well as in membrane models [300]. The methods can be classified into two categories, local and global.

Local tension-modulation methods modulate the speed of sound locally as a function of amplitude. For example, opposite delay cells in a force-wave digital waveguide string can be summed to obtain the instantaneous vertical force across that string sample, and the length of the adjacent propagation delay can be modulated using a first-order allpass filter. In principle the string slope reduces as the local tension increases. (Recall from Chapter 6 or Appendix C that force waves are minus the string tension times slope.)

Global tension-modulation methods [499,498] essentially modulate the string delay-line length as a function of the total energy in the string.