The bow-string interaction of the violin is a highly nonlinear mechanism. When the bow sticks to the string it displaces it. When it tears-off the string two impulses are creasted, one in each direction of the string. These impulses travel along the string, are reflected at its ends and return to the bowing point. Only the impule traveling first towards the bridge is then strong enough to again tear-off the string from the bow.
Only because this impulse tears-off the bow, the pitch of the tone is determined by the strings length, as this impulse first need to travel all along the string. But if the bow force is so strong that this impulse cannot tear-off the bow the bow will displace the string further until the tension of the string is too high to further hold the string. Then the pitch of the string is determined by the bowing force. This technique is known as subharmonic playing as very low tones can be played below the pitch of the low g-string.
In a Finite-Difference Time Domain model of the string/bow the playing pressure is changed steadealy from zero to a maximum and down to zero again. In the correlogram below, showing only harmonic overtone spectra, three phases can be seen. With very low pressure the regular sawtooth or Helmholtz motion is not reached yet. Then from a certain threshold on the Helmholtz appears. During this Helmholtz regime with increasing pressure the pitch is going down slightly which is known as the flattening effect. Then again at a certain threshold the regime changes and the pitch is determined by the bowing pressure. When retuning to zero bowing pressure the regimes are found again, still the threshold points are different. This is a hysteresis effect known from self-organizied systems.
Correlogram of the model with in- and decreasing bowing pressure
In a whole body Finite-Difference model of the violin the role of the different parts of the violin body in tone production can be seen. The sound of different parts is considerably different:
The movement of bridge and top plate over one Helmholtz cycle:
Double-slip motion of cello bowing produces an interesting blurred sound. It is measured with this bowing machine at the University of Applied Sciences Hamburg. During one Helmholtz cycle a second tear-off the bow from the string appears. The time point of this tear-off changes over time
When tracking the peaks within this cycle it appears that the middle peak, the tear-off, is not perfectly in the middle of the cycle. Therefore two periodicities appear areound the second partial, one a bit smaller, one a bit larger than the periodicity of this second partial. Also the interval changes in time. this produces the blurred timbre of the sound.
Feeding the cello double-slilp sound into a cochlear model, at the frequency of the second partial interspike-intervals (ISI) appear (red and blue curves are bowing pressure and velocity).
Fitting the periodicities from the time series (red and blue) with the cochlear ISI the perfectly fit together. Therefore the double-slip motion is coded in the ISI.
Bader, R.: Whole geometry Finite-Difference modeling of the violin. Proceedings of the Forum Acusticum 2005, 629-634, 2005.