A first step towards the design of full virtual string instruments is the modelling of musical strings. To this end, we set an objective to design a versatile linear string synthesis algorithm, based on an accurate physical model.
One such model must be able to produce a wide variety of different sounds. Additionally, the user needs to be able to input custom physical parameters for the string (e.g. stiffness, tension, mass…) and we must also focus on designing physical models for different types of excitation.
The current string model can be simply plucked, but also struck by one or more hammers, bowed, and stopped with left-hand fingers. The excitation parameters (force, position…) can vary with time, in order to reproduce a wide panel of musical gestures. Let us dive in further on some of the aspects of the model.
The bowed string model
In the bowed string model , the transverse motion of a linear string is simulated in two directions, also known as polarisations. Dynamic models of a bow and left hand fingers are included, as well as a rigid fingerboard underneath the string. In one polarisation, a nonlinear collision model  is used for the normal forces exerted by the dynamic bow hair, left hand fingers, and fingerboard. These normal forces result in tangential friction forces in the other polarisation, allowing the string to be both set into motion by the bow, and captured between left hand fingers and the rigid fingerboard.
The gestural control of this model is defined by 2 sets of parameters (3 for the bow, 2 for the left hand finger), that can all be varied along the simulation:
- Bow parameters:
- – bow position along the string, closer or further away from the bridge
- – bow downwards force (usually called “bowing pressure” by musicians)
- – bow tangential force, which will determine the bowing velocity
- Finger parameters:
- – finger position along the string
- – finger downwards force
The presence of left hand fingers and a fingerboard allows to simulate a range of bowed string gestures. The initial transient of the bowing of a stopped note is shown below: the left hand finger (in blue) captures the string against the fingerboard (in grey), then the bow (pink) is lowered onto the string, and pushed across to set the string into motion.
The oscillation regime generally desired by musicians is the Helmholtz motion, shown below.
For a given bow-bridge distance and bow velocity, if the player presses the bow too strongly, the model produces raucous motion, shown below.
In some cases, so-called anomalous low frequencies may appear, producing subharmonic tones, shown below.
When the bow becomes too fast for a normal force too small, the static friction is not high enough for the bow to keep the string captured throughout a whole period, and multiple slipping occurs, resulting in surface sound, shown below:
Both the finger and bow can be moved along the simulation. If the finger slides or oscillates along the string, we can simulate glissando and vibrato gestures:
When the bow lands too fast on the string, it naturally bounces off. Refined control of this mechanism could allow for simulation of spiccato gestures.
The finger model includes nonlinear damping, so that the finger pulp absorbs some of the string vibrations. The following sound example is a plucked open string sound, followed by a pluck of the same string, this time stopped by a finger (as shown in the video):
Losses in linear strings
The nature of the sound decay in plucked musical strings is complex , and has a perceptually important role. An aspect of the work performed on string modelling has been the design of a frequency-dependent damping model, depending on physical parameters of the air and the string itself, in the time domain. The translation of frequency-dependent decay times to a time domain model, which is the NESS approach, is not a straightforward problem; however, we have found relatively efficient ways to accurately represent this loss profile.
The two following synthesised sound examples come from a plucked violin string model. The first one exhibits extremely simplified, frequency-independent losses; the second one makes used of the new, accurate time-domain damping model.
 C. Desvages, S. Bilbao. Two-polarisation physical model of bowed strings with nonlinear contact and friction forces, and application to gesture-based sound synthesis, Appl. Sci. 2016, 6, 135.
 C. Desvages, S. Bilbao, M. Ducceschi. Improved frequency-dependent damping for time domain modelling of linear string vibration, Proc. Int. Congr. Acoust. (ICA), Buenos Aires, Argentina, 2016.