# Efficient synthesis of tension modulation in strings and membranes based on energy estimation

This is the companion webpage of the paper

F. Avanzini, R. Marogna, and B. Bank. Efficient synthesis of tension modulation in strings and membranes based on energy estimation. J. Acoust. Soc. Am., 131(1):897-206, Jan. 2012.

### Abstract

String and membrane vibrations cannot be considered as linear above a certain amplitude, due to the variation in string or membrane tension. A relevant special case is when the tension is spatially constant, and varies in time only in dependence of the overall string length or membrane surface. The most apparent perceptual effect of this tension modulation phenomenon is the exponential decay of pitch in time. Pitch glides due to tension modulation are an important timbral characteristic of several musical instruments, including the electric guitar and tom-tom drum sounds, and many ethnic instruments. This paper presents a unified formulation to the tension modulation problem for 1D (string) and 2D (membrane) cases. In addition, it shows that the short-time average of the tension variation, which is responsible for pitch glides, is linearly proportional to system energy. This proportionality allows the efficient physics-based sound synthesis of pitch glides. The proposed models require only slightly more computational resources compared to linear models, as opposed to earlier tension-modulated models of higher complexity.
### Sound examples

All the following sound examples have been synthesized from Matlab simulations. For the plucked string examples, the sound signals represent the string velocity at the plucking point. For the struck membrane examples, the sound signals represent the membrane displacement at the hit point. No radiation model is considered.
#### Plucked string

Three sound examples of a rigidly terminated, tension-modulated string, obtained from a finite-difference model (see paper).

Model parameters simulate a electric guitar string, *65* cm long and having a tension at rest of *100* N. The string is plucked at point 0.12 (normalized with respect to the total length), and the pluck is modeled as a triangle-shaped initial displacement distribution with a peak value of *10* mm (this large value produces a very clear pitch glide effect).

These examples demonstrate the use of both downsampled energy computation and the energy storage model for estimating tension modulation, for the particular case of the string.
- Complete tension-modulated string model
- Efficient tension-modulated model using the quasistatic component of the nonlinear tension, estimated from string energy with a downsampling factor of 32
- Efficient tension-modulated model, using the "energy storage model" to estimate string energy and hence nonlinear tension

#### Struck rectangular membrane

Three sound examples of a rigidly terminated, tension-modulated rectangular membrane, obtained using modal synthesis and an impact force model that simulates interaction with a drumstick (see paper).

The size of the membrane is *32x25* cm^{2} and its tension at rest is *1500* N/m. It is struck at point *(0.2, 0.1)* (values normalized with respect to the *x* and *y* dimensions), and the impact velocity of the drumstick is *18* m/s, which is close to the highest dynamic levels in drum playing.

These examples demonstrate the use of downsampled energy computation for estimating tension modulation, for the particular case of the rectangular membrane.
- Complete tension-modulated rectangular membrane model
- Approximate tension-modulated model using the quasistatic component of the nonlinear tension, estimated from membrane energy
- Efficient tension-modulated model using the quasistatic component of the nonlinear tension, estimated from membrane energy with a downsampling factor of 32

#### Struck circular membrane

Two sound examples of a rigidly terminated, tension-modulated circular membrane, obtained using modal synthesis and an impact force model that simulates interaction with a drumstick (see paper).

The radius of the membrane is *16* cm and its tension at rest is *3000* N/m. It is struck at point *0.5* (value normalized with respect to the radius), and the impact velocity of the drumstick is *16* m/s, which is still close to the highest dynamic levels in drum playing.

These examples demonstrate the use of the energy storage model for estimating tension modulation, for the particular case of the circular membrane.
- Complete tension-modulated circular membrane model
- Efficient tension-modulated model, using the "energy storage model" to estimate membrane energy and hence nonlinear tension