This is the sound examples page of the paper

Balázs Bank and László Sujbert, "Generation of longitudinal vibrations in piano strings: From physics to sound synthesis,'' The Journal of the Acoustical Society of America, vol. 117, no. 4, pp. 2268-2278, Apr. 2005.

Abstract

Longitudinal vibration of piano strings greatly contributes to the distinctive character of low piano notes. In this paper a simplified modal model is developed, which describes the generation of phantom partials and longitudinal free modes jointly. The model is based on the simplification that the coupling from the transverse vibration to the longitudinal polarization is unidirectional. The modal formulation makes it possible to predict the prominent components of longitudinal vibration as a function of transverse modal frequencies. This provides a qualitative insight into the generation of longitudinal vibration, while the model is still capable of explaining the empirical results of earlier works. The semi-quantitative agreement with measurement results implies that the main source of phantom partials is the transverse to longitudinal coupling, while the string termination and the longitudinal to transverse coupling have only small influence. The results suggest that the longitudinal component of the tone can be treated as a quasi-harmonic spectrum with formantlike peaks at the longitudinal modal frequencies. The model is further simplified and applied for the real-time synthesis of piano sound with convincing sonic results.

Sound examples

Original

A recorded C2 piano tone first unaltered, then the transversal components resynthesized by additive synthesis (the longitudinal components and the attack transients are filtered out), and last, the residual containing longitudinal components and attack transients. This example is presented to show the preceptual importance of these components.

C2 Original

Relative importance of free and forced components (Sec. IV A)

The following examples were generated by the composite string model. Then the longitudinal component was resynthesized by additive synthesis. This made it possible to separate the contribution of the free modes and the phantom partials.

G1 without longitudinal modeling
G1 free response
G1 forced response (phantom partials)
G1 free and forced response

It can be heard that the fast decaying longitudinal modes are masked by the transversal partials after a short time, while the forced response (phantom partials) remains significant. Note that the pitch of the longitudinal component can be heard also in that case when only the phantom partials are present. We beleive that these examples show that the forced component has a larger perceptual significance and that the pitch preception is mainly determined by the phantom partials.

The relative contribution strongly depends on the decay times of the longitudinal modes. Setting the decay times to the double of the measured ones (ceteris paribus) emphasizes the free response:

G1 without longitudinal modeling
G1 free response
G1 forced response (phantom partials)
G1 free and forced response

However, it is still true that the forced response has larger contribution to the characteristic sound.

The composite model (Sec. V B)

The examples generated by the composite string model based on finite-difference modeling for the transversal vibration and nonlinearly excited resonators for the longitudinal motion:

C2 without longitudinal modeling
C2 with longitudinal modeling
G1 without longitudinal modeling
G1 with longitudinal modeling

G1 dynamics (piano to forte) without longitudinal modeling
G1 dynamics (piano to forte) with longitudinal modeling

The sound examples without the longitudinal modeling are presented for easier comparison. These were calculated by using the same parameters for the transversal string model and for the soundboard, but the longitudinal force at the bridge was set to zero. The examples show that the longitudinal component remains an inherent part of the tone, while it is still possible to perceive its pitch, similarly as in the case of real piano tones.

The resonator based string model (Sec. V C)

This model sounds almost the same as the composite model of Sec. V B at a lower comutational cost (less than 50%):

G1 composite string model
G1 resonator based string model

Other online publications on piano modeling