nlm: Real-Time Non-linear Modal Synthesis in Max

Rodrigo Diaz; Rodrigo Constanzo; Mark Sandler

cs.SDMar 10, 2026

nlm: Real-Time Non-linear Modal Synthesis in Max

Authors: Rodrigo Diaz, Rodrigo Constanzo, Mark Sandler

Organizations: Queen Mary University of London · Royal Northern College of Music

Abstract

We present nlm, a set of Max externals that enable efficient real-time non-linear modal synthesis for strings, membranes, and plates. The externals, implemented in C++, offer interactive control of physical parameters, allow the loading of custom modal data, and provide multichannel output. By integrating interactive physical-modelling capabilities into a familiar environment, nlm lowers the barrier for composers, performers, and sound designers to explore the expressive potential of non-linear modal synthesis. The externals are available as open-source software at https://github.com/rodrigodzf/nlm.

Figures & tables

Parameter	Attribute	String	Plate
$E$	youngs_modulus	$\mathrm{GPa}$
$\rho_{\text{m}}$	density	$\mathrm{kg}\text{/}{\mathrm{m}}^{3}$
$L$ , $L_{x}$ , $L_{y}$	lx , ly	$\mathrm{m}$
$h$	thickness	—	$\mathrm{m}$
$A$	cross_sectional_area	${\mathrm{mm}}^{2}$	—
$I$	moment_of_inertia	${\mathrm{mm}}^{4}$	$\frac{h^{3}}{12}$

Table 1: Physical parameters for string and membrane/plate models and their attributes in Max. We use moment of inertia following the definitions in [ 3 ] . For the string

I=\kappa^{2}A

where

\kappa

is the radius of gyration. Note that the density attribute for the externals is the mass density, not the material density used in ( 1 ). The bending stiffness

D

is computed internally and it is not exposed as an attribute directly.

Figure 1: Screenshot of the nlm.plate ~ external using the dk.impulsegeneratorcomplex and live input.

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Explicit and Stable Pseudospectral Time-Domain Method for the Föppl-von Kármán Equations

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Acoustics and Audio Group, University of Edinburgh, Edinburgh, UK · STMS (UMR9912), IRCAM, CNRS, Sorbonne Universit´e, Paris, France

PDE Solving Structural Mechanics Spectral Methods Physics-Based Simulation

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Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.

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