## Tesi etd-05072015-051356 |

Thesis type

Tesi di laurea magistrale

Author

RAPISARDI, GIACOMO

URN

etd-05072015-051356

Title

Frequency Analysis of a Nonlinear Hamiltonian System, application to Sound Synthesis

Struttura

FISICA

Corso di studi

FISICA

Supervisors

**relatore**Sozzi, Marco

Parole chiave

- nonlinear dynamics
- chaos
- sound synthesis

Data inizio appello

28/05/2015;

Consultabilità

Completa

Riassunto analitico

This work arises from the interest to create a connection beetween the two different fields of Sound Synthesis.

Historically speaking the first goal of sound synthesis was to imitate the sound of musical instruments. For this reason the most popular kinds of synthesis are meant to recreate the frequency spectrum of a given audio signal. These kinds of synthesis can be distinguished in a simple mathematical way: linear synthesis (LS) and nonlinear synthesis (NLS).

The main difference between LS and NLS is the following: linear techniques do not generate any frequencies which were not in the input signals whereas nonlinear techniques can generate new

frequencies. Furthermore, from a practical point of view, LS is easy to manipulate,

but requires a significant amount of input data in order to create a complex sound. This technique is often adequate for sounds whose harmonics do not vary over time, but unfortunately the spectra of most instruments vary considerably in time. On the other hand NLS allows to easily obtain dynamic spectra, but requires a less intuitive programming compared to LS.

Unfortunately the use of NLS (and synthesis in general) with the aim of recreating the dynamic spectrum of a musical instrument ended as soon as advanced samplers were developed together with the growth of computer technology.

Nowadays sound synthesis is not meant to imitate the sound of real instruments, but it's rather used to create sounds that are interesting beacuse they're actually really different from those which can be produced by any real instrument.

The sound synthesis we investigate in this thesis work is interesting because it's able to generate a very large variety of timbres by using a very small number of external parameters (basically just one). The algorithm that generates the signal is strongly nonlinear, but the control parameters are very few, so this feature should overcome the not-user-friendly aspect of NLS. In order to achieve this goal we have studied the dynamical features of a 2D discrete map derived from a 4D nonlinear Hamiltonian system, whose possible trajectories in phase space are determined from both the initial conditions and an additional external parameter. Every trajectory is interpreted as a discrete-time sequence; each value of this sequence can be written to a DAC (Digital-to-Analog Converter) at a given sample frequency.The result is a real-time audio representation of a trajectory of the dynamical system we're dealing with.

This kind of synthesis was studied heuristically in the past and named as Functional Iteration Synthesis (FIS) by A. Di Scipio in 1999.

The chosen mathematical map for this purpose was the one derived from the natural 2D Poincaré Section of the Gravitational Billiard, a nonlinear Hamiltonian system which concerns the motion of a point particle moving in a symmetric wedge subject to a constant gravitational field pointing downwards along the direction of the axis of symmetry.

The reason of this choice lies in the fact that this system is known to exhibit a remarkably complex behavior despite of having only two degrees of freedom. In fact, in order to achieve an interesting synthesizer using FIS we first need a dynamical system whose phase space is

extremely rich in different trajectories, and that's exactly the case of the gravitational billiard.

The aim of this thesis work is to give a more rigorous approach to FIS: in order to fulfill this task modern techniques for describing Complex Hamiltonian Systems' dynamics have been applied.

The advantages of a FIS synthesizer based on the gravitational billiard map are finally discussed.

Historically speaking the first goal of sound synthesis was to imitate the sound of musical instruments. For this reason the most popular kinds of synthesis are meant to recreate the frequency spectrum of a given audio signal. These kinds of synthesis can be distinguished in a simple mathematical way: linear synthesis (LS) and nonlinear synthesis (NLS).

The main difference between LS and NLS is the following: linear techniques do not generate any frequencies which were not in the input signals whereas nonlinear techniques can generate new

frequencies. Furthermore, from a practical point of view, LS is easy to manipulate,

but requires a significant amount of input data in order to create a complex sound. This technique is often adequate for sounds whose harmonics do not vary over time, but unfortunately the spectra of most instruments vary considerably in time. On the other hand NLS allows to easily obtain dynamic spectra, but requires a less intuitive programming compared to LS.

Unfortunately the use of NLS (and synthesis in general) with the aim of recreating the dynamic spectrum of a musical instrument ended as soon as advanced samplers were developed together with the growth of computer technology.

Nowadays sound synthesis is not meant to imitate the sound of real instruments, but it's rather used to create sounds that are interesting beacuse they're actually really different from those which can be produced by any real instrument.

The sound synthesis we investigate in this thesis work is interesting because it's able to generate a very large variety of timbres by using a very small number of external parameters (basically just one). The algorithm that generates the signal is strongly nonlinear, but the control parameters are very few, so this feature should overcome the not-user-friendly aspect of NLS. In order to achieve this goal we have studied the dynamical features of a 2D discrete map derived from a 4D nonlinear Hamiltonian system, whose possible trajectories in phase space are determined from both the initial conditions and an additional external parameter. Every trajectory is interpreted as a discrete-time sequence; each value of this sequence can be written to a DAC (Digital-to-Analog Converter) at a given sample frequency.The result is a real-time audio representation of a trajectory of the dynamical system we're dealing with.

This kind of synthesis was studied heuristically in the past and named as Functional Iteration Synthesis (FIS) by A. Di Scipio in 1999.

The chosen mathematical map for this purpose was the one derived from the natural 2D Poincaré Section of the Gravitational Billiard, a nonlinear Hamiltonian system which concerns the motion of a point particle moving in a symmetric wedge subject to a constant gravitational field pointing downwards along the direction of the axis of symmetry.

The reason of this choice lies in the fact that this system is known to exhibit a remarkably complex behavior despite of having only two degrees of freedom. In fact, in order to achieve an interesting synthesizer using FIS we first need a dynamical system whose phase space is

extremely rich in different trajectories, and that's exactly the case of the gravitational billiard.

The aim of this thesis work is to give a more rigorous approach to FIS: in order to fulfill this task modern techniques for describing Complex Hamiltonian Systems' dynamics have been applied.

The advantages of a FIS synthesizer based on the gravitational billiard map are finally discussed.

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