EE2F2 - Music Technology

11. Physical Modelling

Introduction Some ‘expressive instruments don’t sound very

convincing when sampled Examples: wind or bowed stringed instruments

Reasons When a performer plays a real instrument, every note

sounds slightly different – sampled notes all sound the same Also, the transition between notes is not sudden but gradual

Instead of sampling, physical modelling techniques build a computer simulation of the physical processes of an instrument

The model can then be ‘played’ with an appropriate controller and should sound more realistic

Classic Physical Modelling The science of acoustics is all about how things

vibrate A commonly used numerical technique to model

vibrating bodies is finite element analysis It works by representing complex solid objects

by a matrix of discrete points The problem is quantised If the density of the points is large enough,

vibrations in complex bodies can be simulated by many simple linear equations

Vibrating String Example

A taut string can be simulated by a row of small masses connected by ideal springs

If each individual spring is assumed to be always straight, the simulation becomes very simple

The movement of each mass can be calculated using Hooke’s law for the adjacent springs Newton’s 2nd law of motion

Vibrating String in Action

The sound produced by a vibrating string depends on the velocity of the elements

This can be read directly from the model

Examples:

Sum of all elements

Single element, mid-string

Single element, end of string

Planes and Volumes To simulate a plane (e.g. the

skin of a drum), use a 2-dimensional grid of elements

A volume (e.g. a solid bar or an enclosure of air) is a 3-dimensional grid

The equations are the same regardless

All that changes (depending on the material) are:

The mass of the elements The tension of the springs The frictional retardation

Pros and Cons

Pros Using finite-element analysis (or similar techniques)

any shaped instrument made from any material can be modelled

If the numbers are right, the sound can be indistinguishable from the real thing

Cons You need a lot of patience to program in all those

element positions and parameters You need a big computer to simulate them in real

time Currently, not technically feasible

Functional Physical Modelling For reasons of user-friendliness and computational demands, there is an

urgent need to simplify the classic approach By way of example, consider the string again. Given the properties of the string, we can predict known resonant modes:

Fundamental 1st Harmonic 2nd Harmonic Sum of Harmonics

Functional Modelling Cont.

Any initial pluck displacement (the boundary condition) can be expressed as a sum of weighted sine waves

The weight of each sine wave determines how much that harmonic will be excited

If the behaviour of the harmonics is known beforehand, the behaviour following any initial displacement can be easily predicted by adding them together in the right proportions

Fundamental 1st Harmonic 2nd Harmonic

+ +

Source-Resonator Model

A simplified way of thinking of the plucked string is the source-resonator model

Source: The initial displacement of the string Resonator: A filter that resonates according to the modes of the

string This model can be applied to a wide range of

instruments NB. Sometimes, the resonator output modifies the

source. In these cases feedback is required.

Source Resonator Output

Feedback (when required)

Source-Resonator Model

Source Resonator Output

Feedback (when required)

Source spectru

m

frequency

Resonator

Response

frequency

Fundamental mode

Harmonics

Source-Resonator Examples Piano

Source: The hammer displacing the piano string Resonator: The modes of vibration of the string multiplied

by the frequency response of the sound-board Flute

Source: The noise-like rush of air over the mouthpiece Resonator: The resonant modes of the pipe

Trumpet Source: The vibrations of the performers lips Resonator: The resonant modes of the tube, modified by

the effects of the flare at the end Feedback: In this case, the resonance of the pipe feeds

back to the source

Sound Examples

Bowed Violins

Plucked guitar quintet

Flute (with ‘overblowing’)

Pros and Cons

Pros Potentially, produces the most realistic synthesised

sounds around Responds in the same way as the real thing Can be used to synthesise fictional instruments by

breaking a few laws of physics! Cons

Can be very difficult to play (if you’re a rubbish violinist, you’ll also be a rubbish virtual violinist)

Currently, not easy to program – poor user interface

Physical Modelling Summary

Very realistic sounds High computational complexity

(especially using classic modelling) Can be difficult to play

The Future of Synthesis Additive Methods

Elaborate additive synthesis techniques allow easy time and pitch stretching and morphing

Could turn out much easier to play than physically modelled instruments

Processor intensive at the moment Physical Modelling

Modelling environments must be made more friendly Modelling of fictional instruments

The Human Voice Speech and music synthesis combined!

Microsoft’s best effort!

Music Projects

Current Projects (BEng/MEng) Microcontroller based MIDI devices

Pitch-to-MIDI conversion Subtractive synthesiser Controller pedal

Additive synthesis for data compression Bass-servo (in conjunction with Linn)

Music Projects Future Projects

More microcontroller based devices FM synthesis Wind controller

Signal processing Effects processing Automatic transcription Physical modelling Analysis and re-synthesis of sounds

Course Summary

Recording Technology Multi-track recording and mixing Effects MIDI & Sequencers

Sampling & Synthesis Subtractive and Additive Synthesis (+FM a

bit) Sampling and Sample+Synthesis Physical Modelling