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 2 nd 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: Fundamental1 st Harmonic2 nd HarmonicSum 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 Fundamental1 st Harmonic2 nd 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. SourceResonator Output Feedback (when required)

Source-Resonator Model SourceResonator Output Feedback (when required) Source spectrum 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