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- presented by Prof Trevor Cox

Principles of Sound Synthesis

This article aims to discuss principles, techniques and popular equipment to synthesise musical instrument sounds.

The Structure of Sound

Sound is the perceived vibration (oscillation) of air resulting from the vibration of a sound source (e.g. guitar sound board, speaker cone, hair dryer, etc). We can describe such regular (periodic) vibration in terms of the sum of simpler vibrations (harmonics). In other words any periodic oscillation and hence resulting waveform can be described in terms of the sum of its harmonics. Each harmonic being a simple sine wave (often called a pure tone) with it’s own respective frequency and amplitude. The graphs below shows how a simple pure tone varies with respect to time.

pure tone

The graph below shows a more complex waveform comprised of three pure tones of different respective frequencies and amplitudes.

complex waveform

The relative amplitudes and frequencies of harmonics with respect to time define timbre – i.e. the sound we hear. The graph below shows how the plotting amplitude against frequency of a given waveform gives us a visual representation of it’s harmonic content.

Frequency spectrum

Though note the harmonic content will change over time if the waveform doesn't repeat. In other words if it isn't periodic.  

Synthesis provides a means of constructing timbres to emulate existing instruments and create new sounds.  There are two approaches to synthesising these timbres

1) Analysis of the sound itself and directly trying to emulate it.

2) Analysis of the physical workings of the musical instrument and trying to model its sound generation.

Acoustic Sound Generation

Typically most musical instruments can be considered as consisting of three parts

String instrument

  • Excitation Source – this gives the system energy, for instance a violin bow, plectrum, hammer or player’s breath.
  • Wave-guide – this is the main part of the instrument that oscillates (for instance the string of a guitar or the air column in a flute). When the system oscillates in a steady periodic fashion it produces a complex waveform that can be expressed in terms of a fundamental frequency and harmonics.
  • Resonator – this primarily takes energy away from the wave-guide to produce sound. The resonator is typically the sound box, bell or body of the instrument. The resonator will oscillate in it’s own way in sympathy with the wave-guide so changing the oscillation of the wave-guide and modifying the resulting timbre. The sympathetic oscillating properties of the resonator are often referred to as formants. These formants have an important bearing on the quality of the sound produced.

frequency spectra

When we synthesis sound we will need to consider levels of excitation, oscillation/harmonics and how the system releases energy to be heard as sound. We will firstly review some classic synthesis techniques and then explore some relatively new ‘state of the art’ avenues for sound synthesis.

Additive Synthesis

A C3 Hammond Organ

The classic Hammond C3 shown allows the musician to build the harmonic content of a sound by adding in ‘flutes’ (approximating to sine waves) of progressively higher frequencies. This is done by pulling out drawbars to varying extents, so increasing the signal levels of the respective harmonics.

Frequency spectrum

Hammond organs are designed to emulate the steady state of forced response instruments such as brass, woodwinds and bowed strings. Little effort is made to emulate the transient response though the player can choose for the second or third harmonics to be more prominent at an earlier time.

Frequency spectrum

Vibrato can be achieved from the spinning horn of a Leslie speaker. When the horn rotates the resultant Doppler shift of frequency superimposes on the original signal giving a pleasing phasing effect. More sophisticated computer based additive synthesisers exist giving the programmer greater control over harmonic and temporal parameters.

Subtractive Synthesis

A minimoog.

The classic minimoog shown provides a number of harmonically rich oscillators such saw-tooths, pulses and square waves, which can be filtered to emulate the timbre of real instruments.

Transient and steady state responses of the instrument are emulated by a time dependant voltage envelope usually consisting of attack, decay, sustain and release sections whose respective durations can be controlled by the user.

Attack-sustain-release

Such an envelope can also control the filter section modifying the filtering of the waveforms with respect to time. Vibrato can be achieved by modulating of the main oscillator with a low frequency oscillator. Two oscillators of similar frequencies (one slightly detuned) together give a pleasing phasing effect and add more weight to the sound. Although not too bad at brass and strings the Moog doesn’t do a particularly good job at impersonating anything but itself.

Wave equations beyond the 1D case

For many types of instruments where the wave-guide has more than one dimension (e.g. plates, bars and membranes) standing waves may be possible along a number of axes. For instance we could show a rectangular plate

Plate

has an increase in the number of modes of vibration well beyond a harmonic series. The range of possible in-harmonic frequencies further increase when we consider bars, bells, membranes and circular plates, the latter having so many in-harmonic frequencies that no distinct fundamental dominates.

Frequency spectrum

Simply constructing or filtering a harmonic series will be insufficient to model these important cases. Analysis of a well-made bell will show that although some ‘overtones’ do not correspond to a harmonic series they may correspond to other consonant intervals such as thirds and minor thirds giving the sound a musically useful character.

Frequency Modulation

Yamaha DX7

A Yamaha DX7

In theory, any instrument sound can be emulated with six sinusoids operating as frequency modulators of each other.  For instance given a carrier sine wave, its wave shape can be radically changed by a second modulating wave serving as the input to the function.

For instance consider two sine waves of different respective amplitudes ( and ) and frequencies ( and ). We can modulate one with the other as follows

Looking at the frequency spectrum of x…

Frequency spectrum

We can see extra overtones are added to more familiar harmonics f, 2f, 3f, 4f, etc. Having just one modulating oscillator does not give spectacular results.  Usually an FM synth will have 4 to 6 oscillators (‘operators’) that can be routed in a variety of ways (‘algorithms’) the carrier always being the last in the modulation chain. The technique is notoriously hard to implement, most musicians opting for the subtle tweaking of presets rather than full on programming. Though the technique does well at impersonating, electric pianos, bells and xylophones; acoustic pianos and guitars don't sound so convincing.

Sampling

Organ

A Fairlight – once the price of a house you can now have better sampling on your PC for nothing.

The Fairlight shown here is an early and expensive pioneer of the sampler (following on from earlier tape based mellotrons). Here the sound of a given note of a musical instrument is digitally recorded. When the user presses the corresponding key on an electronic keyboard the sound is replayed. If the user presses a different key at a different key strike pressure to sound is pitch shifted (by changing speed of playback) and volume modified appropriately. The problem with this is that the resulting timbre will not reflect that of a real instrument. For instance if you sample a sung ‘hello’ at middle C, the voice will sound distinctly munch-kin like when played just a few notes higher. As real musical instruments vary considerably with pitch and volume dynamics so samplers need to compensate for this with multiple samples for various pitch ranges and keyboard strike pressures. The result is, the more information recorded the more convincing the sound. This causes a problem if memory is limited since ten seconds of uncompressed stereo CD quality audio will take up a 1MB. However, by streaming data directly from a hard disk sample files of the order of gigabytes can be created. The technique is employed by Nemesys’s ‘Gigasampler ' software.

Software synth

Nemesys’s ‘Gigasampler ' software.

Modern samplers are getting better at pianos and strings though still fail to deliver convincing guitar sounds. The non linearities of the excitation of a plucked string and the timbral changes between frets help define the sound so much that sampling alone fails. Another way to avoid the munch-kin effect during pitch shifting is to try to preserve duration of the signal with stretching algorithms and more importantly the formants of the resonator. Such techniques are employed by Roland’s much acclaimed ‘Variphrase’ samplers and studio effects units like Antare’s ‘autotune’ (used to try a make boy and girl bands who can’t sing sound less obnoxious – as if that’s possible).

Variphase

Roland’s VP9000 VariPhrase sampler

All the above techniques have serious problems when trying to deliver the sounds of real musical instruments. The classic instruments so far shown have…

1) Hardware restrictions, the oscillators, envelops, filters, memory, etc are pre-wired giving the musician little scope to modify and hybridise techniques.

2) Oversimplified modelling of parameters

what about temporal variations of components?

what about non linearities?

do the instruments thoroughly account for damping, feedback and second order resonance?

3) The parameters used tend to have no relation to physical parameters of instrument

Virtual Modular Synthesis

Rather than using specific hardware, relatively ordinary PCs with popular sound cards have the ability to emulate oscillators, ADSRs, filters, frequency modulators, samplers, etc in real time. The generic nature of PCs allows users to connect up virtual synthesiser components in many familiar and totally new ways via intuitive user interfaces. The result can be very impressive, for instance it's hard to tell the difference between a real minimoog and a virtual minimoog created in Native-Instruments Generator software. Synthesiser components can be selected from menus, dragged around, inter-connected and assigned virtual controllers such as knobs and sliders.

The recent development and popularity of virtual modular synths has been facilitated by

Popularity of personal computers - which are faster, cheaper and more amenable than ever.

Ever faster CPUs, recently Intel and AMD have released 1Ghz processors for PCs.

Generic Application Programming Interfaces (DirectX, VST2.0) so that software developers can talk to operating systems and applications to create sounds at a high level in generic terms rather than in a hardware specific way. For instance, if a developer wanted to write a chorus/flange effect. With DirectX function calls that chorus effect can be integrated into numerous different sequencers like Cubase and Cakewalk, sound editors like Sound Forge and Cool Edit and even virtual synthesisers like VAZ and Generator.

need for flexibility and invention

Desire for inexpensive simulations of retro kit (virtual moogs, hammonds, DX7s, etc)

Examples include Native Instrument's Generator, Seer System's Reality,  Syn-C Modular, Dreamstation and Vaz Modular. If you are interested in synthesis and have access to a reasonable computer and midi keyboard I strongly recommend you visit http://www.hitsquad.com/smm where you can download perfectly usable shareware versions of SynC and Dreamstation as well as countless demos and some freeware.

Here's Syn-C modular used to construct a virtual Hammond Organ.

synth-c

Sync Modular – impressive modular synthesis shareware.

Digital Wave-guide Modelling

This is recent development in sound synthesis where by the physics behind musical instruments is applied indirectly to sound generation. The whole point of this module has been to explain some of the physics of musical instruments so that in the future you may apply your knowledge to improving or creating musical sounds. Physical modelling is the application of principles learned.

Instead of creating the sound directly with oscillators, models of the physical processes that produce sound are created. The result is

more expressive and realistic sounds

a technique ideal for software only implementation on generic PCs

no need for dedicated hardware or lots of memory

The main difficulty is the potential huge demands on the CPU without optimisations. However, PCs are getting faster.

Brute force approach

Physical modelling in it's purest form would use numerical methods to solve equations of motion with respect to boundary conditions.  We have spent a lot of time discussing the equations of motion (wave equations) for a variety of systems and examined in some depth the nature of excitation and boundary conditions. We could implement many of the maths learned directly, effective recreating musical instruments in the computer, though this can be computationally expensive.

Better Approach

Partway solve equations for changing parameters with

lookup tables

lumped processes

novel algorithms

Delay Lines

Model string as wave-guide

waveguide

When we pluck a string we create waves that move along the string. The above diagram shows how arrays of values called ‘Delay Lines’ can be used to represent a virtual string (wave-guide). Values representing energy move along the arrays being modified (filtered) at Bridge and Nut terminations. High frequency energy may be filtered more than low frequency energy hence the nature of oscillation is changed by the string terminations over time.

Some of the energy may be taken from the system at a given point to be converted to sound. For instance for modelling an electric guitar

Yes

Steve Howe of Yes

String plucked at given position (energy added to delay line)

The virtual string vibrates (energy moves along delay lines and filtered at terminations)

Virtual pickups tap the energy (data from a point along the array is extracted)

We can further create an electric guitar sound by…

Feeding energy to a virtual distortion pedal along virtual cables to virtual amp

Add delayed energy back into the delay line to create feedback loop

VL1

The Yamaha VL1 (1994) was first commercial physical modelling synthesiser

There is a whole gamut of physics that can be applied to physical modelling Physical modelling gives the programmer much more control over how a system will respond with different levels of excitation. This gives the musician access to a much more expressive and realistic sounding instrument.

Ironically early hammonds and moogs are now being physically modelled as instruments in their own right. The tone wheels of a hammond organ and analog circuits of a real moog exhibit non-linear behaviours that give these instruments character and depth. These two classic cases show how a bad synthesiser of the sound of real musical instruments is by no means a bad musical instrument.

Re-use of material permitted provided it is clearly labelled "(c) University of Salford, www.acoustics.salford.ac.uk"