Sound synthesis
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Sound Synthesis. Part II: Oscillators, Additive Synthesis & Modulation. Plan. Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary. AMP. FREQ. PHASE. WF. Simple Oscillator. Oscillator 3 strategies.

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Sound Synthesis

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Sound synthesis

Sound Synthesis

Part II: Oscillators, Additive Synthesis & Modulation


Sound synthesis

Plan

  • Simple Oscillator (wavetable)

  • Envelope control

  • Simple Instrument (Helmholtz)

  • Additive Synthesis

  • Modulation

  • Summary

AMP

FREQ

PHASE

WF


Simple oscillator

Simple Oscillator

Oscillator

3 strategies

Mathematical equation based oscillator

Wavetable oscillator

IIR-Based oscillator

  • Solve math function for each sample

  • Ex: y = sin(x)

  • + Accurate

  • Inefficient

  •  Non real-time applications

  • Pre-computed and stored in memory

  • + Fast (Look-up table)

  • Memory

Unstable filter that generates waveform of desired amplitude and frequency.

+ Fast

+ Memory efficient

 Sound synthesis


Wavetable oscillator

Wavetable Oscillator

  • Example of a wavetable (N = 16)

  • Store N values sampled over one cycle

  • Phase increment: SI=N f0/fs


Wavetable oscillator example

Wavetable Oscillator (example)

  • Parameters

    • N = 16

    • F0 = 220

    • Fs = 1kHz

    • SI = 16 * 220/1000SI = 3.52

  • Increase quality:

    • Increase sampling rate

    • interpolate


Wavetable oscillator distortions

Wavetable Oscillator Distortions

  • Quantization:Eg, pure tone F0=440Hz, Fs=8,192Hz

    • Truncate N=16

    • Truncate N=32

    • Truncate N=512

  • Interpolation: truncate, mean, linear

  • Aliasing


Wavetable oscillator interpolation

Wavetable OscillatorInterpolation

  • Truncation (0th level interpolation)


Wavetable oscillator interpolation 2

Wavetable Oscillator Interpolation (2)

  • Rounding (slightly better 0th order)


Wavetable oscillator interpolation 3

Wavetable Oscillator Interpolation (3)

  • Linear (First order interpolation)


Wavetable oscillator interpolation 4

Wavetable Oscillator – Interpolation (4)

  • Quadratic (Second order interpolation)


Wavetable oscillator interpolation 5

Wavetable Oscillator Interpolation (5)

  • Cubic (Third order interpolation)


Wavetable oscillator interpolation 6

Wavetable Oscillator Interpolation (6)

  • Signal to (interpolation) Noise Ratio (SNR)(eg, pure tone F0=220Hz, Fs=8,192Hz)

    • Truncation: SNR = 6 k – 11 dB

    • Rounding: SNR = 6 k – 5 dB

    • Linear:SNR = 12 (k – 1) dB(Moore, 1977; Hartman, 1987)(k = log2(N) and N is the table length)

  • Conclusion: For increasing quality, increase number of samples, and use interpolation.


Wavetable oscillator interpolation 7

Wavetable Oscillator Interpolation (7)

  • Pure tone F0=440Hz, Fs=8,192Hz

    • Truncate N=16

    • Truncate N=32

    • Truncate N=512


Wavetable oscillator aliasing

Wavetable Oscillator – Aliasing

  • Aliasing: One of the biggest problem for modern digital sound synthesisers (sampling freq fs=48kHz, Nyquist freq fn=fs/2=24kHz).

  • How to avoid aliasing?

    • Storing a band-limited version of the waveform in the table (in memory)

    • Or, generate an aliasing-free signal from frequency-limited Fourier series representation.


Aliasing 2

Aliasing (2)

  • Several sinusoids can fit a set of samples.

  • Aliasing when sampling rate is low! Example:

    • Signal:f0 = 0.9Hz(red)

    • Sampling at:fs = 1Hz, Nyquist freq fn = 0.5Hz

    • perceived fa=|n*fs-f0|=0.1Hz(blue) (n such that fa < fn)


Aliasing 3

Aliasing (3)

  • Square wave, 563 Hz fundamental, 48kHz sampling rate.

Generated using “perfect” squarewaveform

Generated using a limited Fourier series.


Sound synthesis

Plan

  • Simple Oscillator (wavetable)

  • Envelope control

  • Simple Instrument (Helmholtz)

  • Additive Synthesis

  • Modulation

  • Summary

AMP

FREQ

PHASE

WF


Time envelope 1

Time Envelope (1)

  • ADSR Envelope

    • Attack

    • Decay

    • Sustain

    • Release

  • Important is:

    • Duration

    • Shape

      • Linear

      • Exponential

      • Other (functional, table)


Linear vs exponential envelope

Linear vs. Exponential Envelope

A) Linear

B) Exponential

  • Recall:“amplitude perception is (nearly) logarithmic”

    • linear decay  logarithmic (perceived) fading

    • Exponential decay  linear (perceived) fading

  • Note: Exponential decay never reaches zero  set min value


Oscillator as an envelope generator

Oscillator as an Envelope Generator

A

fm

  • Advantages:

    • wavetable interpolated shape.

    • Easy encoding of several repetitions.

  • Drawback:

    • attack and decay times are affected by overall duration!

  • Alternative:

    • interpolated function generator

fc


Sound synthesis

Plan

  • Simple Oscillator (wavetable)

  • Envelope control

  • Simple Instrument (Helmholtz)

  • Additive Synthesis

  • Modulation

  • Summary

AMP

FREQ

PHASE

WF


Simple instrument

Simple Instrument

  • Helmholtz model

    • Waveform

    • Constant frequency

    • Envelope

  • Envelope feeds varying amplitude to the oscillator.

AMP

DURATION

ATTACK

DECAY

ASD Envelope

FREQ

AMP

PHASE


Simple instrument 2

Simple Instrument (2)

  • Envelope generator used as a signal processor.

  • Oscillator feeds varying amplitude to the envelope generator.

  • Allows to process the amplitude of a natural (recorded) sound through an envelope.

AMP

FREQ

PHASE

DURATION

AMP

ATTACK

DECAY

ASD Envelope


Limitations of the simple instrument

Limitations of the Simple Instrument

  • Helmholtz model

    • Waveform

    • Constant frequency

    • Envelope

  • Limitations:

    • Amplitudes of all spectral components vary simultaneously.

    • All spectral components are perfect (integer) harmonics.

  • ... unlike real sounds!

AMP

DURATION

ATTACK

DECAY

ASD Envelope

FREQ

AMP

PHASE


Sound synthesis

Plan

  • Simple Oscillator (wavetable)

  • Envelope control

  • Simple Instrument (Helmholtz)

  • Additive Synthesis

  • Modulation

  • Summary

AMP

FREQ

PHASE

WF


Types of synthesis

Types of synthesis


Additive synthesis

Additive Synthesis

FREQ

FREQ

FREQ

+


Additive synthesis 2

Additive Synthesis (2)

  • Analysis: Frequency and amplitude envelopes can be obtained from analysis (spectrogram)

  • Flexibility: Virtually any sound can be synthesised.

  • Allows for the generation of new, natural sounding functions.

  • Quality: Can realize sounds that are “indistinguishable from real tones by skilled musicians” (Risset, Computer Study of Trumpet Tones, 1966)


Additive synthesis 3

Additive Synthesis (3)

  • But...

    • Require large amount of data to describe a sound

      • Each oscillator requires two functions

    • Functions are only valid for limited rangeof pitch and loudness!

      • Analysis for a given pitch and loudness will not give the same timbre when extrapolated for different pitch and loudness.

  • Requires very large library of function sets!

  • Just too much control?


Sound synthesis

Plan

  • Simple Oscillator (wavetable)

  • Envelope control

  • Simple Instrument (Helmholtz)

  • Additive Synthesis

  • Modulation

  • Summary

AMP

FREQ

PHASE

WF


Modulation

Modulation

  • Modulation:“Alteration of amplitude, phase or frequency of an oscillator, in accordance to another signal” (Dodge & Jerse, 1997)

  • Vocabulary:

    • Carrier oscillator: modulated oscillator

    • Carrier wave: modulated signal (prior to modulation)

  • Spectral components of modulated signal:

    • Carrier components: come only from carrier

    • Sidebands: come from both carrier & modularion


Amplitude modulation

Amplitude Modulation

  • Carrier:

    • Frequency: fc

  • Modulating

    • Frequency: fm

    • Amplitude m*AMP

  • Modulation index: m

    • m=0 no modulation

    • m>0 modulation

    • m=1 full modulation

m*AMP

AMP

fm

+

fc

AMP


Amplitude modulation 2

Amplitude Modulation (2)

  • Carrier frequency fc

    • Unaffected by modulation index

  • Sidebands fc+/-fm

    • Amplitude m/2*AMP

    • Energy split equally between lower/higher

    • When m=1, sidebands 6dB below carrier

  • Perception

    • If fm>10Hz -> two tones, additional loudness.

    • If fm<10Hz -> tremolo

Amplitude

AMP

m/2*AMP

fc-fm

fc

fc+fm

Frequency

Pure tone fc=220Hz

Tremolo fc=220Hz, fm=6Hz, m=1


Amplitude modulation 3

Amplitude Modulation (3)


Ring modulation

Ring Modulation

A

A

A

fm

fc

fm

  • Modulation is applied directly to carrier’s amplitude.

    • A=0  no signal!

  • Alters frequency!

  • If both sinusoidals:

    • Only sidebands:fc-fm and fc+fm!

    • Amplitude A/2

  • Eq. to signal multiplication

fc

*

Amplitude

A/2

fc-fm

fc

fc+fm

Frequency


Vibrato modulation

Vibrato Modulation

  • Modulating signal applied to the carrier’s frequency.

  • “Slight wavering of pitch”

  • Pitch varying between fc-v <= fv <= fc+v

  • Average is <fv> = fc

  • Eg, fc=220Hz

    • Pure tone

    • Vibrato fv=6Hz, v=0.05fc

v

fm

fc

+

A

fv


Vibrato modulation 2

Vibrato Modulation (2)


Sound synthesis

Plan

  • Simple Oscillator (wavetable)

  • Envelope control

  • Simple Instrument (Helmholtz)

  • Additive Synthesis

  • Modulation

  • Summary

AMP

FREQ

PHASE

WF


Additional reading

Additional Reading

  • C. Dodge, C., & Jerse, T. A. (1997). Computer Music: Synthesis, Composition, and Performance. Schrimer, UK.(see chapter 4)


Sound synthesis

v

fc

fm

AMP

DURATION

ATTACK

DECAY

ASD Envelope

+

fv

AMP


Sound synthesis

AMP

DURATION

ATTACK

m*AMP

ASD Envelope

fm

AMP

+

fc

AMP


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