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Sound Synthesis

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Sound Synthesis

Part II: Oscillators, Additive Synthesis & Modulation

- Simple Oscillator (wavetable)
- Envelope control
- Simple Instrument (Helmholtz)
- Additive Synthesis
- Modulation
- Summary

AMP

FREQ

PHASE

WF

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

- Example of a wavetable (N = 16)
- Store N values sampled over one cycle
- Phase increment: SI=N f0/fs

- Parameters
- N = 16
- F0 = 220
- Fs = 1kHz
- SI = 16 * 220/1000SI = 3.52

- Increase quality:
- Increase sampling rate
- interpolate

- Quantization:Eg, pure tone F0=440Hz, Fs=8,192Hz
- Truncate N=16
- Truncate N=32
- Truncate N=512

- Interpolation: truncate, mean, linear
- Aliasing

- Truncation (0th level interpolation)

- Rounding (slightly better 0th order)

- Linear (First order interpolation)

- Quadratic (Second order interpolation)

- Cubic (Third order interpolation)

- 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.

- Pure tone F0=440Hz, Fs=8,192Hz
- Truncate N=16
- Truncate N=32
- Truncate N=512

- 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.

- 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)

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

Generated using “perfect” squarewaveform

Generated using a limited Fourier series.

- Simple Oscillator (wavetable)
- Envelope control
- Simple Instrument (Helmholtz)
- Additive Synthesis
- Modulation
- Summary

AMP

FREQ

PHASE

WF

- ADSR Envelope
- Attack
- Decay
- Sustain
- Release

- Important is:
- Duration
- Shape
- Linear
- Exponential
- Other (functional, table)

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

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

- Simple Oscillator (wavetable)
- Envelope control
- Simple Instrument (Helmholtz)
- Additive Synthesis
- Modulation
- Summary

AMP

FREQ

PHASE

WF

- Helmholtz model
- Waveform
- Constant frequency
- Envelope

- Envelope feeds varying amplitude to the oscillator.

AMP

DURATION

ATTACK

DECAY

ASD Envelope

FREQ

AMP

PHASE

- 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

- 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

- Simple Oscillator (wavetable)
- Envelope control
- Simple Instrument (Helmholtz)
- Additive Synthesis
- Modulation
- Summary

AMP

FREQ

PHASE

WF

FREQ

FREQ

FREQ

+

- 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)

- 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.

- Require large amount of data to describe a sound
- Requires very large library of function sets!
- Just too much control?

- Simple Oscillator (wavetable)
- Envelope control
- Simple Instrument (Helmholtz)
- Additive Synthesis
- Modulation
- Summary

AMP

FREQ

PHASE

WF

- 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

- 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

- 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

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

- 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

- Simple Oscillator (wavetable)
- Envelope control
- Simple Instrument (Helmholtz)
- Additive Synthesis
- Modulation
- Summary

AMP

FREQ

PHASE

WF

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

v

fc

fm

AMP

DURATION

ATTACK

DECAY

ASD Envelope

+

fv

AMP

AMP

DURATION

ATTACK

m*AMP

ASD Envelope

fm

AMP

+

fc

AMP