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Sound Synthesis

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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  1. Sound Synthesis Part II: Oscillators, Additive Synthesis & Modulation

  2. Plan • Simple Oscillator (wavetable) • Envelope control • Simple Instrument (Helmholtz) • Additive Synthesis • Modulation • Summary AMP FREQ PHASE WF

  3. 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

  4. Wavetable Oscillator • Example of a wavetable (N = 16) • Store N values sampled over one cycle • Phase increment: SI=N f0/fs

  5. Wavetable Oscillator (example) • Parameters • N = 16 • F0 = 220 • Fs = 1kHz • SI = 16 * 220/1000SI = 3.52 • Increase quality: • Increase sampling rate • interpolate

  6. 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

  7. Wavetable OscillatorInterpolation • Truncation (0th level interpolation)

  8. Wavetable Oscillator Interpolation (2) • Rounding (slightly better 0th order)

  9. Wavetable Oscillator Interpolation (3) • Linear (First order interpolation)

  10. Wavetable Oscillator – Interpolation (4) • Quadratic (Second order interpolation)

  11. Wavetable Oscillator Interpolation (5) • Cubic (Third order interpolation)

  12. 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.

  13. Wavetable Oscillator Interpolation (7) • Pure tone F0=440Hz, Fs=8,192Hz • Truncate N=16 • Truncate N=32 • Truncate N=512

  14. 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.

  15. 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)

  16. Aliasing (3) • Square wave, 563 Hz fundamental, 48kHz sampling rate. Generated using “perfect” square waveform Generated using a limited Fourier series.

  17. Plan • Simple Oscillator (wavetable) • Envelope control • Simple Instrument (Helmholtz) • Additive Synthesis • Modulation • Summary AMP FREQ PHASE WF

  18. Time Envelope (1) • ADSR Envelope • Attack • Decay • Sustain • Release • Important is: • Duration • Shape • Linear • Exponential • Other (functional, table)

  19. 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

  20. 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

  21. Plan • Simple Oscillator (wavetable) • Envelope control • Simple Instrument (Helmholtz) • Additive Synthesis • Modulation • Summary AMP FREQ PHASE WF

  22. Simple Instrument • Helmholtz model • Waveform • Constant frequency • Envelope • Envelope feeds varying amplitude to the oscillator. AMP DURATION ATTACK DECAY ASD Envelope FREQ AMP PHASE

  23. 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

  24. 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

  25. Plan • Simple Oscillator (wavetable) • Envelope control • Simple Instrument (Helmholtz) • Additive Synthesis • Modulation • Summary AMP FREQ PHASE WF

  26. Types of synthesis

  27. Additive Synthesis FREQ FREQ FREQ +

  28. 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)

  29. 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?

  30. Plan • Simple Oscillator (wavetable) • Envelope control • Simple Instrument (Helmholtz) • Additive Synthesis • Modulation • Summary AMP FREQ PHASE WF

  31. 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

  32. 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

  33. 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

  34. Amplitude Modulation (3)

  35. 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

  36. 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

  37. Vibrato Modulation (2)

  38. Plan • Simple Oscillator (wavetable) • Envelope control • Simple Instrument (Helmholtz) • Additive Synthesis • Modulation • Summary AMP FREQ PHASE WF

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

  40. v fc fm AMP DURATION ATTACK DECAY ASD Envelope + fv AMP

  41. AMP DURATION ATTACK m*AMP ASD Envelope fm AMP + fc AMP

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