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

3. Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE

4. Simple Oscillator Oscillator 3 strategies 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

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

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

7. 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

8. Wavetable Oscillator Interpolation Truncation (0 th level interpolation)

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

10. Wavetable Oscillator Interpolation (3) Linear (First order interpolation)

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

12. Wavetable Oscillator Interpolation (5) Cubic (Third order interpolation)

13. 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.

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

15. 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.

16. 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)

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

18. Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE

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

20. Linear vs. Exponential Envelope 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) LinearB) Exponential

21. Oscillator as an Envelope Generator Advantages: –wavetable interpolated shape. –Easy encoding of several repetitions. Drawback: –attack and decay times are affected by overall duration! Alternative: –interpolated function generator fc A fm

22. Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE

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

24. 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. AMPFREQ PHASE ASD Envelope AMP ATTACK DURATION DECAY

25. 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! ASD Envelope AMP FREQ PHASE AMP ATTACK DURATION DECAY

26. Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE

27. Types of synthesis Sound Synthesis Additive synthesis Distortion techniques Subtractive synthesis Granular synthesis Analysis based Physical modelling

28. Additive Synthesis FREQ +

29. 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)

30. Additive Synthesis (3) But... –Require large amount of data to describe a sound Each oscillator requires two functions –Functions are only valid for limited range of 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?

31. Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE

32. 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

33. 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 AMP fc m*AMP fm AMP +

34. 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 tremolo m/2*AMP AMP fc-fmfc+fmfc Amplitude Frequency Pure tone fc=220Hz Tremolo fc=220Hz, fm=6Hz, m=1

35. Amplitude Modulation (3)

36. Ring Modulation 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 A fm A/2 fc-fmfc+fmfc Amplitude Frequency fc A fm A *

37. Vibrato Modulation Modulating signal applied to the carrier’s frequency. “Slight wavering of pitch” Pitch varying between fc-v <= fv <= fc+v Average is = fc Eg, fc=220Hz –Pure tone –Vibrato fv=6Hz, v=0.05fc fc fm A + v fv

38. Vibrato Modulation (2)

39. Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE

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

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

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