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Sound Synthesis Part II: Oscillators, Additive Synthesis & Modulation

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

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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

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

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

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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

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Wavetable Oscillator Interpolation Truncation (0 th level interpolation)

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

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

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

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

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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.

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

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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.

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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)

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

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

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

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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

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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

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

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

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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

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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

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

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

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Additive Synthesis FREQ +

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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)

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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?

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

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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

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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 +

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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

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Amplitude Modulation (3)

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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 *

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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

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Vibrato Modulation (2)

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

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

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fc fm + v fv ASD Envelope AMP ATTACK DURATION DECAY

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

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