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Sound Synthesis Part III: Distortion-based synthesis (FM & Waveshaping)

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Plan Distortion-based synthesis –FM synthesis –Waveshaping Matlab code for synthesis Example of synthesized composition (Risset) Summary

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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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Distortion (aka nonlinear) Synthesis Additive synthesis is powerful & flexible BUT requires many simple oscillators. Plethoric and non-intuitive controls on the sound spectrum. Using fewer oscillators, with more components. Complex spectra can be obtained by distorting simple oscillators.

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Plan Distortion-based synthesis –FM synthesis –Waveshaping Substractive synthesis Summary

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FM Synthesis Pioneered by John Chowning (1973) “Alteration of distortion of the frequency of an oscillator in accordance with the amplitude of a modulating signal.” Similar to Vibrato modulation, but using audible modulating frequency and larger width. Generate a large number of timbres easily controlled.

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FM Synthesis (2) Modulated frequency in fc-d <= fv <= fc+d Peak frequency deviation d (in Hz) Vibrato: small d & fm FM Synth: larger d & fm Note: if d>fc, then fv can be < 0 (negative phase applied!) Rich spectrum from sine waves only! fc fm A + d fv Modulating oscillator Carrier oscillator

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FM Synthesis (3) Spectrum: fc + k.fm, for k an integer. Sidebands increase with d Modulation index I=d/fm Spectrum amplitude: –Fc+k.fm Jk(I) –where Jk is the kth Bessel function –Sidebands significant ~ for |k|<I+1 (rule of thumb) Bessel Functions J1(X), J2(X) and J3(X) fc-fm fc+fm fc Magnitude Frequency fc+2.fm fc-2.fmfc-3.fm fc+3.fm

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FM Synthesis (4) Bessel functions can be negative! Odd number, low- frequency components have opposite sign J -1 (I) = -J 1 (I) Negative spectrum represents out-of-phase components. Combination of in- phase and out-of-phase components of same amplitude cancels out. fc-fm fc+fm fc magnitude fc+2.fm fc-2.fmfc-3.fm fc+3.fm Bessel Functions J1(X), J2(X) and J3(X)

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FM Synthesis: Dynamic Spectrum Dynamic spectrum can be obtained by applying envelopes to both carrier and modulating signals. I is the max value of the modulation index More limited than additive synthesis, but requires less components. I*fm fm + ASD Envelope fc A fo

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FM Synthesis: Dynamic Spectrum TIME FREQUENCY

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FM Synthesis: Instrument Design Envelopes functions F1 & F2 change the timbre of the sound (John Chowning, 1973) A) Bell-like timbre B) Wood drum sound C) Brass-like timbre fm + fc A fo F1 F2 1/d fm*I 1/d (John Chowning, 1973)

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FM Synthesis: Instrument Design (2) A) Bell-like timbre –D = 15s –Fc = 200Hz –Fm = 280Hz –I = 10 fm + fc A fo F1 F2 1/d fm*I 1/d (John Chowning, 1973) 1 d F1 = F2

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FM Synthesis: Instrument A’s Spectrogram (bell)

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FM Synthesis: Instrument Design (3) B) Wood drum sound –D=0.2 s –Fc=80Hz –Fm=55Hz –I=25 fm + fc A fo F1 F2 1/d fm*I 1/d (John Chowning, 1973) 1 d F1 1 d F2

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FM Synthesis: Instrument B’s Spectrogram (drum)

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FM Synthesis: Instrument Design (4) C) Brass-like timbre –D=0.6 –fc=440Hz –Fm=440Hz –I=5 (or 3) fm + fc A fo F1 F2 1/d fm*I 1/d (John Chowning, 1973) 1 d F1 = F2

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FM Synthesis: Instrument C’s Spectrogram (brass)

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FM Synthesis Instrument Design (5) D) Clarinet-like timbre –d = 0.5 s. –Fc = 900 Hz –Fm = 600 Hz –I = 2 –J = 4 fm + fc A fo F1F2 1/d fm*(I-J) 1/d (John Chowning, 1973) + J*fm 1 d F1 1 d F2

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FM Synthesis: Instrument D’s Spectrogram (clarinet)

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FM Synthesis: Instruments Summary A) BELL-LIKEB) WOODEN DRUM C) BRASS-LIKED) CLARINET

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Plan Distortion-based synthesis –FM synthesis –Waveshaping Matlab code for synthesis Example of synthesized composition (Risset) Summary

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Waveshaping (1): Definition Idea: “realizes spectra that have dynamic evolution of their components.” More computationally efficient than additive (like FM) Allows generation of band-limited spectrum with a set max harmonic number (unlike FM) f0 A WS waveshaper

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Waveshaping (2): Transfer Function Waveshaper: nonlinear processor that alters the shape of the waveform passing through. Defined by a transfer function: ys=f(y0) The shape of the output waveform (and spectrum) changes with amplitude of the input signal! INPUT OUTPUT TRANSFER FUNCTION RESULT CARRIER

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Transfer function Low amplitude ~ linear response High amplitude Nonlinear response

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Waveshaping (3): Transfer Function INPUT OUTPUT TRANSFER FUNCTION RESULT CARRIER

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Waveshaping (4): Some Transfer Functions in out in out in out A) LinearB) Odd C) Even in out D) Discontinuous in out E) Sharp points in out F) Ripple

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Waveshaping: Linear Transfer Functions Eg, f(x) = 0.5x Reduces amplitude by half. No distortion, no harmonic added. in out f(x) = 0.5*x

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Waveshaping: Odd Transfer Functions Definition: f(-x) = -f(x) Eg, f(x) = x^3 Nearly linear for low amplitude ( low distortion) Highly non-linear for high amplitude ( high distortion) Generates only odd- numbered harmonics! in out f(x) = x^3

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Waveshaping: Even Transfer Functions Definition: f(-x) = f(x) Eg. f(x) = 2*x^2-3*x^4 Creates only even harmonics Even transfer function doubles the fundamental frequency raises the pitch by one octave! in out f(x) = 2*x^2-3*x^4

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Waveshaping: Other Transfer Functions Discontinuities and sharp points will create more harmonics than a smooth function Generally causes aliasing! For this reason, it is common to use polynomial transfer functions f(x) = d0+d1*x+d2*x^2+......+dn*x^n in out D) Discontinuous in out E) Sharp points

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Waveshaping: Polynomial Transfer Functions Divh0h1h2h3h4h5h6h7h8h9 x^00.51 x^111 x^2221 x^3431 x^48641 x^5161051 x^632201561 x^764352171 x^812870562881 See Dodge & Jerse Eg, for a pure tone of amplitude 1, f(x)=x^5 will generate? What about f(x) = 2*x^2 – 3*x^4 ?

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Waveshaping: Polynomial Transfer Functions The transfer function: f(x) = 2*x^2 – 3*x^4 applied to a pure tone (sinusoidal) carrier of amplitude 1, will generate the following harmonics: –H0 = 2*2/2 - 3*6/8 –H2 = 2*1/2 – 3*4/8 –H4 = -3*1/8

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Waveshaping: Spectral matching For a pure tone cosine carrier of unit amplitude (A=1) – also called distortion index, Applying a transfer function based on a Chebyshev polynomial T k (x) will generate only the kth harmonic General form for Chebyshev polynomials: T k+1 (x) = 2xT k (x)-T k-1 (x)

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Waveshaping: Chebyshev Polynomials T 0 (x) = 1 T 1 (x) = x T 2 (x) = 2x^2-1 T 3 (x) = 4x^3-3x T 4 (x) = 8x^4-8x^2+1 T 5 (x) = 16x^5-20x^3+5x T 6 (x) = 32x^6-48x^4+18x^2-1 Etc.

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Waveshaping: Using Chebyshev Polynomials For example, to generate only 1 st, 2 nd, 4 th & 5 th harmonics with amplitudes 5, 1, 4, and 3, respectively. We can design the transfer function: F(x) = 5*T 1 (x)+1*T 2 (x)+4*T 4 (x)+3*T 5 (x)

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Waveshaping: Instrument Design Instrument for clarinet-like timbre From Jean-Claude Risset’s Introductory Catalogue of Computer Synthesized Sounds. fm + ASD Envelope 256 F(x) * A 255 0.64 0.085

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Waveshaping: Instrument Transfer Function

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Waveshaping: Instrument Spectrogram

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Waveshaping: Amplitude Scaling Distortion index A affects sound’s amplitude, and there by loudness. Similar to acoustic instruments: louder play usually generates more overtones. Excessive loudness variation excessive bright tones. Solution: apply a scaling function to the amplitude R(A,x) = F(Ax)S(A) f0 A F(x)S(A) *

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Plan Distortion-based synthesis –FM synthesis –Waveshaping Matlab code for synthesis Example of synthesized composition (Risset) Summary

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Matlab Code Examples for all sounds are available here: http://info.ee.surrey.ac.uk/Personal/N.Pugeau lt/MediacastingL3_2012/http://info.ee.surrey.ac.uk/Personal/N.Pugeau lt/MediacastingL3_2012/ Parameters can be tweaked to evaluate their influence. Matlab is convenient for plotting results > spectrogram( y, ntime, noverlap, nfft, fs ) > spectrogram( y, 256, 250, 250, fs ) Can play sounds using the command: > sound( y, fs )

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Plan Distortion-based synthesis –FM synthesis –Waveshaping Matlab code for synthesis Example of synthesized composition (Risset) Summary

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Plan Distortion-based synthesis –FM synthesis –Waveshaping Matlab code for synthesis Example of synthesized composition (Risset) Summary

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Additional Reading C. Dodge, C., & Jerse, T. A. (1997). Computer Music: Synthesis, Composition, and Performance. Schrimer, UK. (see chapters 5 and 6) Matlab examples for all sounds & instruments are available here: http://info.ee.surrey.ac.uk/Personal/N.Pug eault/MediacastingL3_2012/http://info.ee.surrey.ac.uk/Personal/N.Pug eault/MediacastingL3_2012/

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