Copyright © 2011 by Denny Lin1 Computer Music Synthesis Chapter 6 Based on “Excerpt from Designing Sound” by Andy Farnell Slides by Denny Lin

Copyright © 2011 by Denny Lin2 Shaping sound 6.1 Amplitude dependent signal shaping 6.2 Periodic functions 6.3 Other functions 6.4 Time dependent signal shaping

Copyright © 2011 by Denny Lin3 6.1 Amplitude dependent signal shaping Arithmetic is used to scale, shift, and invert signals Addition and multiplication are commutative Subtraction and division are not commutative It is better to multiply by decimal fractions and reserve divide for processing variable signals

Copyright © 2011 by Denny Lin4 Scaling and Shifting a signal Scaling a signal: –Multiply a signal by a fixed amount –Typically used to control amplitude (volume) Shifting a signal: –Add to a signal by a fixed amount –Moves signal up or down to correct its swing

Copyright © 2011 by Denny Lin5 Inverting and Complementing a Signal Inverting a signal: –Multiply by -1 –Changes phase by 180 degrees (  ) Complementing a signal: –Given signal a, its complement is 1 – a –Has same direction as signal inverse, but its signal polarity is retained, and defined between 0.0 and 1.0

Copyright © 2011 by Denny Lin6 Signal reciprocal Given a signal a, its reciprocal is 1/a When signal a is very large, its complement 1/a is close to 0 When signal a is close to 0, its complement 1/a is very large

Copyright © 2011 by Denny Lin7 Limiting a signal Use the max~ object to specify a minimum possible value –max~ 0 specifies lower limit of 0 Use the min~ object to specify a maximum possible value –min~ 0 specifies upper limit of 0

Copyright © 2011 by Denny Lin8 Wave Shaping and Clipping A phasor can be shaped into any other waveform A cosine waveform can be easily shaped into a square wave Use the clip~ object to limit the output within a specified range

Copyright © 2011 by Denny Lin9 Generating triangle waves (I) Use phasor~ to generate signal Shift down by 0.5: [-~ 0.5] Keep only signal between -0.5 to 0: [clip~ ] Invert signal and double the amplitude: [*~ 2] Add left (graph B) and right (graph C) branches: [+~] Re-center: [-~ 0.25] and normalize: [*~ 4] triangle waves

Copyright © 2011 by Denny Lin10 Generating triangle waves (II) Use phasor~ to generate signal Get complement of signal by inverting: [*~ -1] and shifting up: [+~ 1] signal Take the minima of the two signals (graph A) and graph B): [min~] Re-center: [-~ 0.25] and normalize: [*~ 4] triangle waves

Copyright © 2011 by Denny Lin11 Squaring and roots Multiplying a signal by itself is equivalent to squaring a signal –Amplitude scaled according to its original size –Output is always positive Use the sqrt~ object to find the square root of a signal

Copyright © 2011 by Denny Lin12 Curved envelopes Rising or falling control signals can be used to produce curves: –Linear –Squared (2 nd power) –Quartic (4 th power) All curves take the same amount of time to reach 0 More squaring operations causes faster initial signal decay

Copyright © 2011 by Denny Lin Periodic functions Wrapping ranges –Doubling the amplitude of phasor and wrapping the signal, in turn doubles the frequency Can obtain an exact number of phasor cycles from a line (see fig. 6.13) –Specify a slope using the vline~ object –Output from vline~ is a 1/1ms = 1000Hz signal –Multiply by 2 and wrapping the signal produces a periodic 2000Hz signal

Copyright © 2011 by Denny Lin14 Cosine function A cosine oscillator can be derived from a phasor~ object The phasor~ object always produces a uni-polar signal in the range 0.0 to 1.0 The cos~ object produces a bipolar waveform in the range -1.0 to 1.0

Copyright © 2011 by Denny Lin15 Getting Cosine Wave from Phasor PhasorCosine 0.0 (x 360° = 0°) Top of cycle 0.25 (x 360° = 90°) Crosses zero going down 0.5 (x 360° = 180°) Bottom of cycle 0.75 (x 360° = 270°) Crosses zero going up 1.0 (x 360° = 360°) At original position

Copyright © 2011 by Denny Lin Other functions Polynomials are expressed as sums of different power terms Resulting slopes can be useful for creating envelopes Best to start with a polynomial that has a known shape; get new coefficients

Copyright © 2011 by Denny Lin17 Expressions Used to create objects that expresses how a signal is processed Signal inlets 1, 2, and 3, are sent to variables $V1, $V2, and $V3 Less efficient than built-in objects, and more difficult to read

Copyright © 2011 by Denny Lin Time dependent signal shaping Delay Phase cancellation Filters –User friendly filters –Integration –Differentiation

Copyright © 2011 by Denny Lin19 Delay Shifts a signal in time, used for effects such as reverb and chorus The delwrite~ and delread~ objects should be used as a pair

Copyright © 2011 by Denny Lin20 Phase cancellation Can use a delay to create a signal that is 180 degrees out of phase (anti-phase signal) with respect to the original signal Mixing the original with the anti- phase signals, causes phase cancellation, so output is 0 When the two signals are in phase, the output is 2 times the original signal Output amplitude may depend on the delay (given a fixed frequency), or the frequency (given a fixed delay) Controlling this effect is equivalent to filtering the signal

Copyright © 2011 by Denny Lin21 Filters When the amplitude is reinforced by the coincidence of signal delay and period, a pole is formed When the delay time is half the period causing phase cancellation, a zero is formed Filters can be created by controlling which frequencies are amplified and which are cancelled

Copyright © 2011 by Denny Lin22 The rpole~ object Low-pass filters an audio signal fed to its left inlet; can act as an integrator Recursive filter The first argument or audio signal fed to its right inlet, defines the real-valued filter co-efficient a[n] in: y[n] = y[n-1] + a[n] * x[n] –where y[n] is the output –x[n] is the input –Filter is not stable when |a[n]| > 1

Copyright © 2011 by Denny Lin23 The rzero~ object High-pass filters an audio signal fed to its left inlet; can act as a differentiator Non-recursive filter The first argument or audio signal fed to its right inlet, defines the real-valued filter co-efficient a[n] in: y[n] = x[n] - a[n] * x[n-1] –where y[n] is the output –x[n] is the input –Filter is always stable

Copyright © 2011 by Denny Lin24 User-Friendly Filters Common filters are the low-pass, high-pass, band-pass, and band- reject filters: –Low-pass: allows low frequencies to pass –High-pass: allows high frequencies to pass –Band-pass: allows frequencies within a range to pass –Band-reject: reject frequencies within a range

Copyright © 2011 by Denny Lin25 Integration Integration computes the area under a curve, and can be used to shape a waveform Use the rpole~ filter to perform signal integration Integrating a square wave produces a triangle wave A filter can be seen as the most fundamental signal generator

Copyright © 2011 by Denny Lin26 Differentiation Differentiation is the inverse of integration Calculates the instantaneous slope of a signal Use the rzero~ filter to perform signal differentiation Differentiating a phasor produces an impulse spike