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 Lin 1

Computer Music Synthesis

Chapter 6Based on “Excerpt from Designing Sound”

by Andy FarnellSlides by Denny Lin


Shaping sound

• 6.1 Amplitude dependent signal shaping

• 6.2 Periodic functions

• 6.3 Other functions

• 6.4 Time dependent signal shaping


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


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


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


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


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


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


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~ -0.5 0]• 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


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


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


Curved envelopes

• Rising or falling control signals can be used to produce curves:– Linear– Squared (2nd power)– Quartic (4th power)

• All curves take the same amount of time to reach 0

• More squaring operations causes faster initial signal decay


6.2 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


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


Getting Cosine Wave from PhasorPhasor Cosine

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


6.3 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


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


6.4 Time dependent signal shaping

• Delay

• Phase cancellation

• Filters– User friendly filters– Integration– Differentiation


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


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


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


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


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


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


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


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

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

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