Acoustics of Music

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Acoustics of Music

• Dr Ian Drumm

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Digital Techniques

• Aims
• An introduction to Granular Synthesis
• Learning Outcomes
• Granular Synthesis
• Grains as elements of sound
• Control parameters
• Applications
• Wave Shaping
• Shaping or transfer function
• Chebychev Polynomials

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Granular SynthesisBasic Concept

• At tlt20ms we would not perceive successive tone
  bursts as single events
• Idea suggests sound composed of successive
  elements of sound grains (not physically
  correct)
• A grain is in effect a shaped tone bust
• More complex sounds from lots and lots of grains

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Grains Amplitude Envelope

• Attack, Sustain and Decay sections
• Three part linear
• Gaussian
• Pulse sound like wood block taps
• Narrow envelopes crackling and popping

s is standard deviation µ is mean
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Implementation

• Waveform generator X Envelope generator
• Control on a grain by grain basis makes this
  technique powerful

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Control Data

• Method comes into its own if grains vary on a
  grain by grain basis
• Individual grain parameters we might want to vary
  include
• duration
• envelope type
• amplitude
• frequency
• waveform type
• If the waveform is synthetic you will need at
  least an extra parameter for index into a look
  table
• If the waveform is a sample you might want to
  specify start position in the sample and the
  wavefile it is taken from
• High grain densities give an explosion in the
  number of control parameters required

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Sophisticated implementation

• Need automatic high level scheme
• Specify a few global parameters
• e.g. change of spectrum with time hence system
  takes care of generating all the different grains
  for us
• Several such high level schemes suggested tend to
  allow experiment/artistic control rather that
  analytical/scientific control

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Example of a high level scheme

• AGS (Asynchronous granular synthesis)
• Sound produced described by the pattern of
  granules on the time/frequency plain.
• Patterns described as clouds.
• (cumulus, stratus, etc).
• Cloud parameters include
• Start time and duration
• Bandwidth
• Grain Density
• Amplitude envelope
• Spatial distribution of grains within the cloud

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Cloud Shapes
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Macroscopically varying grain parameters within
the cloud

• Grain duration shorter grain durations have
  higher bandwidths resultant sonic textures
  perceived as crackling long grain durations
  give smoother sounds. An AGS system will let you
  vary durations via mathematical functions that
  can be dependant on random numbers or time,
  frequency, etc.
• Grain waveform waveforms that dont change can
  be thought of as monochrome. Transchrome and
  polychrome variations in cloud give richer and
  dynamic sounds.

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AGS Control Parameters
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Advantages / Disadvantages

• Basic concept attractive (but not quite correct)
• Not suited to having an analysis stage (because
  not generally true acoustic theory)
• Therefore not a good emulative technique for
  classic musical timbres
• Excels at some natural sounds difficult to
  produce with other methods (e.g. crackling fire,
  water gurgling, wind gusts, explosions).

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Wave Shaping

• Pass signal trough a distortion box
• Stored table in computer memory
• Simple case
• X is input signal
• X used to index into table W i.e. W(x)
• Table W is shaping or transfer function

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Simplest Example

• N.B. amplitude envelope to control degree of
  distortion

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Wave Shaping as emulative technique

• Many instruments have amplitude dependant timbres
• bandwidth of trumpet increases as sound builds up
• bandwidth of plucked string decrease as sound
  decays
• Pluck, strike, bow or blow instrument harder
  gives richer spectrum
• Wave Shaping gives similar result the greater
  the signal amplitude the richer the spectrum

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How to obtain desired spectrum

• Consider an input function
• Clearly a signal of twice the frequency would be
• Also using basic trig identities we know given
  the input amplitude is 1.
• Hence if we put into transfer function
• then the resulting output corresponds to a
  second harmonic.

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Chebychev Functions

• Can predict result of wave shaping for a cosine
  that varies between -1 and 1
• By applying the kth Chebychev function Tk we get
  cosine at kth harmonic if x/-1
• For example this will give three harmonics
• If we put result in transfer function

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• Chebychev Polynomials