Acoustics of Music

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Acoustics of Music

• Dr Ian Drumm

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Modulation Synthesis

• Aims
• To introduce modulation as a potential means of
  synthesis
• Learning Outcomes
• AM, FM Synthesis
• Advantages and Disadvantages

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Introduction

• Additive techniques resource heavy (50
  oscillators
• Subtractive no so good at a-periodic timbres
  (bells, percussion, etc)
• If one or more oscillators are used to control
  the amplitude or frequency of another oscillator
  then rich and complex timbres can be achieved
  with far fewer oscillators.

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Amplitude Modulation

• Variant of this technique called Ring
  Modulation often provided as an effect
• Wider applications from broadcasting radio
  signals to the creation of Darlek voices.

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Amplitude modulation

• Essentially a signal called a carrier has its
  amplitude multiplied by another signal called a
  modulator.

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Simple Case

• Bipolar sinusoids i.e. oscillate about y0
• Apply trig identity
• New complex waveform, spectrum has two new
  frequencies called sidebands

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• Example 100Hz carrier X 20Hz modulator gives new
  complex waveform whose spectrum consists of two
  new frequencies at 80Hz and 120Hz
• More generally

.
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What if the modulator has a higher frequency than
the carrier?

• For example
• New waveform shows frequencies (sidebands) at 500
  300Hz
• Negative sign of -300Hz can be thought of as a
  phase shift

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Uni-polar amplitude modulation

• M(t) oscillates about y1
• Lets try this with a 100Hz carrier and 20Hz
  modulator.
• Two side bands as before
• Carrier also in resultant signal at twice
  amplitude
• The result of this is a smoother less distorted
  signal

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Adding greater control to amplitude modulation

• Amplitude and frequency of the carrier and
  modulator doesnt have to be static over time,
  for example
• Control the overall level of the carrier with an
  envelope Ac(t)
• Vary the size of the sidebands with a modulation
  index I(t)

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Frequency Modulation

• Can generate the rich frequency spectrums of most
  real instrument timbres
• Using this frequency modulation technique one can
  in theory produce any audible timbre with only
  six oscillators.
• Effectively invented by John Chowning in the
  seventies
• Taken up in the eighties by Yamaha being the
  basis of the classic DX range of syntheses and
  OPL range of chips that feature in many sound
  cards and mobile devices.

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FM Theory

• Modulator varies the phase of carrier
• Modulation amplitude often written as an index I

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Examples

• For example fc100, fm20, Ac1, I1
• A new complex waveform is created with a whole
  range of frequency components (sidebands) at
  20Hz, 40Hz, 60Hz, 80Hz,100Hz,120Hz, 140Hz and
  180Hz

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• Changing modulation index changes size of
  sidebands
• The modulation index can be given in terms of
  desired frequency deviation from the carrier
  frequency

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A subjectively useful parameter

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FM Bandwidth

• Hence it can also be shown that
• Hence FM Bandwidth is controlled by modulation
  index
• Remember bandwidth was an important subject
  feature of timbre recognition
• Note there is no reason why modulation index
  cant be varied with time, being controlled by an
  envelope or another oscillator.

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Amplitude of FM Sidebands

• It can be shown that
• Amplitude of each sideband whos order is denoted
  by n1,2,3 can be given by the Bessel function
  of first kind
• Bessel function of the first kind is
• Though easiest to find Bessel value from table

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FM Examples

• Good brass sounds fcfm, I varies between 0 and
  7. Apply envelope shape to A and I.
• Crude clarinet fcfm/2, gives odd harmonics
• Bell 5/7 ratio of carrier to modulator frequency
  both decay exponentially Imax 5

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Bell Example

• close all clear all
• sampleRate20000
• tlinspace(0,3,3sampleRate)
• fc200 fm2007/5 Aexp(-0.5t) I5A
• wc2pifc
• wm2pifm
• vA.cos(wctI.cos(wmt))
• sound(v,sampleRate)

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DX7

• 32 Algorithms

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Some extra reading

• Double carrier FM
• Can simulate 2D/3D waveguides and Formants
• Dave Morrills trumpets
• John Chownings singing voice)
• Portamento stops timbre from being too uniform.