Physics of Music Lecture 4: Analysis and Synthesis of Tones

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Physics of MusicLecture 4 Analysis and
Synthesis of Tones

• Berg Stork, Chapter 4
• Fourier Analysis
• Pure Tones
• Noise

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Frequency and Intensity scales

• Frequency
• Each Octave is a factor of two change in
  Frequency (Pythagorous)
• Musical Octave sounds in unison
• Intensity
• 3 dB (deci-Bel) is a factor of 2 change in
  Intensity 3.0110.log(2)

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Joseph Fourier (1768-1830)

• (1807) Any periodic wave of frequency f can be
  constructed as a superposition of standard
  (particularly sine cosine) waves of frequency
  f, 2f, 3f, 4f,
• Any non-periodic wave can be constructed as a
  superposition of standard waves of all
  frequencies.

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Wavelets

• A new formalism for describing signals that are
  simultaneously localized in frequency and time
• See also Spectral View mode of CoolEdit.

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Digital Sampling

• CD sampling is 44100 Samples/sec
• Highest possible frequency is 22050 Hz
• Nyquist condition sampling at 44100 /sec gives
  same output for 22050 Hz, 66150 Hz
• CoolEdit2000 Fast Fourier Transform (FFT)
• e.g. 4096 frequency samples, Frequency resolution
  is 22050 Hz / 4096 5.4 Hz
• Sound fragment of length T
• Lowest possible frequency discernible 1/T
• A bassoon can not be played as fast as a piccolo

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Fourier Synthesis Mathematical functions

• Square wave (frequency f) BS Fig 4-5
• Frequencies f, 3f, 5f,
• Amplitudes 1, 1/3, 1/5,
• Sawtooth (ramp) BS Fig 4-6
• Frequencies f, 2f, 3f, 4f,
• Amplitudes 1, ½, 1/3, ¼, 1/5,
• Repetitive Pulse BS Fig. 4-7
• Frequencies f, 2f, 3f, 4f, 5f,
• Amplitudes 1, 1, 1, 1, 1,

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Triangle Wave
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Square Wave (Synthesized)
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Soprano Recorder, C above Middle C
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PanPipes Noisy Triangle Wave
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Chalumeau.wav

• Fundamental at 155.7 Hz D
• (I was trying to build D)
• Expect odd harmonics at
• 3x155.7 Hz 467 Hz (A)
• 5x155.7 Hz 778.5 Hz (G)
• 7x155.7 Hz 1090 Hz
• 9x155.7 Hz 1401 Hz
• Observed frequencies
• 467 Hz, 768 Hz, 1041 Hz, 1284 Hz

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Non Periodic Sound Sources

• Real music (vs. synthesized sound)
• Vibrato, tremolo, attack, decay, etc.
• Inharmonicities
• Noise
• Air Flow (flute)
• Instrument noises (keys, valves)
• Articulation transients
• Sung Consonants
• Percussion

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White Noise Pink Noise(see Noise.wav)

• White Noise Equal (random) intensity at all
  frequencies
• Pink Noise Equal acoustic power in each octave.
  Intensity diminishes 3dB with each octave.
• Integrated acoustic energy intensity times
  bandwidth. Each octave is twice as large in
  frequency range as octave below.

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Sample of White Pink Noise
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Frequency Spectra White and Pink (1/f) Noise
White Noise
Pink Noise
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White Noise, Normal Distribution

• White noise has a well defined mean and standard
  deviation.
• S.d. square root of variance
• Variance mean of the square of the deviations
  from the mean
• Precision of measurement of mean improves with 1
  over square root of number of samples.
• Public opinion polls have a precision of 3 with
  just 1000 people questioned
• 0.03 1/square root (1000)

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Pink Noise, 1/f-Noise

• Pink Noise does not have a well defined mean or
  variance.
• If weather follows a Pink Noise distribution,
  then there is no such thing as climate.
• 100-yr flood is a flood expected once in 100
  years, and 10 x worse than the average flood in
  any 10 year period.

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Frequency Spectrum of Symphonic Music (Pink
Noise?)

• A. Dvorak, Concerto for Cello Orchestra
• 1st 210

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Timbre / Tone Quality

• Attack / Decay Transients
• Spectrum of driving force
• Reed, Lips, Bowing, Hammer, Plucking
• Harmonic response of each instrument
• Clarinet has some even harmonics (slightly
  conical bore)
• Sounding box/board of String instruments
• Formants (especially human voice)
• Inharmonicities
• Finite string size stretches Piano octaves.
• Wind harmonics not necessarily pure integers

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Resonancesin open and closed tubes
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Musical Instrument