of Physics Music Lecture 4: Analysis and Synthesis of Tones
Berg & Stork, Chapter 4 Fourier Analysis Pure Tones Noise
Frequency and Intensity scales
Frequency:
Each Octave is a factor of two change in Frequency (Pythagorous) Musical Octave sounds in unison 3 dB (deci-Bel) is a factor of 2 change in Intensity: 3.01=10.*log(2)
Intensity:
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Joseph Fourier (17681830)
(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
W av e
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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1.5
1
0.5
Am plitude
0 0 -0.5 2 4 6 8 10 12
-1
-1.5
-2
-2.5
Tim e
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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 e.g. 4096 frequency samples, Frequency resolution is 22050 Hz / 4096 = 5.4 Hz Lowest possible frequency discernible = 1/T A bassoon can not be played as fast as a piccolo
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CoolEdit2000 Fast Fourier Transform (FFT)
Sound fragment of length T:
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Fourier Synthesis: Mathematical functions
Square wave (frequency f) B&S Fig 4-5:
Frequencies f, 3f, 5f, Amplitudes 1, 1/3, 1/5, Frequencies f, 2f, 3f, 4f, Amplitudes 1, , 1/3, , 1/5, Frequencies f, 2f, 3f, 4f, 5f, Amplitudes 1, 1, 1, 1, 1,
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Sawtooth (ramp) B&S Fig 4-6:
Repetitive Pulse B&S Fig. 4-7:
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Triangle Wave
Sum of Three Harmonics
1.5
1
Wave Amplitude
0.5
0 0 -0.5 3.14 6.28
N=1 N=3 N=5 Sum
-1
-1.5
Time
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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) 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 1284 Hz, Hz
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Expect odd harmonics at
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Non Periodic Sound Sources
Real music (vs. synthesized sound)
Vibrato, tremolo, attack, decay, etc. Inharmonicities Air Flow (flute) Instrument noises (keys, valves) Articulation transients Sung Consonants Percussion
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Noise
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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 2:10
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Timbre / Tone Quality
Attack / Decay Transients Spectrum of driving force
Reed, Lips, Bowing, Hammer, Plucking Clarinet has some even harmonics (slightly conical
bore)
Harmonic response of each instrument
Sounding box/board of String instruments Formants (especially human voice) Finite string size stretches Piano octaves. Wind harmonics not necessarily pure integers
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Inharmonicities:
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Resonances in open and closed tubes
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Musical Instrument
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