Title: Physics of Music Lecture 4: Analysis and Synthesis of Tones
1Physics of MusicLecture 4 Analysis and
Synthesis of Tones
- Berg Stork, Chapter 4
- Fourier Analysis
- Pure Tones
- Noise
2Frequency 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)
3Joseph 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.
4Digital Sampling
- CD sampling is 44100 Samples/sec
- Highest possible frequency is 22050 Hz
- CoolEdit2000 Fast Fourier Transform (FFT)
- e.g. 4096 samples, Frequency resolution is 44100
Hz / 4096 10.8 Hz
5Fourier 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,
6Triangle Wave
7Square Wave (Synthesized)
8Soprano Recorder, C above Middle C
9PanPipes Noisy Triangle Wave
10Chalumeau
- 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
11Non 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
12White 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.
13Sample of White Pink Noise
14Frequency Spectra White and Pink (1/f) Noise
White Noise
Pink Noise
15White 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)
16Pink 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.
17Frequency Spectrum of Symphonic Music (Pink
Noise?)
- A. Dvorak, Concerto for Cello Orchestra
- 1st 210
18Timbre / 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
19Resonancesin open and closed tubes
20Musical Instrument