Transcript
PHYS207: Physics of MusicVladimir (Vladi) Chaloupka
Professor of PhysicsAdjunct Professor, School of Music
Affiliate, Virginia Merrill Bloedel Hearing Research CenterAffiliate faculty, DXARTS
Adjunct Professor, Henry M. Jackson School of International Studies
A coherent(?) synthesis:
Music, Science and Human Affairs,
With Exuberance and Humility
Physics of Music => Physics AND Music
3 0-A-EEP 1 14 envir+cep 1 1 23 0-C-ART 1 1 24 0-C-ASTR 2 24 0-C-BIOCHM 2 24 C sci_cmp E 1 3 42 0-C-DXARTS 1 2 34 Econ+BA 1 1 22 JSIS+pol.sci. 1 1 2 1 52 German+class. 1 1 24 0-C-HIST 1 14 0-C-MATH 2 22 0-C-MUSIC 1 3 42 0-C-PHYS 1 4 2 1 81 0-C-PREART 1 1 3 51 0-C-PREHUM 2 2 41 0-C-PREMAJ 4 1 51 0-C-PRESCI 4 1 1 61 0-C-PSOCS 1 5 2 81 0-D-ESS 1 13 0-J-E E 1 1 21 0-J-PRENGR 3 5 1 9
8018 21 19 21 1 80
PHYS 207 enrollment 2010
Preview of the Syllabus
• And the square feet per person:
536 square feet
vs.
(536 feet) squared
=> An important goal of the course: improve (or install) your BS detector
What is Music?• Simplest definition: music is organized sound,
with the main ingredients:
• Rhythm
• Melody
• Harmony
• Counterpoint: unity of melody / harmony / rhythm
• And (at least with Bach): unity of reason and emotions
Music and Science, with Exuberance and Humility
• Pythagoras’ integers• Kepler’s Harmonia Mundi• Superstring Theory: all elementary particles as
“modes of vibration” of the same string
(ergo: “Princeton String quartet”)• Laser Interferometer Space Antenna: “listening to the
gravitational Symphony of the Universe”• Music as an example of emergent complexity: parts of
Art of Fugue “sound like parts of the Mandelbrot set”• Goedel Escher Bach• Exuberance and Humility: Two Pipe Organs
J.S.Bach as Amadeus
• The central Theme of Amadeus (play/movie) applied to Bach
• The Bach genetic phenomenon
• Bach myths: BACH = 14 JSBACH = 41
even (from a doctoral Thesis [sic]):
“the Unfinished fugue breaks off at bar 239 because 2+3+9 = 14” !
Number of (male) Bach’s doing music at any particular year
… finally I realized that to me, Goedel and Escher and Bach were only shadows cast in different directions by some central solid essence.
Douglas Hofstadter
Goedel Escher Bach Hofstadter
• A musico-logical fugue in English
• Goedel Undecidability Theorem:
“In every sufficiently powerful formal system, there are propositions which are true, but not provable within the system”
• (i.e. “Truth if more than Provability”)
• Relief provided by fanciful Dialogues
Hofstadter’s GEB Dialogues(in the spirit of Lewis Carroll)
• …..• Meaning and Form in Mathematics• Sonata for Unaccompanied Achilles• Figure and Ground• Chromatic Fantasy, and Feud• Brains and Thoughts• English French German Suite• Minds and Thoughts• …..
Mandelbrot Set Tour (optional)
• 1) z(0) = 0
• 2) z(n+1) = z(n)^2 + c and back to 2)
• 3) if z(n) finite then c belongs to the set
Amazingly, this simplest of algorithms results into an object of infinite complexity (and arresting beauty). One cannot but recall Dirac’s claim that the Quantum Electrodynamics explains “most of Physics and all of Chemistry” … Also: the varied copies of Mandelbrot “body” are reminiscent of various versions of Art of Fugue theme, and the filaments are like the secondary motifs …
Einstein as Scientist, Musician and Prophet
• Einstein as scientist: Recently we celebrated the Centenary of Einstein’s Annus Mirabilis
• Einstein as musician: from a review: “Einstein plays excellently. However, his world-wide fame is undeserved. There are many violinists who are just as good.”
• Einstein as prophet: “Nuclear weapons changed everything except our way of thinking.”
• Farinelli: who was mobbed in 18th Century ?
Exuberance and Humility in Music and Science
Left: The pipe organ at the St. Marks Cathedral in Seattle
Above: the 1743(Bach was just composing the Art of Fugue then!) instrument at the College of William and Mary in Williamsburg.
What is Physics of Music(Musical Acoustics)
• Investigation of the relationships between the perceptual and physical attributes of musical sound
• Basic correlations:
• Loudness <-> Intensity
• Pitch <-> Frequency
• Timbre (“color”) <-> Spectrum
Examples of more complex questions:
“What is the role of imperfections
in creating the perception of perfection?”
“What is the role of sound in Music?”
Resonance:
Example: a mass on a spring
Great playground for Elementary Physics:
Newton: F = ma Restoring force of a spring: F = -kx
Equation of motion: ma = -kx
Energy: Kinetic = ½ mv2
Potential: 1/2kx2
Total = constant (“conserved”)
After calculations (not really difficult):
resonance at f = (1/2pi) sqrt(k/m)
Resonance corresponds to a peak in the response of the system to a periodic stimulus at a given frequency
More complicated systems have more than one resonant frequency; each of them corresponds to a mode of vibration; each mode is characterized by its nodes
Practical examples:Child on a swingCar stuck in snowTacoma Narrows bridge collapseVibration of the violin string …..
Waves:
Waves are disturbances propagating in space
[ ] Mechanical / electromagnetic / gravitational / quantum /.
[ ] Longitudinal / transverse
[ ] 1d / 2d / 3d / …
NB: consequence for intensity = f(distance)
[ ] traveling wave
“standing wave” = a mode of vibration
= a superposition of traveling waves
[ ] reflection off the – fixed end
- free end
Elementary Physics of Music
• Vibration of a string: the slowest (fundamental) mode has a “node” at both ends
• Faster modes have additional nodes in between• It is not difficult to determine the frequencies of the
modes: fn = n*f1 where f1 = v/2L
• in general, the frequency spectrum of an arbitrary periodic vibration of the string will consist of equidistant peaks at the above frequencies – this is often called a ”harmonic” spectrum:
• harmonic spectrum periodic sound
The most difficult math we will use in PHYS207
Mode 1: L=λ/2
Mode 2: L=2 λ/2
Mode 3: L=3 λ/2
Mode 3: L=4 λ/2
Mode n: L=nλ/2 i.e. 1/λ =n/2L n=1,2,3,…
Now: wavelength = distance traveled in one period: λ = v T i.e. T = λ/v
And frequency is the inverse of period: f=1/T = v/λ = v (n/2L) = n(v/2L)
So by a sequence of simple (almost trivial) steps, we have obtained an important and far-reaching result:
Frequency of the n-th mode is f(n) = fn = n f(1)
where the fundamental frequency f(1) = f1 = v/2L
f = f(1)
f = 2 f(1)
f = 3 f(1)
f = 4 f(1)
λL
Vibration modes of a system with 2 and 3 transverse degrees of freedom.
Vibration of a string can be understood as superposition of traveling waves, and/or
as modes of vibration of a system with infinite number of degrees of freedom.
Modes of vibration of a string
• Modes of vibration understood as either standing waves, or as resonances of a system with infinite number of degrees of freedom
N N
N
N N
N
N
N
Mode 1 f1 = v/2L
Mode 2 f2 = 2*v/2L
Mode 3 f3 = 3*v/2L
Mode 4 f4 = 4*v/2L
Example: Spectra of two tones
a) Note C
b) Note G
frequency
intensity
intensity
0 f 2f 3f 4f 5f 6f….. octave 5th 4th Major 3rd minor 3rd
i.e. the harmonic overtones of a simple tone contains the musically consonant intervals(we will learn about the intervals soon …)
Consequences of these extremely simple considerations are actually far-reaching:
• any periodic sound is a mix of several “harmonics”, equidistantly spaced in frequency
In the first approximation:• increasing the “amounts” uniformly corresponds to
louder sound • changing the frequency of the fundamental
corresponds to changing the pitch• using different proportion of fundamental / second/
third / … harmonics means changing the “timbre”, i.e. the “sound color”.
• As we will see, these consideration also determine consonance vs. dissonance
Modes of vibration of a membrane are not harmonic =>
The sound is not periodic => there is no definite pitch
Possible spectrum:
Heisenberg Uncertainty Principle
V = v(sound) Vs = v(source of sound)
Shock wave if Vs > v
Fourier synthesis
waveform spectrum
Instrument 1: f1 = 440 Hz
Instrument 2: f1=440 Hz
f1=1175 Hz
waveform spectrum
Instrument 3: f1=58 Hz
f1 = 196 Hz
f1 = 440 Hz
Beats and the Critical Region• When two coherent sound waves superimpose,
they will go in phase and out of phase at a rate corresponding to the difference of the two original frequencies, producing “beats” with a frequency f(beats) = |f1 – f2|
• A simple DEMO varying the beat frequency demonstrates the existence of a “critical region” where your brain is no longer able to count the beats, yet the frequency difference is not yet large enough for you to perceive two independent sounds. Two sounds with such frequency difference produce a rough, unpleasant sensation.
Beats between tones of slightly different frequencies f1 and f2
f1
f2
f1 and f2
Consonance and Dissonance• Combination of the two above ingredients cannot but
remind you of the Pythagoras’ recognition that tones with frequencies in ratio of small integers are consonant.
• Example: musical “fifth”: an interval with the frequency ratio of the fundamentals of the two tones 3:2 From the well known mathematical theorem:
3 times 2 = 2 times 3• we conclude something quite non-trivial: every 3rd
harmonic of the bottom tone will coincide with every second harmonic of the upper tone. Even when the “fifth” is slightly mistuned, this will results in slow beats, not the unpleasant roughness. And the other harmonics (5th, 7th etc) will be so far from each other that they will be “out of the critical region”, and therefore they will not produce any roughness either.
Spectra of two tones in musical “fifth” (frequency ratio 3:2)
a) Note C
b) Note G
frequency
intensity
intensity
• the smaller the integers involved are, the more justified is the above reasoning.
• Therefore, the “unison” (frequency ratio 1:1) is the most consonant (and also quite boring) interval, followed by the fifth (3:2), fourth (4:3), Major 3rd (5:4) and minor 3rd (6:5).
• That just about does it for the consonant intervals (the Major and minor 6ths are just complements of the minor and Major 3rds).
• The “theoretical” frequency ratios for the dissonant intervals (such as 16:9) should be taken with a (large) grain of salt.
The ear, as see by
A physician
A physicist
An electrical engineer/c.sci.
Intensity -> Sound Intensity Level -> Loudness Level -> Loudness
W/m2 -> dB -> phons -> sones
Sound intensity as perceived by the humans ear/brain
=> loudness
approx.: L = 2 (LL-40)/10
Addition of sounds:
• “coherent” sounds: add amplitudes, square the result to get the intensity
• “incoherent sounds”:
[ ] within critical band: add intensities
[ ] outside critical band: add loudnesses
[ ] way outside: perceive two (or more) independent sounds
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