From Bach to Einstein and Beyond
Celebration of Physics AND Music.
Vladimir (“Vladi”) Chaloupka
Emeritus Professor of Physics
University of Washington
www.faculty.washington.edu/vladi
Our celebration actually starts well before Bach's time.
Unfortunately, after Einstein, the story acquires a tinge of a grave
concern. But fortunately, we will stick to celebrating tonight, and
try not worry too much, besides a very brief mention of the Basic
Problem.
One day more than 2,500 years ago, professor Pythagoras –
not satisfied with his often-irritating high-school-like triangle,
not satisfied even with his (ir)rational numbers –
felt the need to understand why some musical intervals sound
consonant and some others are dissonant. And he discovered
something that we still today teach students of Physics of Music,
and by doing so he laid a cornerstone not only for the Physics of
Music, but for Science as such.
Let us start with the dreadful “right triangle”:
Let opposite side = a, adjacent side = b, then hypotenuse c= sqrt(a2+b2)
Now let us consider vibration of a weight on a spring:
a minute of thought will convince you that it is equivalent to a projection of a uniform circular
motion:
And it quickly turns out that if we associate the square of the opposite side to be proportional to
the position x, and the adjacent side to be proportional to the velocity, the Pythagoras’ a2+b2 = c2
will turn into
½ kx2 + ½ mv2 = E = const
i.e. the equation of the conservation of energy – the foundation of all physics
Not bad, Professor Pythagoras, not bad at all!
Next, we continue with Professor Pythagoras’s rational numbers:
all numbers that can be written as a ratio of finite integers, i.e. r=n/m
examples: 3/2 = 1.5 7/4 = 1.75 2/50 = 0.04 or 2521/7000 =
= 0.360142857142857142857142857142857142857.....
It was truly devastating for the worshippers of the rational numbers to discover
that sqrt(2) is not rational, it cannot be expressed as m/n
Sqrt(2) = 1.4142135623730950488016887242097...... =
14142135623730950488016887242097.......
= ----------------------------------------------------------10000000000000000000000000000000.......
(For that matter, pi is not rational, either. It has an infinite number of digits, and
most likely every pattern you can think of is contained in there somewhere….
Proof is not easy here, but the proof that sqrt(2) is not a rational number should
be in everyone's intellectual toolbox.)
But we digress. Let us see if there is some Physics of Music in there …
The Physics and Math of a vibrating string
λ1/2
f1 = 1 f 1
Mode 2: L= 2 λ2/2
f2 = 2 f1
Mode 3: L= 3 λ3/2
f3 = 3 f1
Mode 3: L= 4 λ4/2
f4 = 4 f1
Mode n: L= n λn/2
i.e. 1/λn =n/2L
Mode 1: L=
λ1
λ2
λ3
λ4
L
Now: wavelength = distance traveled in one period:
n=1,2,3,…
λ = vT i.e. T = λ/v
[where v =sqrt(tension/mass density) is velocity of wave traveling on the string]
And frequency is the inverse of period: fn=1/Tn = v/λn = 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 nth mode (nth harmonics] is
with the fundamental frequency f1 = v / 2L
fn = nf1 n = 1,2,3,…
[ with v = sqrt(T/mu) ]
Example: Spectra of two tones with f2/f1 = 3/2
intensity
a) Note C
intensity
b) Note G
frequency
i.e. the chance of any pair of the harmonics of the two tones to be unpleasant is
minimized, and the interval (in this case: a “perfect fifth”) sounds consonant
Because of the Vladi Law of the musical 5th: 3 times 2 = 2 times 3
i.e. “every 3rd harmonic of the bottom tone coincides with every 2nd harmonics of top”
Beats between tones of slightly different frequencies f1 and f2
The beat frequency is fbeats = |f2 – f1|. If it is so low that you can count the beats, it is usually OK (in
fact, sometimes it is welcome!). If it is high enough, it is also OK – you hear two different tones. But
when it is “in between” (i.e. in the “critical band”) it is disturbing.
That, combined with Pythagoras’ m/n rule, is (in first approximation) the explanation of consonance
and dissonance.
f1
f2
f1 and f2
Consequences of these extremely simple
considerations are actually far-reaching:
• any periodic sound is a mix of several/many “harmonics”,
•
•
•
•
•
0
equidistantly spaced in frequency
In the first approximation:
changing the “amounts” corresponds to changing the loudness
changing the frequency of the fundamental corresponds to
changing the pitch
using different proportions of harmonics means changing the
“timbre”, i.e. the “sound color”.
Intervals with f2/f1 = n/m with small n and m are consonant
And the spectrum of even a simple tone contains all consonant
intervals:
f
2f
octave
3f
5th
4f
4th
6f …
5f
Major 3rd
minor 3rd
• All of this was obtained by a sequence of easy steps
• Somewhere, Professor Pythagoras is smiling
The next stop on our brier tour will be with Johannes Kepler, who more than 400 years
ago, led by a musico-mystical motivation (“Harmonia Mundi”) laid another
cornerstone: for the modern science.
Around the year of 1600 AD Kepler spent years of his life on trying to reconcile the
new, precise observations of Tycho de Brahe with the then new Copernican Solarcentered system. Mars data did not fit!
Kepler motivation for this (enormous!) labor has been nothing but rational: he believed
that planets, as they go round, do produces sound inaudible to humans but audible to
God, who therefore created things so that there be harmony (“Harmonia Mundi”).
J.S.Bach: was he a Learned Scholar?
• 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 ” !
And so: no, Bach was not a “Learned Musician”,
• I think he even would have preferred Equal Temperament (if it
was not so difficult to tune without electronics!)
• and no, one should not try to “complete” his Unfinished Fugue
It is hard to believe that, after the superlatives employed in the discussion [of Bach's Art of Fugue] so
far, there would be something even more profound, more beautiful and more moving. And yet I will
argue that the giant, unfinished Fugue 14 is the pinnacle of Bach’s life work, if not of much of Music
in general. in this single composition Bach looks back at least 150 years, and also anticipates music
not to be written for another 150 years - and he does it with beauty and wisdom which takes multiple
listening to even start to appreciate. That is why any attempt at a "completion" is preposterous. The
problem is not a puzzle in counterpoint to be solved, and no one can "complete" the life of a genius.
As in Beethoven's Opus 111, where the Arietta Theme comes back at the end, in its simplest form,
and as in Bach's Goldberg Variations, where the original Aria returns, verbatim, after all the wonders
Bach subjected it to in the Variations, so it seems appropriate to repeat the simplest presentation of
the Kunst der Fuge Theme at the end of the Art of Fugue, to fully realize that Bach indeed went
much beyond Music in this work – he reflected and meditated, and makes us reflect and meditate, on
the inner wisdom of the Heavens and Earth.
PS: Where did I learn to write sentences as the last one? From a letter by Mozart to his sister:
“I hope, my queen, that you are enjoying the highest degree of health, and that now and then or
rather, sometimes, or, better still, occasionally, or even better still, qualche volta, as the Italians say,
you will sacrifice for my benefit some of your important and intimate thoughts, which ever proceed
from that very fine and clear reasoning power, which in addition to your beauty, and particularly from
a woman, and particularly from one of such tender years, almost nothing of the kind is ever
expected, you possess, O queen, as abundantly as to put men and even graybeards to shame.
Now, there you have a well-turned sentence [sic].
Familiarity with a few such expressions makes one appreciate the skill with
which Douglas Hofstadter explains recursions in language in his “Goedel
Escher Bach”:
“The proverbial German phenomenon of the 'verb-at-the-end', about which
droll tales of absentminded professors who would begin a sentence, ramble
on for an entire lecture, and then finish up by rattling off a string of verbs by
which their audience, for whom the stack had long since lost its coherence,
would be totally nonplussed, are told, is an excellent example of linguistic
pushing and popping.”
So, to conclude this literary excursion, one can only say with Mark Twain:
“A German joke is no laughing matter”.
(Incidentally, Mark Twain, in his studies of the German language and culture,
also discovered that ”Wagner's music is better than it sounds.”)
Einstein as Scientist, Musician
and Prophet
• Einstein as scientist: In 2005 we celebrated the
Centenary of Einstein’s Annus Mirabilis
• Einstein as musician: from a review:
“Einstein plays excellently. However, his worldwide fame is undeserved. There are many
violinists who are just as good.”
• Einstein as prophet: “Nuclear weapons
changed everything except our way of thinking.”
Brief Summary of some aspects of Music ad Science:
[ ] Pythagoras/Kepler/Bach/Einstein as already mentioned
[ ] musical acoustics is an excellent way to visualize Quantum
Wave Mechanics
[ ] Superstring Theory: all elementary particles are different
modes of vibration of a string
[ ] ergo: the Princeton String Quartet (the second one)
[ ] Music is an excellent example of Emergent Complexity
(some fugues “look like” part of the Mandelbrot set, see also
Hofstadter Goedel Escher Bach…)
Research advertisement:
If you are (or know)
a Windows programmer
(C, assembly, multicore
programming, Windows
API, Visual Studio,...)
and if you might be
interested helping out in a
Mandelbrot research / art
project
please email vladi@uw.edu
Music is an excellent playground to explore Time:
Schrodinger vs. Wheeler-deWitt
Music leads to Exuberance and Humility, at the same time
...
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.
LISA: Laser Interferometer Space Antenna
Listening to the Gravitational Symphony of the Universe
Will detect the change of the distance as small as 10-11 m !!!
LISA: listening to the gravitational symphony of the Universe
LISA orchestra, soloists and the first 0.000 000 000 000 1 seconds
And finally:
[ ] the symbiosis/synergy helps some musicians (especially
music faculty) to feel better about what they do
[ ] a little music makes lectures on Science and Society less
depressing
Science and Music, with Bach, Darwin and Einstein (and others, of course)
symbolize many aspects and dimensions of infinity and eternity,
exuberance and humility,
and wisdom and hope.
The Andromeda
Galaxy: 2 million
light years away.
The most distant
object visible by
naked eye (you
have to know
where to look, and
find a really dark
place, but the
experience is very
much worth it!)
Note: for details on when and
how to see Andromeda, see
http://www.physics.ucla.edu/
~huffman/m31.html
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 …