From Guido d

1
From Guido dArezzo toWigner of Budapest

• The uncertainty principle
• and
• musical notation
• Tony Bracken
• Collegium Budapest
• (on leave from Department of Mathematics,
• University of Queensland, Brisbane)
• May, 2008

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Any sound is a vibration or rather, many
vibrations per second.
Musical tones are associated with definite
frequencies of vibration A above Middle C
Frequency 440 vibrations per second Middle C
Frequency 261.6 vibrations per second D
below Middle C Frequency 146.8 vibrations
per second and so on.
3
However, in practice
Every musical sound has an uncertain frequency
It is impossible to produce Middle C exactly.
Also, every musical sound has a finite
duration It is impossible to produce a sound
instantaneously.
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The Uncertainty Principle
As the duration of a sound is decreased, the
uncertainty in frequency increases. In order
to decrease the uncertainty in frequency of a
sound, the duration must increase.
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(uncertainty in frequency) X (duration) 1
seconds
vibrations per second
In quantum mechanics
?q ?p ½ h Here
?? ?t

gt
1 ---- 4p
gt
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Why is it so?

To determine a pure tone (with a definite
frequency),
. .
Amplitude
. .
Time
we need the sound to last from the distant past
to the distant future.
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Rectangular note.
If instead we produce a note with a finite
duration such as
Amplitude
Bell-shaped note
Time
or
Amplitude
Time
that is not at all the same thing the note no
longer has a precise pitch.
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Any sound of finite duration contains a spread of
frequencies
Density
Amplitude
Time
Frequency
Amplitude
Density
Frequency
Time
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Similarly
Density
Amplitude
Time
Frequency
Density
Amplitude
Time
Frequency
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Does it matter to the composer or the musician?
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Less important one octave higher
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More important one octave lower
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A musical scale is like the
Richter scale for earthquakes

-- a logarithmic scale.
An earthquake measuring 7.2 on the Richter scale
is ten times the size of an earthquake
measuring 6.2
An earthquake measuring 5.2 is one tenth the size
of an earthquake measuring 6.2, etc.
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Similarly C one octave above Middle C has a
frequency twice that of Middle C. C one octave
below Middle C has a frequency half that of
Middle C.
To go up one octave, double the frequency. To
go down one octave, halve the frequency.
To go up one semi-tone, multiply the frequency
by 2(1/12)
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Converted to a logarithmic scale, the uncertainty
picture looks like this
C
Log. frequency
Middle C
C
Time
Does the Uncertainty Principle have implications
for musical notation?
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Guido dArezzo 993- to 1033
Benedictine monk St. Maur des Fossés, near
Paris Pomposa, near Ferrara Arezzo
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Guido introduced the sol-fa method of teaching
Gregorian chants

• Ut queant laxis
• Resonare fibris
• Mira gestorum
• Famuli tuorum,
• Solve polluti
• Labii reatum,
• Sancte Ioannes.

• Doe, a deer a female deer
• Ray, a drop of golden sun
• Me, a name I call myself
• . . .

That your servants may with relaxed throats sing
the wonders of your deeds, take away sin from
their lips, Saint John
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More important, Guido invented the stave (or
staff ) of musical notation
21
Later refinements were other clefs, also
time-signatures and bars.
3
4
This was the birth of 1000 years of recorded
musical composition
FREQUENCY
TIME
Note that this is a representation of a musical
signal in the time-frequency plane
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The representation of musical tones in the plane
is
23
Or, better
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The duration of notes is determined by the
time-signature, the measures and bars, and by
special marks on the notes
However, the uncertainty in frequency of each
note is not indicated.
Does it matter?
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When composers mark a note on the stave, say an
eighth note at Middle C,
?
they do not ask the musician to produce a note
with a precise duration and a precise frequency
that is impossible, because of the
Uncertainty Principle.
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Rather, what is indicated is that a note of that
duration should be played or sung, with
whatever spread of pitches the instrument
produces.
So in written music the Uncertainty Principle
sits in the background. Its influence is felt
but it is not made explicit.
But now this begs the question Can we show the
time and frequency content of a sound in the
time-frequency plane, in a way that
is consistent with the Uncertainty Principle?
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Eugene Wigner 1902-1995
Budapest Berlin Göttingen Princeton
Plaque at Király ut. 76 Budapest
Wigner pioneered the use of symmetry principles
in quantum physics, and for this he was awarded
the Nobel Prize in 1963.
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In 1932, Wigner proposed a way to indicate
simultaneously in the time-frequency plane,
the characteristics of any sound signal.
Amplitude f(t)
Time t
The Wigner function
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For bell-shaped sounds, the Wigner function is
a simple double-bell
Amplitude
W
Time
Frequency
Time
with a contour-plot as we used earlier
Frequency
Time
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For other sounds, the Wigner function is more
complicated
W
Frequency
Amplitude
Time
Time
Frequency
Time
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Note how the Uncertainty Principle is built in
to the Wigner function
Frequency
Amplitude
Time
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And again
Frequency
Amplitude
Time
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The Wigner function has been called the score of
a signal, but no-one would seriously propose to
use it for musical notation
G
F
Log. frequency
E flat
D
Seconds
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However, the Wigner function is widely used in
more technological uses of signals (e.g.
radar), and also in its original context, in
quantum mechanics, where it is important in
quantum optics, quantum tomography, studies of
the relationship between classical and quantum
physics, ...
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The Wigner function is not well-suited to
describe how sounds can be superposed (to
produce beats, or to produce chords).
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Frequency
Amplitude
Time
Time
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Amplitude

Amplitude
Frequency
Amplitude
Time
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W
Time
Frequency
Time
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However, it is not easy to go from here
to here directly.
Frequency
Time
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The reason is that the Wigner function is a
density, not an amplitude. It is related to the
signal amplitude f in a nonlinear way
Can we define amplitudes in the time-frequency
plane that can be simply added together, that
show the time-frequency characteristics of a
signal, and in terms of which the Wigner function
can be defined?
41
A possibility is to consider the Gabor transform
of the signal
where f is a fixed, reference signal, for
example, a fixed bell-signal.
0
This ? is linear in f , and so such
time-frequency amplitudes can be superposed
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f
Time
?
Frequency
Time
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f
f
1
2
Time
Time

?
?
1
2
Frequency
Frequency
Time
Time
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Now
f f
1 2
Time
1 2

? ?
Frequency
Time
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Another view
f f
1 2
Time
1 2

? ?
Time
Frequency
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In contour
Frequency
Time
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We can recover the corresponding Wigner
functions, working in the time-frequency
domain. Thus
?
Frequency
Time
W
W ? ?
Time

Frequency
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and
1 2

? ?
Frequency
Time
W
W ( ? ? ) ( ? ? )
12

12
1 2
1 2
Time
Frequency
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Final comments

• How does the form of ? depend on the choice of
  f ?
• Is there an optimal choice?
• What characterizes the class of integral
  transforms
• f ?
• for various choices of f ? (Gabor-Bargmann
  transform?)
• How useful is the concept of the amplitude ?
  in quantum
• mechanics? (Schrödingers equation,
  entanglement?)

0
0
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References
Howard Goodall, Big Bangs The story of five
discoveries that changed musical history (London
Vintage, 2002). Wikipedia Web pages on Guido
dArezzo, musical notation, musical scales,
Eugene Wigner. J. Wolfe, Heisenbergs
uncertainty principle and the musicians
uncertainty principle ( www.phys.unsw.edu.au/jw/u
ncertainty.html ). I. Fujita, Uncertainty
principle for temperament ( www.geocities.co.jp/i
myfujita/wtcuncertain.html ). J.J. Wlodarz, On
quantum mechanical phase-space wave functions,
J. Chem. Phys. 100 (1994), 74767480. Go.
Torres-Vega and J.H. Frederick, A quantum
mechanical representation in phase space, J.
Chem. Phys. 98 (1993), 31033120.