Download From Guido d`Arezzo to Wigner of Budapest

From Guido d’Arezzo
to
Wigner of Budapest:
The uncertainty principle
and
musical notation
Tony Bracken
Collegium Budapest
(on leave from Department of Mathematics,
University of Queensland, Brisbane)
May, 2008
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.
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.
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.
(uncertainty in frequency) X (duration) ≈ 1
seconds
vibrations per second
[In quantum mechanics:
Δq Δp >
=½ħ
Here:
Δν Δt >
=
1
---4π
]
Why is it so?
Amplitude
To determine a pure tone (with a definite frequency),
Time
we need the sound to last from the distant past to the distant future.
‘Rectangular’ note.
a finite duration such as:
‘Bell-shaped note’
Amplitude
If instead we produce a note with
or:
Amplitude
Time
Time
that is not at all the same thing – the note no longer has a precise pitch.
Density
Amplitude
Any sound of finite duration contains a spread of frequencies:
Frequency
Density
Amplitude
Time
Time
Frequency
Density
Amplitude
Similarly:
Frequency
Amplitude
Density
Time
Time
Frequency
Does it matter to the composer or the musician?
Less important one octave higher:
More important one octave lower:
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.
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) ]
Converted to a logarithmic scale, the uncertainty picture looks like this:
Log. frequency
C
Middle C
C
Time
Does the Uncertainty Principle have implications for musical notation?
Guido d’Arezzo
993- to 1033+
Benedictine monk:
St. Maur des Fossés, near Paris
Pomposa, near Ferrara
Arezzo
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
More important, Guido invented the stave (or staff ) of musical notation:
Later refinements were other clefs, also time-signatures and bars.
This was the birth of 1000 years of
recorded musical composition:
FREQUENCY
3
4
Note that this is a representation of
a musical signal in the time-frequency
plane :
TIME
The representation of musical tones in the plane is:
Or, better:
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?
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.
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?
Eugene Wigner
1902-1995
Budapest
Berlin
Göttingen
Plaque at Király ut. 76
Budapest
Princeton
Wigner pioneered the use of symmetry principles in quantum physics,
and for this he was awarded the Nobel Prize in 1963.
In 1932, Wigner proposed a way to indicate simultaneously in
Amplitude φ(t)
the time-frequency plane, the characteristics of any sound signal.
Time t
The Wigner function:
Amplitude
For ‘bell-shaped’ sounds, the Wigner function is a simple ‘double-bell’:
W
Time
Frequency
with a contour-plot as we used earlier:
Time
For other sounds, the Wigner function is more complicated:
Amplitude
W
Frequency
Time
Time
Amplitude
Frequency
Note how the Uncertainty Principle is built in to the Wigner function:
Time
Frequency
Amplitude
And again:
Time
The Wigner function has been called the score of a signal, but no-one
would seriously propose to use it for musical notation:
Log. frequency
G
F
E flat
D
Seconds
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, ...
The Wigner function is not well-suited to describe how
sounds can be superposed (to produce beats, or to
produce chords).
Time
Time
Frequency
Amplitude
Frequency
Amplitude
Amplitude
Time
Amplitude
+
W
Frequency
However, it is not easy to go from here to here directly.
Time
The reason is that the Wigner function is a density, not an amplitude.
It is related to the signal amplitude φ 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?
A possibility is to consider the Gabor transform of the signal:
where φ0 is a fixed, reference signal, for example, a fixed bell-signal.
This Ψ is linear in φ , and so such time-frequency amplitudes can
be superposed:
φ
Time
Ψ
+
φ
1
φ
2
Time
Time
Ψ1
+
Ψ2
Ψ1 + Ψ 2
1
2
φ + φ
Now:
Time
2
1
2
Time
1
Ψ + Ψ
φ + φ
Another view:
Frequency
In contour:
Time
We can recover the corresponding Wigner functions,
working in the time-frequency domain. Thus:
Ψ
W
W=Ψ *Ψ
1
Ψ + Ψ
2
and
W12 = ( Ψ1 + Ψ2 ) * ( Ψ 1 + Ψ 2 )
W
12
}
Final comments:
• How does the form of Ψ depend on the choice of φ0 ?
Is there an optimal choice?
• What characterizes the class of integral transforms
φ
Ψ
for various choices of φ ? (Gabor-Bargmann transform?)
0
• How useful is the concept of the amplitude Ψ in quantum
mechanics? (Schrödinger’s equation, entanglement?)
References:
Howard Goodall, Big Bangs: The story of five discoveries that changed musical history
(London: Vintage, 2002).
Wikipedia: Web pages on Guido d’Arezzo, musical notation, musical scales, Eugene Wigner.
J. Wolfe, Heisenberg’s uncertainty principle and the musician’s uncertainty principle
( www.phys.unsw.edu.au/jw/uncertainty.html ).
I. Fujita, Uncertainty principle for temperament
( www.geocities.co.jp/imyfujita/wtcuncertain.html ).
J.J. Wlodarz, On quantum mechanical phase-space wave functions,
J. Chem. Phys. 100 (1994), 7476—7480.
Go. Torres-Vega and J.H. Frederick, A quantum mechanical representation in phase space,
J. Chem. Phys. 98 (1993), 3103—3120.