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

Musical tones are associated with definite frequencies of vibration: A above Middle C: Frequency = 440 vibrations per second Middle C: Frequency = vibrations per second D below Middle C: Frequency = vibrations per second and so on. Any sound is a vibration – or rather, many vibrations per second.

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. However, in practice:

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. The Uncertainty Principle:

(uncertainty in frequency) X (duration) ≈ 1 vibrations per second seconds [In quantum mechanics: Δq Δp ½ ħ Here: Δν Δt ] π >=>= >=>=

To determine a pure tone (with a definite frequency), we need the sound to last from the distant past to the distant future... Time Amplitude Why is it so?

If instead we produce a note with a finite duration such as: that is not at all the same thing – the note no longer has a precise pitch. Amplitude Time Amplitude or: ‘Rectangular’ note. ‘Bell-shaped note’

Any sound of finite duration contains a spread of frequencies: TimeFrequency Time Frequency Amplitude Density

Similarly: TimeFrequency Time Frequency Amplitude Density

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: Time Log. frequency Middle C Does the Uncertainty Principle have implications for musical notation? C C

Guido d’Arezzo 993- to Benedictine monk: St. Maur des Fossés, near Paris Pomposa, near Ferrara Arezzo

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 Guido introduced the sol-fa method of teaching Gregorian chants:

More important, Guido invented the stave (or staff ) of musical notation:

Later refinements were other clefs, also time-signatures and bars. 3 This was the birth of 1000 years of recorded musical composition: Note that this is a representation of a musical signal in the time-frequency plane : 4 TIME FREQUENCY

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.

So in written music the Uncertainty Principle sits in the background. Its influence is felt but it is not made explicit. Rather, what is indicated is that a note of that duration should be played or sung, with whatever spread of pitches the instrument produces. 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 Plaque at Király ut. 76 Budapest Berlin Göttingen 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 the time-frequency plane, the characteristics of any sound signal. Amplitude φ(t) Time t The Wigner function:

For ‘bell-shaped’ sounds, the Wigner function is a simple ‘double-bell’: with a contour-plot as we used earlier: Time Frequency W Time Amplitude Time Frequency

For other sounds, the Wigner function is more complicated: Time Amplitude Time Frequency W Time Frequency

Time Amplitude Note how the Uncertainty Principle is built in to the Wigner function: Frequency

Time Amplitude Frequency And again:

The Wigner function has been called the score of a signal, but no-one would seriously propose to use it for musical notation: Seconds Log. frequency D E flat F G

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 Amplitude Time Frequency

Time Amplitude Frequency +

Time Frequency W

However, it is not easy to go from here Time Frequency to here directly.

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 φ is a fixed, reference signal, for example, a fixed bell-signal. 0 This Ψ is linear in φ, and so such time-frequency amplitudes can be superposed:

Time Frequency Time Ψ φ

+ + φ 1 φ 2 Frequency Ψ 1 Ψ 2

Now: φ + φ 1 2 Time Frequency Ψ + Ψ 1 2

Another view: Time Frequency Ψ + Ψ 1 2 Time 1 2 φ + φ

In contour: Time Frequency

We can recover the corresponding Wigner functions, working in the time-frequency domain. Thus: Time Frequency W Ψ W = Ψ Ψ *

and Time Frequency W Ψ + Ψ 1 2 W = ( Ψ + Ψ ) ( Ψ + Ψ ) 12 *

}

Final comments: How does the form of Ψ depend on the choice of φ ? Is there an optimal choice? What characterizes the class of integral transforms φ Ψ for various choices of φ ? (Gabor-Bargmann transform?) How useful is the concept of the amplitude Ψ in quantum mechanics? (Schrödinger’s equation, entanglement?) 0 0

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 ( ). I. Fujita, Uncertainty principle for temperament ( ). 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.