## Vibrating Strings, Musical Intervals and the Curves of Lissajous

This site allows you to visualize two simultaneous oscillations via the motion of one or two vibrating strings or the Lissajous curves. By changing the wavenumbers of the two strings and the phase between the two oscillations graphs of different musical intervals, beats and interferences are generated. More explanations below.

m and n are the wavenumbers of the first and second string respectively, wavenumber m=1 corresponds to a full sine wave.
φ is the phase between the two oscillations
"⇐ slow fast ⇒" regulates the velocity of the animation
"octave up / down" = "⇑ / ⇓ " doubles or halfes the frequency
START - starts the animation, STOP - ends it

 m: n: φ: π

⇐     slow      fast     ⇒

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### Basics

An ideal oscillation. is described mathematically by a sine curve. A musical interval corresponds to two independent oscillations. The time development of these two independent oscillators, for example two vibrating strings or a pendulum with two independent joints, are described by two standing sine waves. (The case of running waves like for instance radiation or water waves is omitted here since the fixed plucked string serves as the most important example).
These two waves can be represented graphically by
1) the movement of the two strings
2) the movement of one string where the two oscillations are superposed
3) a Lissajous curve

### Lissajous curves

Lissajous curves are named after the french physicist Jules Antoine Lissajous. Their graphs are parametric plots. One oscillation follows the horizontal axis, the other one the vertical axis. Both are sine curves with different frequencies which are given by the corresponding two wavenumbers m and n. The relative phase between the two oscillations determines the starting point of the curve.

### Superposition

The addition of the two strings into a one-string-graph is called a superposition: Both waves are added without influencing each other. As a consequence the following phenomena arise:
Beats: Two waves of almost equal frequencies superpose. Try for example m=20 and n=19 or m=64 n=63 in the one-string-graph. While running press "⇑" to get the full picture. Now regulate phi slowly with the slider.
Interference: Start with m=n=3 in the two-string-graph. While running change slowly the phase with the slider. Now switch to the one-string-graph and do the same. A first surprise may be, that two sine curves which are shifted in phase do exactly add up to a third sine curve. As the phase approaches 1.0 pi complete extinction of the signal occurs. A phase of 0.0 or 2pi will result in maximum amplification. Switching between one-string-graph and two-string-graph explains this easily.

### Lissajous curves in vivo

To get a better understanding of the Lissajous curves the choice "move Lissajous curve" was introduced. This allows you to see a continuous change of a Lissajous curve, while you alter m,n or the phase. For example after selecting "move Lissajous curve" choose m=5 n=6 and change the phase via the slider. The result looks like a three dimensional rotation.
In electrical engineering Lissajous curves are used in oscilloscope measurements to detect very small frequency deviations or phase differences. If you define m=1 n=1 and change the phase via the slider you will see a slowly moving elliptical shape. Try the wame with m=50 n=49.999 . In the one-string-graph you notice the beats in time. In the "move Lissajous curve" you notice an elliptical shape moving automatically.

### Musical intervals and their ratios

A number of example-intervals can be chosen. Numerically as intonation just intonation. is used. However the parameters can be changed easily by entering new values or using the slider.
Examples of intervals ordered by the size of the ratio's denominator (the intervals beecome less and less harmonic so to say):
Unison (german "Prime"):
Both Strings have the same wavenumber. Thus their ratio is 1:1. If the phase is zero, the strings' movement is parallel.
Octave (german "Oktave"):
The second string is fixed at its ends and in the middle as well. The results is a complete sine wave doubling the frequency, thus the ratio between the oscillations of the first and second string is 1:2
Perfect fifth (german "Quint"): Ratio 2:3
To see quint-oscillations with higher periodicity you can change the wavenumbers from 2:3 to 4:6 or 6:9 or even 30:45. Since the ratio is always the same, the same musical interval is described. In the one-string-graph the periodic behaviour becomes more apparent. In the Lissajous graph however you will note no difference at all. This shows in some ways the strength of the Lissajous graph: it filters out "unnecessary" information.
Perfect fourth (german "Quarte"): Ratio 3:4
Note that a fourth added as an interval to a fifth results in an octave. This becames apparent numerically by multiplying the ratio of the fifth with the ratio of the fourth: 2:3 times 3:4 equals 2:4 which is the same as 1:2, the ratio of the octave. Mathematically speaking our acoustical perception works with the logarithmic law. (This phenomenon ist investigated on the field of Psychoacoustics.) For the perfect fourth and fifth this is seen in the following little calculation:
Quart + Quint = Oktave
log(34) + log(23) =  log(34 ⋅ 23) =  log(12)
With our senses we feel an addition, our hands on the string though perform a multiplication of ratios.
Major third (german "grosse Terz"): Ratio 4:5
Major sixth (german "grosse Sexte"): Ratio 3:5
Minor third (german "kleine Terz"): Ratio 5:6
Again if you add the intervals of the Major third and the Minor third you get a perfect fifth. Numerically 4:5 times 5:6 equals 4:6 which is the same as 2:3, the ratio of the fifth. Note that these simple relations describe already almost all of the underlying foundations of occidental music, which is based on the major and minor chords. In logarithms:
log(45) + log(56) =  log(45 ⋅ 56) =  log(23)
gr. Terz + kl. Terz = Quint
Minor sixth (german "kleine Sexte"): Ratio 5:8
Again adding the intervals of a Major third and a Minor sixth results in an octave as well as adding the intervals Minor third and a Major sixth.
Minor seventh (german "kleine Septime"): Ratio 5:9
Major second (german "grosse Sekunde"): Ratio 8:9
Here the first mathematical problem occurs. Subtracting a major second from a minor seventh results in 5:9 divided by 8:9 equals 5:8, which is a major sixth, so far so good. In logarithms:
log(59) - log(89) =  log(59 ⋅ 98) =  log(58)
kl. Septime - gr. Sekunde = gr. Sexte
Equally adding a major second to a minor seventh should result in an octave but numerically we get 5:9 times 8:9 which equals 40:81 which is almost but not quite 1:2 . This is the well known problem of the compatibility of just intonation. In logarithms:
log(59) + log(89) =  log(59 ⋅ 89) =  log(4081)
kl. Septime + gr. Sekunde ≠ Oktave
Major seventh (german "grosse Septime"): Ratio 8:15
Diatonic semitone (german "kleine Sekunde"): Ratio 16:15
Same problems as above ...
Nevertheless as long as n and m are commensurable that is n/m is a rational number, the Lissajous curve and the superposition curve on one string is periodic, that is the curve returns to its starting point. For the Lissajous curve this means that it forms a closed curve. In the case of an irrational quotient n/m however the curve becomes non-periodic, for example the curve representing the tritonus (the only example listed where equal tempered intonation is used thus placing the tritonus numerically exactly in the middle within an octave).

Large oscillations of the planets around the sun in our solar system are shown in Curves of planetary motion in geocentric perspective: Epitrochoids.
For a quantum mechanical treatment of oscillations see Quantum States of Light. This site gives an introduction to the quantum mechanics of the light field explaining notions like the wave packet or the uncertainty relation.
A very nice 3D applet, demonstrating the generation of Lissajous curves by C.K.Ng is Lissajous figures.  Contact:  higobreitenbach@arcor.de
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