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 1 corresponds to half a sine wave, that is a string fixed at its both ends.

phi is the phase between the two oscillations, to effectuate changes of the parameters you have to press the RETURN-key

"slow <----> fast" regulates the velocity of the animation

"octave up / down" doubles or halfes the frequency (for better visibility the velocity is not changed proportionally)

START - starts the animation, STOP - ends it

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

Beats: Two waves of almost equal frequencies superpose. Try for example m=20 and n=19 or start with m=40 n=40 in the one-string-graph. While running regulate n slowly with the slider. Typical beat nodes will arise.

Interference: Start with m=n=3 in the two-string-graph. While running change slowly the phase with the slider (preferably by pressing the arrows of the slider with the tip of the mouse). 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, no extra wiggles or so. Note as the phase approaches 1.0 pi almost complete extinction of the signal occurs. Contrariwise a phase of 0.0 or 2pi will result in maximum amplification. The reason for this is easily understood by comparing the one-string-graph with the two-string-graph. The first is just the sum of two parts of the second.

In electrical engineering Lissajous curves are used in oscilloscope measurements to detect very small frequency deviations or phase differences. Try with m=50 n=49.99 . In the one-string-graph you notice the beats in time. In the "move Lissajous curve" you notice a slowly moving elliptical shape.

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(

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(

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(

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(

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 A Gallery of Quantum States. This site gives an introduction to the quantum mechanics of the light field explaining notions like the wave packet or the uncertainty relation.

For a very nice 3D applet, demonstrating the generation of Lissajous curves by C.K.Ng see Lissajous figures.

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Last modified: Jun. 2013