Whether it be Chladni plates, dishes of water, bells or drumheads, the vast majority of vibrating systems in nature are anharmonic, that is to say that their overtones do not correspond with the harmonic series. Harmonic systems such as the stretched string are very much the exception.
In respect to this project then, is it possible to find a vibrating system that is both harmonic and displays the wealth of dynamic patterning that we saw with vibrating dishes of water, since a vibrating string holds limited appeal as a visual counterpart to harmonic music.
If you remember, I started this reseach by exploring the sonic architecture of a saxophone which proved to be more or less harmonic. Now the bore of a saxophone is conical and since a cone is a part of a sphere, it would seem logical that a sphere should vibrate harmonically.
Does experiment prove this to be the case?
Intriguingly I’ve read that the American polymath Buckminster Fuller immersed vibrating balloons in liquid dye and noted the resultant symetrical patterning on their surfaces, but I’ve yet to track down any further information about his experiments.
I’ve followed the example of Hans Jenny and centred my investigation upon bubbles. Now if a bubble is vibrated with a low frequency sine wave it adopts steady states of vibration as shown below (click on each for moving image).
|2nd harmonic||3rd harmonic||4th harmonic|
|5th harmonic||6th harmonic||7th harmonic|
The emergence of these different shapes can be plotted on a graph (figure 66) where the y-axis shows bubble diameter in centimetres, the x-axis frequency in hertz. The dotted curved lines represent the progression of each of the bubble symmetries shown above. As with the dishes of water earlier we can see evidence of harmonic behaviour; so for a bubble 12cms in diameter the steady states appear at roughly 12, 18, 24, 30, 36, 42, 48 Hz which would coincide with a harmonic series from 2nd to 8th harmonic whose fundamental is 6Hz.
Figure 66 - Click to enlarge
What about subharmonics (undertones)? What would this graph look like if a bubble did indeed vibrate harmonically? Figure 67 shows the behaviour of an idealised harmonic bubble.
Figure 67 - Click to enlarge
We can recognise the lambdoma superimposed upon the graph, with the overtones proceeding horizontally from left to right and the undertones curving upwards from right to left. So, for instance, when 40 Hz is the generating signal, a bubble 12 cms in diameter will vibrate in the steady state of the 8th harmonic, while a bubble 6 cms in diameter will vibrate in the steady state of the 4th harmonic. All vibrational states are clearly proportionally related.
Returning to our experimental bubble graph (figure 66), it appears that this graph bears a strong if truncated resemblance to the idealised version; which would tend to suggest that our initial proposition was correct: a sphere does vibrate harmonically.
With this in mind, the video I have created with Gregory McKneally called 'Spherical Chimes', which utilises vibrating bubble footage, is the sole piece where the music does not just accompany the cymatic visuals but is directly harmonically correlated to the pulsatons of the bubbles.