What vibrating matter seems most conducive to harmonic enquiry? Many have investigated and classified the vibrational patterns of metal plates - Ernst Chladni, the French mathematician Sophie Germain and most comprehensively Mary E. Waller.
I’ve chosen to begin by experimenting with vibrating water. This is because of the sheer range of different vibrational patterns to explore and also because these patterns, unlike the static formations of sand on vibrating plates, are dynamic, ebbing and flowing periodically over time, which makes them suitable for musical accompaniment.
The figure below shows my set-up for filming these cymatic phenomena. A dish of water is placed on the speaker and illuminated by light wihch is diffused through the lens to achieve a uniform disc of light which in turn is then reflected onto the water by the angled glass sheet.
Figure 64
When the water is vibrated by a low frequency sine wave certain distinct steady states of vibration are achieved, dependent upon the amplitude of the signal. So in figure 65 a tone of 20 Hz is slowly increased in volume causing the water to adopt two steady ring states before the pattern evolves into an intricate dynamic form with thirteenfold symmetry.
Figure 65
One issue which is immediately apparent concerning the possible cross-fertilisation between music and visual pattern is the mismatch of scale. The dynamic process of figure 65 has in fact been slowed down about 20 times. The realtime speed is shown in figure 66 where much of the subtleties of form development are lost in the blur of motion.
Figure 66
Added to this, the tone which created this patterning can hardly be called musically complex - intricate dynamics emerge from very simple sonic impulses - and it is not the most captivating of musical experiences listening to a 20 Hz sine wave which has all the musicality of an idling bus engine.
Furthermore the cymatic patterning of more complex musical sound tends to be visually unintelligible; witness figure 61 where Mozart is played to vibrate a dish of water. Any apparent visual patterns are more the result of the music’s dynamics rather than its harmonic content. This I think is also the case with Alexander Wattenhasser’s attempts to bring music and cymatics together (figure 67); as the musicians play the sound is fed directly into the dish of water which splashes about rhythmically.
figure 67
However, while any viable one to one realtime correspondance between the visual and the musical on a creative level appears impossible, the two can still be brought together meaningfully. In the same way as a bass line underpins harmonic movement so the visual element can be conceived as a shadowing of the shifting sonic landscape.
To what extent, however, do the dynamic symmetries which the vibrated water adopts conform to harmonic principles? Figure 68 is an interactive graph of the symmetries resulting from vibrating a dish of water with a sine wave over the range of 6-37 Hz. We can categorize these forms by their nodal diameters (symmetries) and nodal rings. The y-axis marks the nodal diameters, the x-axis the frequency. Progressing horizontally across the graph we encounter successive multiples of nodal rings.
figure 68
Furthermore, If we follow the sequence of each symmetry horizontally we can hear that the frequencies which produce the steady vibrational states adhere to some extent to the harmonic series. So for example in the case of the patterns with sixfold symmetry we can hear that they coincide with the 5th through to 12th harmonic of a harmonic series. However while the behaviour of the water may hint at harmonicity, there seems no way of explaining how the different symmetries interconnect from a harmonic pont of view.
Now I am no scientist and have only a tenuous grasp of the maths underlying these fluid dynamics. Accordingly, I’ve visited several leading acousticians in England to pick their brains. Due to the small wavelengths which characterize these patterns, they are of the opinion that the intricate nature of the patterning is the result of boundary inconsistencies – small vibrational anomalies between the dish and the speaker on which it sits, and imperfections in the wall of the dish. To my mind this does not adequately explain the intimation of harmonicity in the manner that successive nodal ring modes of patterns of the same symmetry procede, and why the progression of self-same forms appears in different home-made dishes and on different sized speaker cones (figure 69).
figure 69
Furthermore, the progression of the symmetries in dishes of different sizes and with different quantities of water is identical.
When dealing with the harmonic nature of vibrating strings we noted that doubling the length of the string halved the resultant frequency. Is there any similar proportional correlation with the vibrating liquid media?
Figure 70 shows the vibrational states of three dishes of water which are proportionally related. Graph (a) corresponds to a dish with 95mm diameter filled with 25ml of water, graph (b) represents the same dish with twice the depth of water (50ml) and graph (c) shows a dish with twice the diameter (190mm) and four times the volume of water (100ml), thus having the same depth as the water in the dish shown graph (a).
The yellow lines on each graph highlight the stable vibrational states which correspond with the 4th harmonic for each symmetry.
Figure 70
As with the doubling of string length so with the doubling of the dish’s diameter from (a) to (c), the resultant frequency appears to be roughly halved: the 4th harmonic of the 5 symmetry in (a) occurs at 16.8 Hz while in (c) it occurs at 8.4 Hz. However, there seems to be no meaningful harmonic link between (a) and (b) when the volume of water in the same dish was doubled.
In conclusion, while vibrating dishes of water with sound certainly brings forth an array of fascinating dynamic figures as can be seen on the gallery pages and provides an arresting visual component for my music videos, they only offer intimations of harmonicity - to find a closer correlation between harmonic music and the visual patterning of vibrating media we must look elsewhere.