posted 11 March 2003 02:11            

malletgong.kym

A free circular plate is a relatively complex instrument to synthesize because the number of overtones out to some limiting frequency grows as the square of that frequency limit. It also involves finding roots of the third derivative of a function composed of Bessel functions of real and imaginary arguments. These are solutions to the wave equation in a round rigid plate, which is a 4th order 2-dimensional partial differential equation. In summary, it is just too difficult to produce a model for Kyma that permits the strike point to be varied as we did for the bars.

Instead of using a cutoff limiting frequency, I chose all the partials that exist in the various vibrational modes that have amplitudes above -20 dB relative to the amplitude of the fundamental (hah! what a concept here!). These partials were amplitude weighted according to the degree they get excited when the plate is struck at the edge. This basically excites all the modes to varying degrees.

That gets us about 13 partials.

Interestingly, the mode (2,1) vibration actually has a vibration frequency below our lowest (0,1) mode. It vibrates at approximately 0.79 times the lowest mode frequency. The mode numbers (m,n) refer to:

m = angular mode. When zero, the vibration shows no nodal lines in the plate. The plate vibrate with large amplitude at the center and edges. When greater than zero, it has no vibration at the center of the plate, and nodal lines form spokes emanating from the center. This number, m, represents how many slices of pie you get in the modal lines.

n = vibrational mode. It runs from 1 to infinity. This is the number of amplitude envelope peaks running from the center of the plate out to the edge. When n = 1, we have the lowest possible vibration modes for each angular mode.

Using the cutoff of -20 dB relative to the fundamental amplitude, we have modes (0,1)..(0,5), (1,1)..(1,2),(2,1)..(2,3),(3,1)..(3,2), and (4,1). So there are quite a few interesting vibrations excited by striking the edge of the plate.

If we had gone to an amplitude cutoff of -30 dB then there would have been more than 35 vibrational modes of different frequencies. That's probably too many to put on Kyma, and to be fair, I would have to compute higher order angular modes than going out to m=4 as we have done here.

None of the vibrational modes are harmonic. This thing is a real monster. I have the polyphony set to 2 or 3. If you can't run this on your Capy, try cutting back on the polyphony.

Even though we can't readily move the strike point along a line from the center to the edge, we can modify the mallet hardness. That helps select out some interesting variations.

Cheers!

- DM

[... why a circular plate gong? and not a rectangular one? Interesting that you should ask...

As it happens the 4th order partial differential equation representing the vibrations of a plate can only be solved in a system with circular symmetry. There is no known solution for a rectangular shape.

Of course, we all know that you can make rectangular gongs if you like. It's just that we don't have the mathematical tools (nobody does!) to solve this seemingly simple case. It is actually simple (relatively speaking) to solve for systems with circular symmetry, and quite impossible for any other shapes.

If somebody knows otherwise, please let me know... I tried for several days to find a rectangular solution, and finally turned to my copy of Courant and Hilbert to find them stating explicitly that no other systems can be solved with known functions. Hmmm... ]

posted 11 March 2003 03:41            

malletgong.kym

I reassigned the root pitch to mode (2,1) which is the lowest among the 5 azimuthal modes. Amplitudes have been corrected.

Interestingly, with the !Slope parameter adjusting the decay of higher partials, this root pitch will decay more slowly (!Slope > 0) or more rapidly (!Slope < 0) than other partials at nearby pitches. Truthfully, it sounds more like a real gong now. But the lowest pitches are in higher order vibrational modes that decay differently from the lowest modes.

I should work one up with a mallet strike near but not at the center. Striking at the center can only excite the non-azimuthal modes of a gong. You need to strike off center to excite the other higher order modes.