Acoustic Modelling 5.....
TWO-DIMENSIONAL MODELS: GONGS, BELLS ETC.
The basic model of a thin string is essentially one dimensional; so I thought what about more dimensions? A rectangular plate must have two modes of resonance at least. Maybe there are resonant modes (standing waves) down the length and the width of the plate. And maybe these interact in someway. A large bell shape probably has many more, circumference, height, modes connected to the shape, a low frequency flexibility resonance for example

The patch diagram below shows a simple experiment I did in two dimensions, and the sound clip reveals the  results! The delay line filter controls are both open, the two delay lines set to delay times in the order of a few milliseconds and the overall loop feedback adjusted to just below the point of a sustained oscillation. The same noise impulse from the envelope generator circuitry, just as in the string model, was used. This patch produces the distinctive sound of a struck bright metallic plate or gong. The sound is rich and extremely lifelike.
2 Delay plate gong patch.
[SOUND EXAMPLE 4:  Sounds from a simple plate gong... MP3 (128kbit/sec)].
[SOUND EXAMPLE 5:   Sounds from a collection of various gongs and bells... MP3 (128kbit/sec)].
So what’s happening?
        When the two differing delays are connected in parallel, then two resonant loops, of delay time (d1 longest say) and (d2), are set up. Next there is a third ‘free’ one produced too, the time taken for a figure of eight pattern round the two delays (d1 + d2). From the split output connection  That’s not all. Also each time the noise pulse goes round the parallel loop the signal is being split and then delayed by different but fixed amounts and added (mixed) together again. So the original noise pulse signal split just once when added becomes two signals separated by (d1 – d2); a fourth ‘free’ side product. This is then looped back. There are four time delay products at work, d1, d2, (d1 + d2), (d1 – d2), produced during each couple of loops. This is accumulative!
 It seemed to me that the realistic plate or gong sound is the result of this myriad of delayed time additions and subtractions which are organised by the resonance of each loop product. So the sound evolves as the initial noise impulse becomes progressively more averaged, this time into ordered harmonic and inharmonic series, having an additive and subtractive relationship: d1 and d2, (d1 + d2) and (d1 – d2). If you’ve programmed synthesisers, maybe this looks a little bit familiar?  There is an interesting similarity to the process of ring modulation; which merely produces a fixed static series of sum and difference components from the harmonics of two input waveforms. With the modelling patch, the sound evolves dynamically.  

Fiddling around with the controls, a wide variety of struck plates, bars, bells, gongs, tin lids, small, large resonant and dead can be produced. Reducing the filter roll off frequency softens the sounds considerably into a more wooden quality. It can also be made more drum like. Using the inverting output of one or more of the delay lines causes little change to the basic bell quality, but there is a definite change into a more hollow sound, presumably due to the lack of even harmonics in the sustained part of the sound. Softening the attack of the noise impulse can produce brushed effects, changing the delay time can slide the pitch;as in sound example 5.
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Using a modular analogue synthesiser