Acoustic Modelling 7.....
Now things take a new direction. The trumpet model was arrived at after a lot of thought and difficulty. The main difficulty of course was in trying to visualise, in the form of connection of electronic modules, the workings of the trumpet. In particular, it was the mouthpiece mechanism, which seemed to be all important. How to just extract the essence of it in a simple patch became the main problem.

In trumpet playing, the player the tenses the lips and then blows air through them starting the lips to vibrate. The initial air pressure forces open the lips. The area of the aperture formed increases and the resistance to air flow falls proportionally. There is a point reached when the increasing stretched lip tension overcomes the force of the outward rushing air and the whole thing collapses again, forming a rapidly and wildly oscillating human air valve. When the trumpet is being played, basically a large bore resonant pipe is placed over the lips valve, to resonate in sympathy at frequencies that correspond to multiples of the length of the tube. The strong resonant modes of the tube makes it possible by varying  air pressure and lips pressure to alter the basic general resonant frequency of the lips valve to be attracted to, home in and pick out the overtones of the tube. This is what makes the brass sound.

What happens in detail in the pipe is the initial air pulse allowed through the lip valve travels (at the speed of sound) down its length, where it suddenly discharges into free air. The sudden big pressure drop into free air partly travels back up the pipe as a negative pressure wave where it meets the mouthpiece mechanism again. Air pressure conditions here are very changeable however depending on how the lip position is regulating the incoming airflow at that instant. A further reflection of some kind will therefore occur from the pressure disturbance and be grossly affected by these airflow conditions. This does 2 things. It generates harmonics as the amount of reflection varies greatly with the lip position and the enriched wave goes back down the pipe again to discharge once more into free air. The whole cycle then repeats generating further delayed harmonics on that until it dies away. If the timing of lip oscillation and tube delay roughly coincide the result is a resonant loop of overlapping and yet decaying sound layers each being cumulatively modified by reflection  at the lips valve.  This is what basically makes that rich trumpet or brass sound.   
So how can a simple analogue model of it be made?  
The Trumpet patch...
In the patch diagram above, electronically, the ‘pipe’ is very simply represented by a single delay line module. If the trumpet’s pipe is roughly about 1 meter long, a round trip up and down the electronic pipe is 2m, so, at the speed of sound, it’s about 6msec of delay. The 2 filters on the right of the patch simulate the trumpet’s horn effect.  Lips are basically rather soft so the driving resonant system might produce something like a pure sine wave. So that’s what I used; a standard sine generator VCO. This is not quite right as the frequency of the VCO is not coupled to the resonance of the ‘pipe’, which is what happens in a real trumpet. It’s a good enough model though to get the sound mechanism to work and that is what I was after.

what about the valve? Well, a voltage controlled amplifier (VCA) seems to fit the bill. It can turn signals up and down depending on the value of the control voltage. If the VCA is turned (biased) partially on by the fixed gain control knob, any signal on the control input will either turn it on further if it’s positive or towards off if it’s negative. The elements that are affected by the ‘valve’ are the sine wave (lips) and the ‘air pressure’ signal coming back up the pipe from the delay module.  There are a few ways to patch this up but I originally just used the simplest and easiest way to very roughly approximate this. The two signals are mixed before going into the both VCA control input and its audio input. The amount of control  the VCA has over that mid gain bias setting is set by the voltage control knob. It can go from nothing to a ‘huge effect’ as it does in the sound clip below. The little switch enables the VCA to handle positive and negative control signals if required (the 4 is 4 quadrant operation) which is what’s needed here. There’s a bit more to it as lips have tension  which act as a limiter for the magnitude of the effect. I originally used the inbuilt limiter in the delay line as a crude way to do this.
The VCA is a multiplier’s with inputs say call X & Y. Let’s call the output Z and M the manual bias (gain knob), then mathematically the VCA is doing this….

  Z = Y( X + M )

In the patch, X and Y are the same signal, so..

  Z = X ( X + M )       or         Z = Xsquared+ XM

This is interesting because the squaring effect generates harmonics and it is also in the delay loop as well. Each time round the short loop (msec) the signal gets repetitively squared, producing an evolving but decaying cascade of higher harmonics. This is really what emulates a real trumpet and makes the trumpet model sound so rich. In real life it’s probably richer than this patch as the lips vibration and the tube resonance are not as rigidly fixed as they are in this rather over simplified model.  Any non-linear element will however generate layered evolving harmonics in a delayed loop. The type of non-linearity and the type of type (stopped, open , horn) and how they are driven or persuaded to self oscillate is what makes wind instruments sound the way they do.   
This is an evolving a trumpet sound from the sine wave as a starting point, slowly adding more of the VCA’s squaring action and delay pipe effect, then there’s a section of a fixed pipe length and glissandi sine tone sounding the series of harmonic resonances of the tube.
[SOUND EXAMPLE 7 : Trumpet sound and glissandi...  MP3 (128kbit/sec)].
[SOUND EXAMPLE  8: The human trumpet!... MP3 (128kbit/sec)].
Twiddling the knobs, the trumpet tone can be varied in several ways. Changing the amount of VCA bias on the squarer softened or hardened the tone. Changing the sinusoidal amplitude had a similar type of effect. Using the low pass filter in the delay line, had the effect of ‘widening’ the bore. Again softening the sound.
By altering this model and its parameters, it’s  possible to make simple models of other members of the brass family. There’s a reasonable tuba sound in the this demo.
After I got the basic trumpet  patch to work, I tried some radical things like replacing the driving sine wave with other waveforms. Surprisingly it didn’t affect the basic sound as much as I expected. Whatever waveform I used a strong characteristic trumpet type sound appeared. In the extreme case, I could even drive the trumpet with a human voice. In this sound clip a human voice is fed though the trumpet model...
[SOUND EXAMPLE  9: Tuba... MP3 (128kbit/sec)].
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Using a modular analogue synthesiser