I have built the world's worst trumpet. The idea was to make an instrument that had natural resonances that are not even remotely close to being harmonically related to one another, and I was reasonably successful. My trumpet, which I will call the WW (world's worst), consists of a 39 oz Maxwell House coffee can with two holes punched in the bottom. I soldered an Olds Ambassador cornet bell in one hole and a piece of tubing in the other. I then slipped a Bach 25 leadpipe onto the tubing and replaced the plastic snap on cover on top of the can. There it was, a WW! The sound is just terrible, and if I squeeze the tube and the crook of the bell together I can produce an oilcan sound while playing.
The measurement setup was the same as I have used for real trumpets. The WW was driven by a small speaker glued onto the rim of a mouthpiece, which also has a tube soldered into a hole leading into the cup to which I attached a small microphone. I also placed a Shure BG4 microphone about 1/2" from the bell. I excited the speaker with a sine wave that was swept from 100 to 2000 Hz in 10 s and built up a spectrum, as measured by both microphones, by adding several sweeps. I did an analysis to search for small integral multiples and came up with a near 2 to 3 match (a fifth) between the 2nd and 3rd resonances, but nothing else. Figure 1. shows the result.
The upper trace shows the resonances as measured by the sound pressure level in the cup of the driving mouthpiece. The markers indicate the closest points that I could find to be simple integral multiples of some false fundamental ("harmonically related"). The overall appearance of the trace is remarkably like that of a real trumpet. The lower trace shows the output at the bell. What a mess!
I then played three notes that seemed to "slot" fairly well using a regular mouthpiece and recorded the resulting noise with high resolution. The next three figures show the resulting spectra. To make the spectra I averaged over a second or so of the central part of each note. Believe me when I say that a WW can produce some interesting transients when starting a note!
As you can see, the principal peaks of the spectra are harmonically related to the "played" notes, as they must be for any periodic sound, no matter how ugly. The three played notes can be identified with the first three natural resonances of the WW. I was disappointed that there are secondary peaks (suppressed by 20 dB or so) that are anharmonic, and therefore aperiodic. These must come from vibrations of the ends of the coffee can. The natural vibrations of a circular membrane are not harmonically related. The frequencies are related to the successive zeros of Bessel functions. That's what makes a banjo have its characteristic sound. Just as was the case with a real trumpet, the natural resonances are sharp compared to the "slotted" played notes.
The frequency of any natural resonances of a trumpet is not necessarily related in any simple way to the frequency of any other resonance. Indeed it takes skill and lots of trials to make them appear to be harmonically related, and it should come as no surprise that no trumpet is perfect. It is a mistake to call the successive resonances of the trumpet harmonics, or even overtones. Harmonics (and overtones) are, properly speaking, the spectral components of a sound and all these components are produced at the same time. We can get notes to sound (slot) near the natural resonances, and those notes will have harmonics that are not related to any resonances of the instrument.
Move over Dave Monette and Cliff Blackburn, the WW has been born!John T. Lynch
© Copyright 2001, John T. Lynch; Ralph J. Jones
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