Why A Trumpet Works the Way It Does
by N. Drozdoff
In several previous articles I've discussed the acoustics of closed end and open end tubes and the overtone series. It is now time to tie this whole discussion together to gain an understanding of brasswinds.
As discussed in those essays, a closed end tube only produces odd multiples of the fundamental or lowest resonant note. A trumpet, however, produces all integer multiples (or, rather, very close to those). Does this mean it is actually an open end resonator? It can't be. As soon as you put your lips up to the mouthpiece you clog up one end of the horn. It is definitely a closed end resonator.
The catch is that a trumpet is not a cylinder! We must be careful when we consider the old tradition of calling a trumpet a cylindrical bore instrument. There is, in fact, only a very small amount of cylindrical tubing in a trumpet. The mouthpiece, from throat opening to the end of its backbore has a very serious taper. The leadpipe has a definite yet gradual taper. There is a venturi at the receiver. The bell section has a very obvious taper. Granted, these tapers are not necessarily conical (based on a cone), but they are anything but cylindrical. All of the previous discussions were based on cylindrical tubes. This oversimplification is necessary to gain a basic understanding of how sound waves (pressure waves) travel in a pipe and reflect off of the ends of the tube thereby providing the basis for resonance and the overtone series. The trumpet is, in fact, a very sophisticated acoustical device. Through the geometric trickery of introducing those tapers the cylindrical resonance regimen can be adjusted to very closey aproximate the resonance regimen of an open end pipe. This is necessary in order to be able to make an instrument that will work with only the three valves. An example of an instrument that doesn't make effective use of this sort of geometric trickery and therefore requires many more holes in order to "fill the gaps," so to speak, is a clarinet.
Let's analyze, qualitatively, what happens as we gradually add taper to a cylindrical closed end resonator. Consider the following closed end series (aproximate pitches). I've tried to comment on their relative tuning compared to our sense of pitch in contemporary western music.
These pitches were lifted directly form John Backus' book,"The Acoustical Foundations of Music," page 263. There is a typo on that chart. He had mis-labeled the line A5 as C6. I have corrected for that here. Of course I am making the assumption that the pitches selected in the chart are, in fact, correct. My perusal of the chart leads me to believe that this is true and that the only typo was the misplaced label, C6.
The addition of a mouthpiece lowers the top six pitches pitches (harmonics number 9, 11,13,15,17,19)thus bringing them into tune with our accepted scales yeilding the following new or adjusted series:
Obviously the addition of the mouthpiece was to lower the upper modes to fit the expected open ended series. Now we need to fix the bottom notes.
Now let's add a bell section by "flaring" the end of the existing tube (note that we are not adding any length to the tube as we did with the mouthpiece). This affects the lower modes by moving them up. The modes raised are 1,3,and 5. The new series arrived at are as follows:
Now if we look at this carefully, we can see that this is the standard overtone series for a low B-flat trumpet. The only overtone that comes through unscathed from our geometric trickery of adding a tapered mouthpiece and flaring the end of the pipe into a bell was the original 7th mode of the closed end cylindrical pipe.
Now we all know about pedal tones. In Claude Gordon's famous book he tells the student that the pedal C (B-flat) is very flat and must be lipped into tune. If we look at the final adjusted series we find that the lowest note is an E2, not a B-flat 2. The rest of the series follows an approximation of an open ended series, but this lowest note seems out of place. This fact explains Gordon's comment. This lowest note is indeed the "fundemental" of an "adjusted series". It is, indeed, much flatter than what would be the fundamental of a true open ended series. The low B-flat 2 is what Backus refers to as a "fictitious fundamental". When we play a pedal C on a standard B-flat trumpet it is in fact very flat because it is supposed to be. Lipping it in tune is forcing a non resonant to come out. Now, I am not saying that this is a bad exercise. In fact, I make a practice of playing entire scales without using the valves as an exercise in developing my "acoustic driver" (my chops).
There is another way of thinking of part of all of this. A trumpet is a closed end resonator with an adjustable acoustic length. That is, the effective length of the instrument is dependent on the note being played. Due to Newton's Law of Inertia the lower notes reflect farther back in the horn. The very lowest notes reflect back near the crook of the bell section. The leakage of the tone occurs at that point. As we play progressively higher and higher notes the reflections occur farther out in the bell section. Eventually, as we play even higher notes the reflections would technically be occuring beyond the end of the bell. At this, point, of course there are, indeed, no more reflections. The trumpet doesn't resonate any more. It has turned into a megaphone. This occurs around a high G or so (in trumpet speak).
This concludes my discussion on the acoustics of brasses. I hope I have been able to put into reasonably understandable terms some rather abstract concepts. Understanding how our instruments work can help in ways that we might no initially imagine. Personally, I have been able to solve a great many problems with my playing and my equipment by coming to a clearer understanding of the acoustics of my instrumet.
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