# Making a “transverse trumpet” (and mostly failing)

In the field of mathematics and music, I work mainly on mathematical models of musical structures such as chords, rhythms, etc. but acoustics is also something that interests me. And so, during the summer vacations I had some time to deal with a little project of mine which began some eight years ago, namely building a “transverse trumpet”. It looks like this:

At the time, I even applied for a patent, although to be honest it was more because I was interested in the process of writing a patent and applying for it than expecting to be making any money from this idea.

Before I continue, I would like to point out two excellent books on the acoustics of musical instruments:

• Acoustique des instruments de musique“, by Antoine Chaigne and Jean Kergomard. This one is French. It gives a very thorough mathematical treatment of the acoustics of various musical instruments (clarinets, flutes, brass, percussions, etc.).
• The Physics of Musical Instruments“, by Neville H. Fletcher and Thomas D. Rossing, which is not as mathematically detailed as the previous one, but focuses more on intuitive explanations and details about musical instruments.

The inspiration for the “transverse trumpet” comes from two musical instruments, the baroque trumpet, and the koncovka. Both are overtone instruments, meaning that the different notes which can be played with these instruments are obtained only by excitation of the different frequencies the main resonating body without modifying its length, for example by the use of valves or holes.

The baroque trumpet is well-known in western classical music. Here are some typical examples:

It is basically a long tube, about 2.5m in lengh (twice the length of modern trumpets), with a mouthpiece at one end and a bell at the other one. For acoustical reasons which I’m not going to detail here, the combined effect of the mouthpiece and the bell leads (when the trumpet is properly designed) to a complete harmonic series (except for the fundamental), i.e. the second and higher resonances have frequencies which are in the proportion 2:3:4:5… The baroque trumpet player accesses these resonances by modulating the tension of his lips.

If we take the fourth resonance as a reference point, then the next four resonances are in ratio

• 5/4, i.e. a pure major third,
• 6/4=3/2, i.e. a pure fifth,
• 7/4, i.e. a harmonic seventh, something between a major sixth and a minor seventh, and
• 8/4=2, an octave.

This five notes are typical of fanfare or military trumpet calls. If we now take the eighth resonance as a reference point, you can check that the next eight resonances form a scale which is almost diatonic, albeit with some peculiar frequency ratios (for example 11/8, which is higher than a usual 4/3 fourth). This register was used during the Baroque era in various concertos, and because the frequencies get very close together, this meant that baroque trumpet players had to have perfect control on their lips to avoid hitting the wrong notes. In fact baroque trumpet players at that time were were very well-paid virtuoso. For a perfect introduction to the baroque trumpet, check out this video, from the Youtube channel Orchestra of the Age of Enlightenment.

The koncovka is a Slovak instrument which is much less well known in comparison. It is simply an end-blown flute without any holes between 40 and 85 cm. The originality of koncovka playing is that notes are obtained by controlling the amount of air blown in the mouthpiece (resonance selection), and closing or opening the other side of the flute. Check out this video for some example of koncovka playing.

By the acoustics of open pipes, if we draw the acoustic wave pressure curve along the length L of the koncovka for the various resonances, we obtain the following diagram (first five resonances represented). The mouthpiece is supposed to be at the right: recall that since it is a flute mouthpiece, it is open and thus the pressure at this point is null. In the open configuration, the pressure at the other end of the flute is also zero.

Now if we close the end of the flute, the value of the pressure at that point will be maximum, and we obtain the classic resonance series of a pipe closed at one end, which is represented below.

You will notice that the series $f_1,f'_2,f_2,f'_3,f_3,...$ have frequency ratios which are identical to that of the baroque trumpet above.

And now we come back to the idea of the “transverse trumpet”. I was particularly interested in the idea of a brass overtone instrument, but building a baroque trumpet from scratch can be quite difficult. The hardest part is probably the bell, which is usually made from a brass sheet rolled and bent to the right shape. I tried, and, well… never again. The koncovka idea of alternatively closing and opening the end of the pipe was interesting but you obviously can’t do that if you already have a trumpet mouthpiece at one end. Contrary to flute mouthpieces, brass mouthpieces actually correspond to acoustic pressure maxima. In other words, if you want to excite a resonance with a brass mouthpiece, you have to ensure that you’re doing it at a pressure maximum. And if you look at the diagrams above you will find plenty of them along the pipe length: hence the idea of putting the mouthpiece transverse to the pipe, which can then be opened and closed at one end.

The picture of my transverse trumpet above corresponds basically to this diagram.

It is appr. 3m long pipe obtained by soldering brass tubing (0.5mm thickness, 16mm diameter) with standard copper plumbing parts. The mouthpiece is put at a fixed distance of 21cm from one end of the tube, and the other end can be closed or opened with a finger. Below are the diagrams with the acoustic pressure curves in both configurations, along with the theoretical frequencies.

 Opened Closed

The theoretical frequencies are calculated by considering a straight pipe with no T-branching, so it’s only an approximation of the real case since the mouthpiece and the T-branch are expected to modify them.

By now, you probably already have noticed why this instrument is doomed to fail. The first pressure maximum only coincides with the mouthpiece position for a narrow range of frequencies: when this is not the case, the resonance is not properly excited and the note is very bad, if it can ever be attained.

In practice, I managed to hit $f_3$ to $f_9$ in the open configuration, and $f'_4$ to $f'_8$ in the closed configuration. The frequencies measurements gave the following values: 222 Hz, 278 Hz, 331 Hz, 389 Hz, 442 Hz, 496 Hz and 550 Hz for the open configuration, and 245 Hz, 305 Hz, 364 Hz, 419 Hz, and 469 Hz for the closed configuration. These frequencies are in good agreement with the theoretical ones.

I’m sure you are waiting for some audio, so here are some sound samples of me trying to playing this instrument. First, the opened configuration:

Then, alternatively opening and closing the tube:

First, the sound is deceptively similar to that of a brass instrument. The resonances at $f_3$ and $f'_4$ are not very stable, which is not surprising considering how we excite them, but in the above resonances I found that the notes were locked easily and the transitions between notes could be quite sharp. Also, I’m more a trombone player and I’m not used to trumpet mouthpieces so I couldn’t hit the high notes easily.

Overall, this is not going to be a success so I don’t think I will be spending a lot of time on it in the future. There are a couple of observations and remarks I’d like to make as a conclusion:

• This is certainly no expert work in the making ! Because I’m using standard plumbing parts, the connections are very crude and not at all optimized for a musical instrument. For example, the T-branch could probably be adapted so that the mouthpiece would be connected in a better way with the main tube. Also, the U-turns are the worst one could come up with: the turns are very sharp and it is known to influence the standing wave inside the tube. It would be better to bend a straight tube as it is done on trombones and trumpets (alas, one does not bend brass tubing as one bends steel, especially when the tube thickness is so small, I’ve learned that the hard way… there is a reason why you put ice or sand in it before bending).
• One could think of adjusting the tube length, so that the position of the pressure maximum always coincide with that of the mouthpiece, for example with a slide. I’ve made some experiments which suggest that if the slide is at the far end, it could well be possible. In theory, it could also be possible if the slide is at the end which is closest to the mouthpiece, but I’ve found that there seems to be an optimal length of tube on that side with respect to sound quality: if it is too long, the sound becomes muffled, although I can’t really explain why.
• Also, I’ve tried putting some additional lengths of tube and holes on the far end, so that one could complete the scale and get a chromatic one.
• I guess changing the diameter of the pipe along its length could be used to control the standing waves of the different resonances and thus the position of the first pressure maximum, but I have no way of building such complex shapes.