Some Background

I became fascinated with how the overtone series, or harmonics, relates to how brass instruments function. Most trumpet players (or brass, really) should notice that as they play higher notes, they become closer together on the staff. What fewer might notice is the pattern, where each octave that one goes up, there becomes twice as many notes (played without valves), and that each newly available note comes in between the notes of the previous octave.

This led me to notice one last thing: trumpets can't play the fundamental frequency for their overtone series. The lowest note one can clearly produce (without using the valves) is a middle C, but the next note up is a G, clearly not the 2nd harmonic. This means that the lowest clear note for a trumpet is the 2nd harmonic, not the fundamental.

However, I have also played the flugelhorn, which has the same tube length as a trumpet, and the same overtone series (the 2nd and 3rd harmonics are still the same frequency C and G as on a trumpet). On a flugelhorn, I found that you can also comfortably drop an octave below middle C, which is the fundamental frequency for the instrument.

The question:

Why can a flugelhorn clearly and easily drop to the fundamental, when a trumpet can not? The main functional difference between the two instruments is that a trumpet has a cylindrical bore while a flugelhorn has a conical bore. Why does this difference in bore have such an impact on the range that each instrument can play? I know that the cylindrical bore has a "warmer sound", but I don't particularly care about that for the purposes of this question.

I did notice a section of this Wikipedia article mentions "whole tube vs. half tube" in relation to the ability of some brass instruments' ability to hit their fundamental, but I am interested in a more in-depth explanation of why that is important.

Visuals of the instruments in question

Trumpet (cylindrical bore)

Flugelhorn (conical bore)

You can clearly see the difference in tubing between the two instruments.The trumpet's diameter only widens along its final approach to the bell, while a flugelhorn's diameter widens along the entire length of its tubing. Otherwise, they play nearly identically (outside of the thing that sparked this entire question).

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    $\begingroup$ I don't have a detailed answer, but it is to do with the pipe width varying along the whole length. I have played a tuba (varying width) and a trombone (constant width) and the tuba easily sounds the fundamental while this is hard on the trombone (though it can be done with some effort). $\endgroup$ – John Rennie Jan 14 '20 at 7:31
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    $\begingroup$ Wikipedia's "whole tube/half-tube" description just sounds wrong to me as a physicist. $\endgroup$ – user4552 Jan 14 '20 at 17:30
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    $\begingroup$ I came across a paper a couple years back that studied standing wave resonance in curved tubes. They were able to correlate an analytical model with experimental results. In short, curvature slightly bends (shifts) the frequency that would be predicted in straight tube harmonics. Looking at the two pictures perhaps this might explain the difference. $\endgroup$ – docscience Jan 14 '20 at 17:43

In a simplified model, the tube is effectively closed by the player's lips at the mouthpiece end. For the theoretical case of a straight tube with completely cylindrical or conical bore, and no bell, the harmonics of the conical bore have frequency ratios 1:2:3:4.... but the cylindrical bore only has the odd harmonics 1:3:5:7....

Those facts are most obvious in woodwind reed instruments. The clarinet has a cylindrical bore and the first harmonic is at the 12th (3 times the fundamental frequency) but almost all other instruments (e.g. the saxophone) have conical bores and the first harmonic is at the octave (2 times the fundamental).

It is also apparent in organ reed pipes, where the pipes designed to imitate trumpet tone are literally conical along their entire length, with no bell.

In "cylindrical bore" brass instruments, the large bell reduces the frequency of each harmonic compared with the simple theory, so the actual frequencies change from 1:3:5:7:... to something close to (a number less than 1):2:4:6:... The actual shape of the "cylindrical" tube is carefully designed to make the higher ratios as close as possible to 4:6:8:... and so far as the player is concerned, these are equivalent to the 2:3:4:... of a conical tube of twice the length. Hence the description of "half tube" instruments which are (approximately) half the length of a "whole tube" instrument that from the players point of view sounds the same harmonics.

However the fundamental frequency can not be accurately "modified" in the same way as the higher harmonics. It is always physically playable, but it may be so out of tune, and its frequency very unstable and depending on the exact tension in the player's lips when blowing, that it is musically useless and hence described as "unplayable" in practice. The embouchure for playing the fundamental may be quite different from playing the higher harmonics, so there is no practical use for learning how to play a note that is completely out of tune in any case!

It's worth noting that the physicist Helmholz who founded the modern study of acoustics in the 19th century got his theory of brass instruments completely wrong, because he didn't realize that the player's lips effectively formed a closed end to the tube and not an open end. And his reputation as a scientist was so great that it was half a century or more before anyone seriously questioned his incorrect theory! Of course musical instrument makers had been constructing practical instruments for centuries before that, and musicians had been playing them, with no theory at all except the empirical observation of the frequencies of the different harmonics.

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    $\begingroup$ Interesting information about Helmholtz! I have seen texts stating that the mouthpiece would be an open end, and I did not understand how this could be the case. $\endgroup$ – Pieter Jan 14 '20 at 7:56
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    $\begingroup$ @Pieter in real life, the mouthpiece it is somewhere in between open and closed, but "closed" is a better simple approximation. The exact shape of the mouthpiece, which tapers in the opposite direction to the bore of the instrument, is also critical, but outside the scope of simple theory. $\endgroup$ – alephzero Jan 14 '20 at 8:00
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    $\begingroup$ That's fascinating. However, @alephzero , you mention that the trumpet will be playing the even harmonics, starting at 4, which will be "equivalent" to starting at the 2nd harmonic and playing all of them. My question is why both instruments will play in the same register, since they have the same length of tubing, if the flugelhorn's 2nd harmonic is actually the trumpet's 4th harmonic. $\endgroup$ – fyrepenguin Jan 14 '20 at 15:22
  • $\begingroup$ I have certainly noticed what you say in the second-to-last paragraph, where one can somewhat produce the fundamental, but it doesn't feel like a solid note. Notes have a tendency to "lock in", except for the fundamental on a trumpet. It's fascinating how similar a flugelhorn and trumpet are, yet have completely different properties. $\endgroup$ – fyrepenguin Jan 14 '20 at 15:25
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    $\begingroup$ It's pretty amazing what humans achieved before we actually understood the underlying mechanisms, like millenia of folk medicine prior to germ theory, animal breeding before genetics, etc. $\endgroup$ – Barmar Jan 14 '20 at 16:22

Yes, it is the conical bore which makes that the boundary conditions near the mouth piece are like those for an open-open cylinder. Here is an explanation with many links.

I like to think of it as the mouth piece in free space, in the center of an imaginary sphere. Now put some imaginary radial conical walls in this sphere. Those won't affect the longitudinal motion of the air in spherical waves from a point source.

I encountered this when I tried to devise a course lab experiment with a plastic vuvuzela trumpet and got initially puzzling results...

  • $\begingroup$ Mouth-piece in free space... $\endgroup$ – Wookie Jan 14 '20 at 20:44
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    $\begingroup$ Can you add some explanation to make this answer more self-contained? $\endgroup$ – eric_kernfeld Jan 15 '20 at 19:27
  • $\begingroup$ For instance, if trombones and euphoniums use the same mouthpiece, how is it that they should have different boundary conditions at the mouthpiece? $\endgroup$ – eric_kernfeld Jan 15 '20 at 19:28

I started on trumpet more than fifty years ago, and have played all the brass in school bands. I can play the fundamental, but of course there are unreachable notes between fundamental with all valves up and second harmonic with all down.

By the way, a trumpet’s second with all up is B♭, not middle C, and with all down, the E below that.

Fundamental is more difficult, but whether the tone merits the adjective “clearly” is opinion-based. In fact, after listening to the Marsalis solo cited in another answer, I would say he played the fundamental very clearly.

  • $\begingroup$ It's not totally opinion based, as there is definitely a difference between playing the fundamental on a flugelhorn (clear tone, easy to hit, note easily locks in) and on a trumpet (muddy tone, not clearly a pitch to hit). I agree that you can sort of produce a note on a trumpet below middle C, but it definitely has a different feel when playing it as opposed to every other note. $\endgroup$ – fyrepenguin Jan 15 '20 at 0:52
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    $\begingroup$ Well, if we don’t want to allow the lowest register, the lowest second harmonic note (as written) is F#. Middle C is all valves up. $\endgroup$ – WGroleau Jan 15 '20 at 1:36
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    $\begingroup$ For the purposes of this question, you can consider the valves irrelevant, as they don't affect the point in the harmonic series that the instruments are playing, only the pitch actually produced. $\endgroup$ – fyrepenguin Jan 15 '20 at 5:19
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    $\begingroup$ @fyrepenguin: Then you should mention in your question that you ignore the valves. I was confused by your claim that it's not possible to play a lower note than middle C. I'm not a good trumpet player by any means but my low F# comes easily and sound pretty clear. I simply say "toh" instead of "tah", with all valves down. $\endgroup$ – Eric Duminil Jan 16 '20 at 9:49
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    $\begingroup$ @EricDuminil I've clarified it in the question. I mentioned that I wasn't talking about valves at one point, and I was had hoped that referring to the overtones/harmonics that it would be understood that I was not dealing with the valves. However, I see your point that I had not explicitly stated that, and have now fixed that. $\endgroup$ – fyrepenguin Jan 17 '20 at 16:37

I think the premises of the question is wrong. You can indeed play the fundamental on a trumpet. It is easier on a flügelhorn compared to a trumpet. If you have noticed the difference, you might also have noticed that the difficulty of playing the fundamental on a cornet lies between that of the flügelhorn and the trumpet.

Just listen to Wynton Marsalis hitting the fundamental while playing a cadenza in Haydns trumpet concerto (@ 5:55).

It has to do with the depth of the cup in the mouthpiece. As a rule of thumb, the deepest mouthpieces are used on the flügel horn. Quite deep is used on cornets, and the shallow mouthpieces are used on trumpets. The fundamental on a trumpet is more easily played on a deep mouthpiece.

  • $\begingroup$ You shouldn't write an answer for a comment just because you don't have the rep to comment. (That's the point of limiting it.) In this case, though, I think you have an answer to the question. Whether the community thinks it's a good one or not, we'll see. $\endgroup$ – Brick Jan 16 '20 at 15:34
  • $\begingroup$ I've slightly restated the title, for clarity, as I do state in the question "Why can a flugelhorn clearly and easily drop to the fundamental, when a trumpet can not?" which is slightly at odds with the question title. I've now restated the title to include "easily", which I will agree is more technically correct. $\endgroup$ – fyrepenguin Jan 17 '20 at 16:39
  • $\begingroup$ Also, I don't think I've ever heard someone hit that note so clearly and (apparently) easily. I seem to recall playing a flugelhorn with a trumpet mouthpiece as well, and still didn't have trouble hitting the fundamental. I'll agree that the depth of the cup probably helps, but it isn't everything. I've never played a cornet, however. $\endgroup$ – fyrepenguin Jan 17 '20 at 16:41

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