How is a "false tone," e.g. on a tuba, formed

Apparently a tuba, as well as maybe other brass instruments, can produce a false tone a fourth above the fundamental, that is a tone with frequency $\frac{4}{3}f_0$. I heard one man do this with his voice too - that is, start on a tone and drop to a subharmonic an octave and a fifth below. How does this happen? It can't be a combination tone of two harmonics, because the frequency then could only be an integer multiple of $f_0$. Thanks!

From the player's point of view it is useful to describe brass instrument harmonics as having the simple frequency relationship 1:2:3:4:... etc (with some practical instrument-dependent corrections to the intonation made by the player, of course) but the underlying physics is somewhat more complicated.

A cylindrical bore instrument (e.g. a trombone, where the bore must be exactly cylindrical because of the slide) with no bell would actually have harmonics in the ratio 1:3:5:... etc. A uniform conical bore with no bell produces 1:2:3:... etc. This is basically the same as for woodwind instruments with reeds - for example compare clarinet (cylindrical bore) which overblows at the 12th (3x the fundamental frequency) with saxophone (conical bore) which overblows at the octave (2x).

The bell tends to lower the frequency of all the resonances, and therefore tends to convert the ratios 3:5:7:... into 2:4:6:... which are the same as 1:2:3... from a tube of half the length. This is the origin of the classification of instruments into "full-tube" and "half-tube", and explains why the some instruments can not produce the fundamental frequency as a "pedal note" - or at best, it is so out of tune as to be useless musically.

But of course real instruments do not have a uniform taper over the whole bore, and the changes in taper are also used to tweak the frequencies of the harmonics and improve the intonation.

Because of their physical size the tubas and euphoniums are likely to be the furthest away from the simple "theoretical" models, and the most obvious way to find out what this "false tone" really is (at least to an engineer!) would be to compare the measured the frequency response curves of instruments that have it, and with those that don't.

I'm not sure if anybody has published the results from that type of investigation specifically into "false tones", but the final plot in Figure 10 in https://www.grc.com/acoustics/an-introduction-to-horn-theory.pdf seems to show a (weak) resonance peak at about the right frequency above the fundamental. If that is the case for real instruments, there is no need to speculate about difference tones, subharmonic excitation, or other nonlinear effects - this is just a real resonant frequency of the system, but one that doesn't match the over-simplified theory of pipe acoustics taught in high school physics!

The original source of Fig 10 is apparently https://secure.aes.org/forum/pubs/journal/?elib=5981 - unfortunately behind a paywall, so I haven't had chance to read it. But the fact that the paper is from ISVR Southampton is a pretty good recommendation that it's not likely to be nonsense.