In a video I saw, a guy was reviewing a prototype trombone, and he said that "the high f is out of tune" (the sixth overtone) I don't understand how a fixed length of tubing (1st position) can produce a out of tune overtone. For what I know, I was taught that a overtone's pitch is related to the length of tubing and that they are slightly out of tune by always a constant amount.
While the total length of a brass instrument is basically fixed, the ratio of conical to cylindrical piping is not set in stone for a trombone. It is true that the 6th partial is generally quite sharp, and needs to be lowered. There are several components that can affect the partials relative to one another, including:
- Mouthpiece cup depth
- Leadpipe (venturi) taper shape & speed
- Overall mouthpiece length
- Overall slide length
- Bore size of the slide
- Speed of the taper in the neckpipe
- Speed of the taper in the tuning slide
- How far out the tuning slide is pulled
- Shape of the bell stem
- Overall length of the bell section
There are some specific examples between variations in models that one can test... for example, a Bach 42 slide is overall around an inch longer than a Shires or an Edwards tenor slide. The stock tuning slide on a Shires has a taper that differs from many other trombones and often "fixes" the D in the 5th partial so it is nearly the same as Bb in the 4th partial. The reason for this is so the D in the 10th partial is in a position closer to 1st. This is achieved by having a taper of a particular shape.
That said, the easiest way to test is to push your tuning slide all the way in and compensate with your handslide. Observe how far out the different partials are. Then, pull your tuning slide all the way out. You will usually find an exaggerated version of the pushed all the way in. This is due to having a longer overall length and possibly some additional taper tubing put into play depending on if the horn has a reversed tuning slide or not. Note that TIS (tuning in slide) horns generally do not have such a variation. This is because the mechanism in the slide is during the conical section. Further addition of length of the instrument which has a uniform bore size does not alter the ratio of conical to cylindrical tubing and therefore the effect upon the overtone series is diminished if not eliminated. However, horns like this are not very common when compared to tuning in bell horns but I believe the merits/demerits of bell vs slide tuning are out of the scope of this question.
So when someone says that a particular partial is "out of tune", what it means is that it is more out of tune than it usually would be. So while these partials are nearly universally out of tune, the degree to which each one is out of tune to one another is not fixed between horn to horn. Nor, actually, on the same horn in different positions. As you move the slide out, the bore size of the instrument increases. In 7th position (on a horn with no F attachment), there are several feet of additional tubing of the inner diameter of the outer slide which is much larger than the inner diameter of the inner slide. Combined that the further out the slide is, the lower the base partial is, so positions get wider the farther you go out. So in 7th position, you would expect to have an exaggerated difference if you were to play, say, a B natural in the 6th partial vs. playing an F in the 6th partial.
This is a massive topic that I'm not really qualified to talk about, but "length of tubing" is a highly idealized model. In actuality, every bend in the tube and every change in bore diameter will knock things away from this ideal. If an antinode for a harmonic sits close to some irregularity like this, the resonance will get modified in some way.
A lot of our 'knowledge' of overtones comes from textbooks describing the behaviour of a perfect, massless string or of an air column in a perfect cylindrical tube. Real instruments are more complicated than that!
The wonderful thing about the trombone is that, unlike trumpet or other valved instruments - there's no great virtue in aiming to build it 'in tune'. EVERY note is infinitely and constantly tunable by the main slide. Even 1st position - slide 'right in' - is normally against a spring stop so that a 'short 1st' is achievable. The seven positions are merely nominal. On any particular instrument the player soon learns where to put the slide to make any particular note in tune.
The ideal model of standing waves in an ideally rigid tube, and of waves on a string, etc are very good at predicting data but nor perfect. They are ideal models. Not only do slight deviation from these models cause differences between actual harmonics and predicted harmonics but the systems don't obey the ideal physics used to solve the equations in the first place. They may match a different ideal prediction.
One case in point is the vibration of stiff rods and plates. They obey a 4th order differential equation, as opposed to the 2dim equation known as the wave equation. The "harmonics" of these systems do not follow the sequence fn = n*f1, which is the equation we want to hear, or build to. You can find examples of stiff systems in xylophones and marimbas for example. But in fact the phenomenon is also present in strings. The body of the brass horn is more like a bell. You can look up bell overtones but in short many of these systems have harmonics that do not follow the simple sequence stated above. Their overtones are mathematically predictable but they are rather "out of tune" relative to what we are used to hearing. These systems can be modified to bring the overtones closer to the desired harmonic sequence. For example bending of carving out some part of the vibrating body will change the intonation of the overtones. In the case of the marimba the resonance tubes that amplify the sound are probably designed to pick up the fundamental, but will vibrate as well.
The idea of a resonance tube is that the tube is supposed to act as an ideal rigid boundary forcing the air to have specific resonance modes. In fact the tube itself will have resonances for exciting vibrations in the actual body. Those may be more like the overtones of a bell or stiff vibrating body and not fall in tune.