Not to belabor the obvious, but the euphonium and trombone are designed to produce sound. To become proficient with one of these instruments, it’s helpful to understand a little bit about the subject of sound.
There are three aspects of sound we need to be aware of.
First, there’s the physics of sound; how it works whether anyone is listening or not. The answer to the old question, “If a tree falls in the forest and no one is around to hear it, does it make a sound?” is “Yes.”
The falling tree creates vibrations in the air. Those vibrations obey the laws of physics. Physics doesn’t care if anyone is listening.
The physics of sound is called acoustics. During a program of private lessons, you would be introduced to more information about acoustics and related topics in the context of building skills on your instrument.
Second, there’s the human perception of sound. Our ears and brains don’t directly process sound as it exists physically. When sound waves reach our auditory canal, the vibrations are converted into signals that travel to the auditory cortex.
Exactly how we experience those sensations depends on factors such as our health, the sensitivity of our ears, our memory, and our psychology.
What we experience as “hearing” is a little different from what’s happening acoustically outside our bodies.
Interestingly, what we hear in our own heads when we speak, sing, or play an instrument is different from what other people hear.
Third, there’s the way sounds are formed and sustained inside our instruments and inside our oral cavity when we play.
This topic relates to the design and acoustical properties of the instruments we play.
This is a particularly important topic for euphonium players to understand, due to the intonation problems inherent in valved brass instruments. Those problems are exacerbated in larger instruments, such as euphonium and tuba.
Without going into too much detail, the issue is that the valve slides are of fixed length, while we need to add a proportional length of tubing to lower the pitch by a given amount. That is, we need to add more length in the lower register than the higher register.
So, the valve tubing lengths are a compromise - too short (sharp) for low notes, a little long (flat) for high notes, and more-or-less okay for the middle register. The intonation differences are easier to hear in larger instruments than smaller ones, so the problem is greatest for tuba and fairly significant for euphonium.
It’s important to understand the design trade-offs the instrument maker made in creating the model of euphonium you are playing, in an attempt to enable the instrument to be played in tune. The valve slides are of different lengths in different models - three-valve, four-valve, five-valve, and four-valve compensating designs.
In addition, instruments with different bore widths have slightly different acoustical properties. Different mouthpieces fitted into the receiver will affect intonation and how free-blowing the instrument is. On euphoniums with adjustable receiver gaps, different gap settings will affect intonation and other properties of the instrument.
The player must make adjustments specific to the type of euphonium as well as their mouthpiece and their embouchure to ensure good intonation. The adjustments may include selecting an appropriate mouthpiece, adjusting the receiver gap (if possible), lipping the pitch up or down, setting the tuning slides to appropriate positions, and choosing alternate fingerings in selected musical passages.
Besides all that, the intonation of the instrument changes as its temperature changes and as you warm up and then grow tired during playing sessions.
Musical acoustics is a multidisciplinary field that includes elements of acoustics, human perception, instrument design, and musical concepts. There’s a good overview on Wikipedia here: Musical Acoustics on Wikipedia.