Stability. To be 'stable'- unwavering. Consistent. Steady. Like a foundation of a house, or a pyramid. Environmental variability will not cause a change in course. I define stability as the ability to remain at a level of homeostasis regardless of the presence of factors of variability.
As a musician (particularly a wind musician), we need an embouchure (when the muscles in our face create a stable formation that can allow a consistent flow of air to pass through an opening (aperture), which creates a consistent vibration in the lips, thus a consistent sound). The muscles in the face need to be coordinated in a very specific manner that allows for consistency as well as flexibility.
This entire concept of stability is a matter of scaling. If we think about anything that is stable it has to be relative to time. A bridge is stable, unless you are referring to a period of a million years. In that case, a bridge is actually not stable at all- it is quite flimsy. For our purposes as musicians, we can relate to things that seem consistent over a period of seconds or minutes as this most closely relates to the amount of time in between breaths or tiny breaks in musical lines. In the grand scheme of life, the amount of stability we need is actually quite small.
As a brass musician, I need to be able to play tones that are unwavering in frequency or timbre or volume. They need to have clarity and efficiency. I shouldn't have to work too hard to make these sounds. I noticed a funny thing- I have been playing trombone for decades and only recently discovered how bad I am at playing really stable tones. I basically have NO stability! In fact, as I work with my students, I am starting to discover that almost none of them have stability in their playing either.
I have experimented with a basic sustained tone. 6 beats of sound and 4 beats of silence (playing 60% of the time) and highly accomplished students with the best of intentions start to notice fatigue after about 40 seconds. Wobbly embouchure, significant jaw movement, wavering pitch, compromised tone quality. It's painfully obvious. It's funny how we play long tones every day and manage to find a way to be 'successful' at long tones but still have no stability. I guess it is not surprising when you think of it- most of the actual demands of playing the trombone only require stability for a second or two before changing notes or having a break (rest or breath) in the sound. No wonder many of us do not have stability!
I have paid keen attention to great players as of late for this very reason and I started noticing the best players do have great stability- they way they play is so smooth that even notes that change pitch etc. appear to be all 'the same' as if there is no movement in the face. I am not sure if they meant to develop stability or not, but they DID, and if you want to sound like them, stability is one skill you need to master.
The body is designed to operate under variable constraints and can usually deal with a changing environment to produce consistent results (product over process). It is fine when the system accommodates tiny adjustments and variability (that is most likely imperceptible), but if you want to play at the highest levels, you have to have an exceptional balance of stability and agility such that there is no audible involuntary deviation from your sound. That means being able to play sustained tones with no 'funny business' in your face. Minimal shifting around in your embouchure in order to sustain tones. It makes sense that this would also help agility as increased shifting around while you play, the more cumbersome it becomes to move around the range of the instrument (thus less agility).
Here's what I started doing to develop stability
You have to start small. Set the metronome at 120 BPM. Take a note that is not 'high' per say, but toward the higher part of your mid range (I started with D4- if I went any higher it would start to sound bad pretty quickly). Play a D4 for 5 beats and rest for 5 beats. Do this for 30 seconds. A minute. 90 seconds. At some point it will become difficult. You'll start to hear, feel, or see (If you are using your phone or a mirror) issues that arise due to microfatigue. The muscles in the face need to be coordinated so specifically in terms of how they work that the slightest measure of fatigue will cause compensation in the weirdest ways. This will start to work against what we are trying to train- true stability!
Embouchure is still a four-letter word
My students know that I do not believe in talking too deeply about the embouchure and even though we are talking about training the face to minimize changes to the embouchure, I still do not think it is worth getting into details. The main concept to address is noticeable changes to the embouchure. You will most likely start with an efficient and productive way to produce sound. That's fine- you don't need to get too specific. Just be aware when the embouchure starts to 'collapse' and do weird things, it is time to stop. If you keep going you may ingrain poor habits and those habits may prevent you from being able to do what thousands of other people can do on a trumpet or a trombone, etc. You have to develop sound in a way that enables you to cover the demands of the repertoire.
Be specific in your efforts
We are talking about the coordinated effort of a very small number of fibers that need to work in a very specific way. Our methods need to be specific. Rather than trying to over-analyze what the face is doing, just keep it simple and minimize the change over the range from high to low or low to high. Keep accurate track of time as you measure your performance in stability exercises and you will notice tiny improvements. Improvement will be slow- like throwing one penny in a jar every day. Get used to it.
Here is a video where I explain stability
Here is a free PDF of the chart I referenced in the video.