In the Cimarosa Concerto, which two of my students were just working on for our year-end recital, there's a passage that repeatedly leaps the octave from middle C to high C. The fingerings are easy but those two notes are both terrifying ones to try to play in tune. Both have extremely short fingerings, vibrating only a few inches of the wood of the oboe. Both are VERY flexible, such that a tiny amount of embouchure motion can easily make them very sharp or very flat. Normally, my students slur to the high note flat, then immediately correct to be in tune, because they have great ears.
That's not the way to really solve the problem, though, is it? In the context of a long note, you can hit it too flat, correct it quickly, and spend lots of lovely time playing it in tune. But if the note is short, as it is in the Cimarosa, and you try that, the correction time itself becomes a much higher percentage of the note and we as an audience become aware of the effort. And unimpressed with the oboist.
The way to solve the problem in your practice room is to NOT adjust the bad intonation. Make the leap, place the note where you think it goes, and just sit there. Analyze whether you are too high or too low, and by how much. Then go back and try again. Predict where you want the note to land, go there, and sit on it so you can see if you are right or wrong. Once you have the feel of the interval, start making the leap faster and faster, but unless you are actually in performance DON'T adjust a bad interval, redo it so it's right. It's the interval you need to practice, not the correction. No one wants to hear the correction.
I attended a group cello recital last weekend. Full disclosure: my daughter PERFORMED in her FIRST EVER cello recital last weekend, and knocked it out of the park with Hot Cross Buns on her 16th size cello. Played the whole thing through without stopping. Took a bow. Was visibly proud, as was I.
In other words, I was 100% prepared to love everything about the hour and a half of student performances, and I did. It's scary to perform in public, but no one cried or ran away and piece after piece went perfectly nicely and left everyone smiling.
It's interesting, though - when you watch student players you begin to realize what's hard about the instrument. The difficulties that professional players don't let you see. What I really noticed was shifting and intonation - because it's exactly the same thing I'd been working on with my own students. It's just much more visible on the cello.
To make a big leap upward on the cello you need to move your entire hand and finger position to a different place on the fingerboard, without any frets or keys or buttons to guide you. You just have to know, on this expanse of wood and string, precisely where to go, and, unsurprisingly, many of the students we heard did not. It takes a ton of practice and experience to hit the notes accurately, especially in a performance situation when nerves come into play.
The students with the best ears corrected themselves right away. They'd hit the note out of tune, and immediately wiggle their finger around until it sounded right, and then keep moving through their piece. Exactly what they SHOULD do in performance. Perfect manners.
But not the way, ultimately, to solve their intonation problem.
I pointed this out to one of my oboists, who was attending, and watched his eyes widen as the problem he'd invisibly been fighting in his mouth became visible, and apparent. It's nice to be able to SEE your way through the problem you've been working on.