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Is it possible? Yes, it is possible to make equivalent positions sound very much like their standard counterparts. But I may be completely wrong here, this is just guesswork on my side. E.g., if the harmony is F major, and you need to play an F3, its overtones would resonate better with the chord if you play it as 4th partial 6th position rather than 3rd partial 1st position.
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Though I'm far below that level, I suppose high level players might be able to use these differences to their advantage, to select "alternate" positions based on musical demands. Is this possible, or even desirable? I've always thought the different partials have slightly different sounds, due to differences in the overtone series, which are due to the physics of the instrument, and thus not something we can control 100%. Paul Faulise uses a similar idea in his routine books by suggesting that bass trombonists use all possible equivalent slide and valve positions to develop flexibility and familiarity with equivalent positions. I use equivalent positions into my warm-ups and scale routines. I believe this is like Joseph Alessi’s “tone cloning”: making every note you play sound exactly like the note you just played. But actively using them to make them sound the same is the only way to develop facility with equivalent positions. Yes, the tone quality to the player will be slightly different, and the horn’s feedback is different. I approach equivalent positions by listening for tone quality and intonation. I don’t have Buddy Baker’s book or Tom Malone’s book. I was fortunate that one of my teachers stressed learning and using them, but also not to think of them as “alternates,” but as equivalent positions. I believe one of the difficulties with “alternate” positions is that they are taught as alternates, almost like second class citizens. I really would have like to have seen Willie Dennis in action, for instance, as I can't think of a way to do his stuff without all the positions. Some parts of the horn have a remarkable performance envelope, others not much. It is interesting how much can be done without many alt positions. Things sometimes don't work well down 5 or up 4 so finding new things of the same sense (inversions with other modifications) can make it happen in the same neighborhood. It may be that a little chromatic thing can make the move in an interesting way. Things to try include working a pattern in one tonality, like dominant, and then end on some part of tonic somehow. And then stuff like 53 42 is subject to endless variation. Others, like 1357 2468 kind of things not so much. Some things work almost everywhere, like turns. So I try to capture the "sense" of it with position/note/rhythm modifications.
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Some stuff just does not work in some places. In addition, as I work patterns, usually scalewise, I will be flexible with their interpretation at different places on the horn. It's a great way to get your ear in tune with them. Really low bar alt practice is to use them in bands on the easy parts. I’d say practice and learn using alternate positions as part of your regular practice routine to improve your ability to use them “on demand” Over time, you can default to alternate position more naturally as using them becomes “second nature”. If I am sight reading I seem to default to whatever I have been able to ingrain for scales etc. I will be more analytical / strategic as to which positions facilitate music expression if I am working on a piece of music I am learning and have the luxury of time to practice it. I typically practice improvised patterns using alternate positions such as some arpeggios and broken scales that lend themselves to “fretting” using muscle memory and a different starting point for the pattern.