"Let’s say I’m playing the first registration of a song and I decide to change tempo. Does that mean I have to go to registration 2, change tempo, save, then go to registration 3 and again change tempo, save?"
If I'm understanding the question, yes, it means that when you originally set up the bank, you will need to set up #1 like you want to start, then make your tempo change, and save that to #2. You can use FREEZE to keep anything else from changing when you use it. Or you can elect to have other factors change. FREEZE is a very powerful function.
Of course Yamaha's archaic system means you have to resave the BANK before you leave it or your changes will be lost.
It's the best way I found to do songs such as Suspicious Minds without using a midi file.
And if you plan to also use any of the other registrations for that song, you'll need to save the proper tempos to them too. Then resave the damned Bank again.
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Also, as I recall, Yamaha doesn't have a button for Tempo Lock, so be sure that parameter is set properly, since it is difficult to do on the fly.
You will probably make a boo-boo or two, but once you have it done, the work will be over.