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I'm spending a lot of time sight reading piano music and one thing that is constantly trying me up is a scenario like this:

A note, D say, is sharp in the left hand. In the same bar, another D, an octave above the sharp one, is also played, this time by the right hand. There is sometimes a natural symbol on the upper note. I get quite confused at that point, since I was expecting it to be natural. I often mentally trip, looking back to see if I had played that D wrong if it ever came up before, since a natural, in my head, undoes an accidental.

First, is this a customary practice? Second, any tips on how to get over my mental hiccup?

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It depends on the style of the music, and also "how you are reading it". If you are reading one note at a time, then it might trip you up. On the other hand if you are reading in "chunks" in terms of the harmonies and chord progressions, it can be a very useful warning.

There are many different conventions used by different publishing houses for these cautionary accidentals - just learn to get used to them. This is just a selection of the options that one professional music notation application can apply automatically - music notation isn't as "simple" as it might first appear to be!

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The customary practice is to have accidentals apply only to the octave they're on. This custom is solid enough that Musescore automatically uses it. (I haven't tested with Finale yet, but I suspect it also uses that custom.)

To get over that mental hiccup, at least you found a good piece with courtesy accidentals that forces you to realize that accidentals apply only to the octave they're in (I prefer all my courtesy accidentals to have parentheses around them, though--does that piece in the question have parentheses around that natural sign?). Find more such pieces if possible and practice with them first.

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I find it quite a useful addition, although strictly speaking, unnecessary. An accidental only refers to other notes in the same octave, after it, in the same bar. However, some people prefer to analyse the music as they go along, and I hope I'd be forgiven for thinking that since one note was sharpened, the same name note in a different octave might just be too. Since the writer has put a natural, it underlines the fact that my supposition would be wrong. But it wouldn't be the first time someone had forgotten to sharpen that other note...

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Quick answer - yes, it is common for 'cautionary' accidentals to be used. Often they will be in brackets. Sometimes they're a matter of house style. Sometimes they're essential. We can argue over whether the lower E in this example technically requires a natural (probably it does). But I have frequently seen this sort of thing 'corrected' in pencil by students or teachers not tuned in to today's pleasantly astringent harmonies!

(I say 'today's'. This sort of thing has been going on for 100 years and more. But there are still teachers who live in a world of simple classicicism and hymn-books. And who would hear one or other of those Es as a 'wrong note'.)

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