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Sorry but it becomes a bit technical
The problem is a PDF can contain both vector objects (maybe text, maybe graphics) which can scale to any size or an embedded 'picture' a
bitmap which will be a specific original size in pixels.
If you can open your PDF in a PDF viewer, (sorry I do not have a windows one around) and check the properties you might find the designated paper size.
This example is A4: Just made it and the text (notation) is a vector. Most likely your music will be an embedded image.
Gimp is a bitmap editor, works in pixels, not vector.
Any vector is converted to a fixed size when you first open the PDF. Gimp will use the value
100 ppi unless you change it to higher resolution. This for example the A4 sheet at
300 ppi to give printing quality.
ImageMagick is no different, the
-density 300 setting is the equivalent.
Of course if the PDF contains the notation as a low quality bitmap 'picture', and you say they are images (of unknown quality) then no amount of adjustment will improve it. In that case it is try-and-see but generally try 300 ppi to start with.
If you can share one of the sheets - dropbox or similar file-share - then it will be possible to give more specific advice.