There are some ways to raise the pitch of the flute overall. Proceed with caution!
The most hazardous is tweaking the embouchure and I highly recommend trying this as you will ruin your flute. However, the maker should be able to adjust it if it is within a certain size range. Making the embouchure hole larger will raise the pitch. On my flutes I aim for a dimension of 11.4mm lengthwise and 10.2mm across the width. My embouchures vary a bunch from this actually. But that seems to be a comfortable size that plays easily without much effort for my flutes.
Avoiding that (don’t do that!!!), there are ways to adjust the pitch overall which are less hazardous. The bottom D can be adjusted in 3 different ways. If the bottom has 2 vent holes the one closer to the embouchure can be enlarged. A more effective way is to simply move the footjoint up. This would require a lathe. If the flute doesn’t have the extended foot then simply subtracting some material from the bottom will shorten the sounding length and raise the pitch. Finally if one has a tapered reamer of the right size the bore at the end can be reamed out a bit.
Once retuned, the octave Ds are checked and adjusted with the cork position. Pushing the cork in has the effect of raising the 2nd octave more than the first.
Then the notes are raised, starting at the E, first octave. If flat the simple remedy is to make the hole a tiny bit larger using a narrow round file or a bit of 120A sandpaper wrapped around something metal the diameter of a bamboo skewer. One should rely upon one’s ear at this point, as well as a tuner (see the paragraph below on tuning with a tuner). Then the F# should be addressed. Good luck on some flutes getting it up higher. You want to avoid making it too large! Sandpaper wrapped around an Xacto Knife handle works well for this note. Then you keep going up the scale.
When you get to C#, be sure to check the highly important c natural fingering too. 0XX 000 in the first octave and 0X0 XXX in both or just the upper are standard though 0XX XX0 works.
Note that when you raise the pitch of a hole, the pitch of the next hole will go up too though to a lesser degree.
Then check the 2nd octave by first blowing each 1st octave note starting with E. If the 2nd octave is flat, the hole needs to be undercut a little bit. This can be done with the sand paper wrapped around the skewer sized rod.
Tuning with the tuner. At least on my Korg Tuners, some notes look sharp when they are in pitch to the ear. A is always about 10 cents sharp, as well as the notesabove it. The 2nd octave is another 15-20 cents sharp. This has more to do with how the tuner hears the flute.
An aside - and a possible explanation for the “Flat Bottom D” that is encountered in antique flutes. The techniques for adjusting the bottom D are little known to the average flute consumer and for most, beyond their capabilities. However, as the pitches went from 435 or less to 440 the need arose to sharpen the flutes to that pitch standard. Thus they went after the embouchure, in some cases destroying the balance. They most certainly (and quite skillfully) adjusted the fingerholes. Its hard or impossible to tell if these are original as cut in the factory. I propose that the bottom D is an indicator that these modifications have been done. They just couldn’t adjust the most important note! But this is within the capabilities of most makers and there are a lot of us it seems these days!
Casey