Here is my philosophy for adjusting the head cork or plug - and a peek into my tuning and voicing style:
I learned early on in my career from observation that the position of the head joint cork or plug on these conically bored flutes should be set to where the tone quality is best. On my flutes this ends up being a deeper setting (up to 25mm) than what is considered the standard setting on Modern flutes (19mm).
19th century authors on the flute such as Rockstro considered the ideal position (center of embouchure to face of the plug) equal to the bore diameter, which has remained the orthodoxy since. But even he mentioned better tone can be had in the 1st octave bottom notes by moving the cork farther out.
In general moving the cork out flattens the 2nd octave with respect to the first while making the first easier to play and richer in tone. Moving it in destroys the bottom and makes the 2nd octave and higher octaves easier to play. The Cuban Charanga players take this practice to an extreme and move the cork so that it is 3.7mm (on average) away from the edge of the embouchure, which they have commonly enlarged. This allows playing in the 3rd and 4th registers but the bottom registers are trashed. Its no problem as they never use these lower registers!
On a modern flute made in metal with its cylindrical bore and large tone holes there is no way to adjust except by the compromise position of the cork. On our tapered wooden flutes however, we have the wall thickness and generally smaller holes which allow us to work with some additional parameters. I never adopted the orthodoxy and have always used a farther out position for the plugs on my flutes to benefit tone quality. I suspect this was also the case for some of the 19th century makers such as Boosey, Prowse and Rudall.
If you set your plug at 19mm after cleaning or oiling the bore, you may have simply moved it away from its “factory” setting.
When tuning up a flute body, I first voice the embouchure or used an already voiced head joint, and size each of the holes to pitch in the 1st octave by first drilling these undersized and then reaming each individually to pitch. Increased diameter = sharper. The hole size seems to control the 1st octave pitch and as long as the diameter isn’t increased by additional tasks (undercutting and finishing the surfaces) the 1st octave pitches remain stable. For the bottom D the pitch can be tuned by keeping a strict attention to sounding length (521 on my Standard flute). It can also be reamed from the end on a 6 hole flute to sharpen if necessary and on some bores this improves the tone. If all else fails, length can be removed from the bottom end of the foot joint. I frequently have to do this when prototyping. On a flute with an extended foot joint, increasing the size of the two vent holes will have a similar effect but not be as effective. In this case the scaling may be off. This flat bottom D phenomenon could be solved by simply moving the foot joint up. But this means shortening the middle joint (or lower middle joint on a 4 piece flute) and recutting the tenon. Sometimes the tenon will be a few mm shorter than the corresponding socket. If so, making the tenon use all of that socket length will make a difference.
At this point the 2nd octave will play flat, especially in the upper notes of the scale which are more sensitive. The 2nd octave is tuned by undercutting each hole to the degree needed. The more undercutting the sharper the 2nd octave pitch. Smaller holes (E and A) will be more sensitive to these changes than the larger holes. The amount of wall thickness will also have a bearing on this. Thinner walls seem to tune in with less undercutting sometimes. Undercutting also improves the overall tone quality, evenness and response of the flute.
Note that you should rely on your hearing to some degree instead of an electronic tuner to determine pitches. The tuners hear a flute in a way that is different than humans. What will look in pitch in the 2nd octave will actually sound flat to my ears. The 2nd octave should generally look 20-30 cents sharp. Also, the upper notes of the scale (B, C# especially) should also look sharp. People sometimes contact me saying their flutes are out of tune based on their tuners. I ask them if they have a modern flute and many do - and have them play that into the tuner. They discover a similar degree of deviations from pitch on the tuner and are surprised.
We should be playing for our ears, not some arbitrary electronic device!
So if on a flute that is otherwise working well but some notes seem sharp in the 2nd octave, there are some strategies that can be used. Pushing the plug out can bring down these higher notes and commonly increase the resonance of the instrument. But if these notes remain sharp in the 2nd octave, they can be adjusted down by careful application of resin to fill in some of the undercutting. Hot Stuff Superglue works well for this. I pour some onto a piece of plastic, and then use a bamboo skewer to pick some of it up and paint the inside wall of the finger hole and build it up in layers, testing as I go (and being careful not to glue my fingers to the finger holes!).
Just some practicalities for other makers: for undercutting tone holes some sandpaper (Garnet A120) wrapped around a narrow round file works. In my workshop I do the final shaping of the undercutting this way. But to rough these to shape I use a reverse tapered bit (approx 1/4" diam) at the end of a #45 Foredom handpiece for the larger holes and a 1/8" cylindrical bit for the smaller holes. Some holes respond better with a straight-sided undercut versus one that flares towards the bore. There is no general rule as one works with different bore shapes. Its best learned by iterative practice - tuning and voicing literally hundreds of flutes!
Casey