|Where should the holes go?
They can be placed with any number of criteria in mind: handability, aesthetic appeal, sound, etc. This page will look at hole placement from the point-of-view of 'Flat Spots'. It might be helpful to review the subject of nodes and tuning to get a grasp of why it might be advantageous to position the holes at certain locations and not others. Right between the flow and pressure nodes is what we might call a flat spot--a point of dynamic quiescence along the air column.
These locations present opportunity for hole placement because what happens in the vicinity of a flat spot has the least effect on sound. By 'happens' I mean any change in the geometry of the bore. If you want to do something to the bore (widen, narrow, drill a hole through the wall, etc.) and want your modification to have the least effect--do it at a flat spot. All of the following is intended to be applied to a tube--a straight non-varying bore. If the cross-sectional area of the bore varies then all bets are off as it becomes a harder problem to identify locations suitable for holes. That being said standard flat spots are a good place to start with any bore.
So where are these Flat Spots?
Figure 2 is a detailed 'map' of the acoustical properties of an air column, identifying the flat spots of the first and second octaves that are closest to the foot of the flute. Although the flat spots don't line up exactly, they're close enough to locate the holes between the two octaves. 'Traditional Holes' (above) indicates the hole location of traditional shaks--if there's such a thing as an average traditional shak. Relative to traditional, the first, second and third flat spot holes move up and the fourth and fifth holes are lowered. Placing holes in these locations will ensure the least adverse effect on the sound.
The idea implicit in Flat Spot Hole placement is that holes can have precise locations which correspond to the flat spots of the air column. Tuning is accomplished by sizing a hole rather than moving it. It doesn't work out that the holes are all the same size. The first three will be smaller and the last two larger. With this hole placement method, holes have an absolute location but not an absolute size. Flat spot holes have a tighter grouping that more traditional layouts which makes their use in longer flutes attractive.
See The Synthesis for a final flute design.