Basic Calculations for Flute Making

Below is a medium length description of the details; the reader is advised
to skip it if you're not interested.

There are a few important physical quantities you need to
keep in mind.  First, let's define some measurements:
  2a = bore diameter in the vicinity of the hole
  2b = hole diameter
  2s = hole spacing (e.g. from this hole to the next open hole)
  t  = physical length of hole, e.g. thickness of wood
  te = effective acoustical length of open hole, approx. te=t + 1.5b
  x1 = distance, in a conical bore, from the hole to the hypothetical
       apex of the cone.

Benade defined some length corrections for various cases, and I will 
list these below.  These length corrections describe flattening effects
which must be subtracted from each hole's idealized position (measuring
from the mouthpiece end).

First, if only one hole is open, then define D as the distance from 
the center of this hole to the open end of the bore.  The effective
bore is not perfectly open at the hole, but appears to extend beyond
the hole by a distance Cs ("s" for single hole)
                    te
  Cs =    ----------------------------
          (2b/2a)^2 +  te*(1/D + 1/x1)

If the topmost open hole is followed by at least one more open hole,
then the bore from the open hole downward can be usefully represented
as a semi-infinite lattice constructed of similar holes and bore.
The open-hole lattice correction, Co, is

  Co = 2s * (0.5) * [(1 + 4(te/2s)(2a/2b)^2)^0.5 - 1]

But, if there are any closed holes _north_ of the first open hole
(i.e. in the main bore before the first open hole), they also provide
a flattening effect.  Each closed hole adds the correction Cc

  Cc = (0.25) t (2b/2a)^2

Another very important physical quantity is the "cutoff frequency" fc.
This marks the boundary between low frequencies, which are reflected
by the tone holes back into the instrument to form strong resonances,
and high frequencies which leak freely out through the tone hole row.
The cutoff frequency also marks the division between the low-frequency
isotropic radiation pattern (energy radiates in all directions equally)
and the high frequency directional radiation patterns. 

Anyway, fc can be estimated by
        c    2b       1
  fc = --- * -- * -----------   
       2pi   2a   (te*2s)^0.5

_IF_ this expression is evaluated for all the tone holes and the numbers
are consistent across the instrument, _THEN_ this is a useful 
approximation.
Virtually all reasonable musical instruments with tone holes share this
feature.

So, if for example one were going to design an instrument from scratch, 
here's a sketch of the process:

1. Calculate the ideal lengths of the bore for each note, assuming no
   holes and the bore were going to be cut off clean (and no end effects)
2. Choose a cutoff frequency (fc=1500 Hz is a good start for treble
   instruments; it scales pretty well with the base pitch of the 
instrument).
3. Choose tone hole sizes (2b) and spacings (2s) which are manageable
   by the hands of the player, consistent with fc.
4. From the bottom of the instrument, working up, apply the corrections
   for each hole relative to the ideal lengths, starting with Cs for the 
   first hole and then Co after that.
5. Now you pretty much know the rough layout, so now calculate the closed
   hole corrections, adding up the effects for all the closed holes for
   each fingering.
6. Now repeat steps 4 and 5 until you don't get much change from one 
   iteration to the next.
This works quite nicely on a spreadsheet, for example.

Good luck.  Feel free to  drop me a 
line if you have any questions. Hoekje@uni.edu