The wavelength of the sound produced is determined by the flute tube and tone holes. For a pipe of length L, open at both ends, and ignoring end effects, the wavelength of the sound is twice the length of the tube. The frequency produced is given by dividing the speed of sound (345 m/s) by the wavelength.
The method described here is to estimate an effective length for a real (cylindrical) tube taking into account end effects, the size of the tone holes, etc.
If the fundamental (all tone holes closed) corresponds to an effective length Leo, then the desired effective lengths for a six (or fewer) hole flute to produce notes of the (Western major) scale are given by:
Units of Leo
The actual physical length of the flute will be shorter than Leo and the distance to the tone holes (from the blow end) will be shorter than the values calculated in the table.
As described by Benade, the end effects (at both the blow hole and at the first open hole) act (approximately) as an additional length which must be subtracted from the calculated effective lengths to get the physical lengths desired. As long as you don't deviate too much from typical flute proportions, these approximate calculations should get you pretty close. If you are serious about getting your flute in tune, plan on making at least two flutes.
First, make your flute with no tone holes and adjust the length
to match the desired lowest note. The blow hole is typically
1/2 to 2/3 the tube's inner diameter, and the stopper will be
about an equal distance from the center of the blow hole.
Then get out your ruler and measure the following:
Lb can be determined approximately by:
The actual position for each hole will depend on the tone hole size and the position of any other open tone holes. Hence, you have some leeway to choose one or the other of these. Smaller tone holes will give a mellower sound and larger holes give a brighter (and louder) sound. Your holes do not need to be all the same size, so it is possible to make some choices which affect the ergonomics of your flute. Below I will assume that all the tone holes are fixed in size, and only their positions are to be adjusted. If this results in tone hole positions which are uncomfortable (or unusable) then adjust one or more of the tone hole sizes and recompute.
Now compute the first corrections:
For each of these calculations, you will need to repeat them several times. For example, for the first tone hole, an initial value of D is used which was computed from the previous approximate positions. Once the correction is found, compute the new value of D and recompute the correction. Continue until the answer doesn't change much. For the other holes, recompute s using the corrected position to obtain a better correction. Do this one hole at a time. If you correct all the holes, then go back and redo the calculations for all the holes, you can get into numerical trouble in some cases.
Now drill your holes. If a note is a bit flat, you can enlarge the first open hole (from the blow end) a bit to sharpen it. If you drill your holes a little small to begin with, you might be able to bring your flute into tune without having to make a second one.
Note: if you have particular thick walls on your flute (like some wood flutes) you will need to add another correction. Use
Now check the tuning of your flute, and estimate adjustments to make your second flute. That is, if your note is 3% flat, move the hole 3% closer to the blow hole, etc. A frequency counter is best for this, but if necessary, you can do it "by ear" (e.g. by comparing to another instrument which is in tune and listening to the beats). If you see systematic problems (e.g. Re is a little flat, Me a little flatter, and by the time you get to Ti it is very flat) then your value of Lb can be adjusted to fix all the holes at once.
For a fipple flute (e.g. a recorder) the correction at the mouth can be approximated by
If you just want to calculate, or need to check your calculations, then see the