Step 9: Further improvements: adding another higher octave
Again, the answer came from physics. How we can do that? Simple: creating an hole in the tubes.
Wait, don't take your driller too soon. We need some calculation, obviously.
First, if we make a single hole, its size doesn't matter. If you want to add two or more holes for each tube, than the things became complex. So we'll drill only one hole.
We can calculate the distance of the hole center from the open end using the same formula I've shown in previous step (see first image). This time we know the frequency (if you don't then go to the second step). All we need to do is take the frequency of the pipe, double it two times and calculate the value of L. For example, the D4 has a frequency of 293.7 Hz, so the D6 has a frequency of 1174.7 Hz. So the hole center must be 7.06 cm from the open end. If you make a large hole (like 5 mm of diameter), the approximation will be better than a smaller hole.
However, reaching the central tubes with your finger can be difficult, so again finding a way to keep closed all holes and open them simultaneously is a great idea. I'm thinking about the key system of a flute (or sax, or similar systems), but I'm not so good in making such precise works...
This method only works when you can also open the ends, because the note produced is noticeably different if the end is open or closed. This is due to harmonics produced by the section behind the hole. So, the best thing (maybe it's only a dream) is a mechanism that open both ends and holes with a single lever, and only the ends with another lever. This way you can have a pan flute whose range is, for example, C4-C7, three octaves!