I would assume "wood" as a 3D printer material would be a something like ground cellulose fibers in a suitable matrix. Not exactly wood as we know it in flute world, but maybe more marketable as closer to a natural material than a plastic flute?

my algorithms will accept a request for 8 holes but depending on the musical key, and constraints on finger stretch and hole-size it may not find a good solution. I haven't really tried yet.


I humbly thank you for the inspiring words. I am sure that as my first endeavor it is nowhere near the quality of a flute made with the proven traditions. But maybe a glimpse of what could be possible, and I am having fun with it.


haha. sort of. it is actually PLA/Wood composite it talks about. usually about 30-50% sawdust, but still primarily plastic. sandable, stainable, lower density. not especially easy to print with.

Some more thoughts on modelling and optimization...

At least one member here has no use for mathematical modelling, believing that physical prototypes are the only way to be sure what is actually going to happen. Another member sees value in mathematical modelling, but has no use for algorithmic optimization, believing that only an experienced flutemaker knows which direction to go to improve a design. I respect both of these people, and see merit in their opinions. Modelling and optimization still have their limitations, and those of us attempting them shouldn't be blind to those limitations.

In particular, what scope do you have for manual adjustments to a printed flute, post-fabrication?

The time taken by global optimization is inherently exponential in the number of dimensions. If you have a good starting point, for fine-tuning an existing design, a good local optimizer can be far more efficient than relying exclusively on a global optimizer. It's nice to have a choice of optimizers, to see which one gives the best results in a reasonable time.

In any case, stick with optimizers that respect bounds or constraints on the geometry, so you don't go chasing after physically impractical designs, like 20 mm finger holes. This is a particular issue if you want to design, say, an 8-hole flute where the bottom two holes are always-open vent holes. The tuning alone doesn't put enough constraints on where to put the vent holes or how big to make them, so you'll probably need to impose artificial constraints on the geometry to make the optimization tractable.

As far as post-fabrication adjustments, it is certainly something I would like to explore in time. For now, you could say I am trying to provide a solid foundation for myself that will get me 95% of the way there.

As far as local optimization, your point is well taken. Previous to the particle-swarm global optimization, I was making use of the L-BFGS local optimization technique. That had pretty good results most of the time actually and would only rarely get stuck in local minima. The main issue was that a well-conditioned Hessian matrix was difficult to obtain. Before that, I was using an unmodified BFGS but the memory requirements exploded since it keeps a historical record of the path taken to compute the partial derivatives of the objective function. in each case, I do use a box-constraint for the tone hole diameters to keep the search restricted. Each hole gets its own diameter constraint based on the finger intended to cover it. The finger-stretch is also defined per-hole, but in the objective function as an exponential error (so it has a little give but not much). You are right, tuning alone doesn't do it. Another part of the objective function tries to compromise between maximizing tone hole size and making all holes approximately the same size. Each error term is weighted and summed for the whole of the flute. sounds like we are on the same page, my friend. Hadn't thought of supporting vent holes, that is a good idea too.

The nice thing about the particle-swarm optimization is that it actually runs quite quickly by employing multiple agents that independently search for local minima and "communicate" their findings with each other to pick the best minimum it can. You can be sure that if the optimization becomes intractable, I will have no choice but to scale back.

Very interesting. How does the program adress certain fingerings? Like if you want to have oxxooo, oxxxox or oxoxxx for C natural. Or third-octave fingerings?
I optimized my own hole-layout according to finger length -- I moved the 2nd hole from the top a bit further down because of that. So the top 3 holes are not evenly spaced which makes the fingering feel more natural to me. I also try to optimize the lower three holes so that spacing is even. I think a similar hole size for all holes cannot be achieved on a cylindrical bore without having a very uneven hole-spacing for the lower hand.

My program does not address forked fingerings as you describe at all, yet. I feel it is still quite early in development - a quite simple flute.
for tone hole spacing, each has a minimum and maximum allowed spacing between the hole and the previous hole (if any). Each gets a different set of values per hole (based on my own ability to stretch). This also allows the program to put extra space between holes 3 & 4, for example, to place the socket/tenon (so the programmatic hole placement also determines the location of the "break" indirectly). And you seem to be right about the lower end, can't make that last hole larger without moving it further down away from the other holes (or deviating the bore or tone-hole height)
But, I am not directly controlling the layout as you are. For example, I could increase the maximum distance between holes 1 & 2, but that doesn't necessarily mean the program will move the two holes apart - just that it could, should the formula say it leads to a more "optimal" flute (with "optimal" being used loosely here - my program is still somewhat simple)

WIDesigner can handle forked fingerings. It builds a model of the instrument from the bottom up, a section of tube, a tonehole, another section of tube, another tonehole, .... with other pieces at the top for the excitation mechanism and the head space. Each tube section can be cylindrical or conical, each tonehole can be open or closed. For each note, WIDesigner uses the instrument model to predict how close the given fingering will play to the target frequency. This means the target frequency can be in any register, first, second, third, .... For example, it can work with OXXOOO as a useful fingering for high D on a whistle.

I know. But it seems much too complicated. I use just a simple app called "DIY flute" that runs on my phone to calculate holes. The tuning comes out great, even though the overall length of the flute or whistle doesn't seem to be right. So I first cut the embouchure, then tune the bell note and then drill the holes. By now (after about 6 yrs of instrument building) I have a pretty good idea which hole sizes work for my preferred fingerings. But WIDesigner is certainly great when doing other things than just making simple 6-hole practice flutes or tin whistles, like I do. For me the functionality is overkill. Even though I'd love to give it another try one day to make a flute with venting holes, as "DIY flute" is limited to 6 holes.