I’m thinking of 3D printing a tapered/parabolic headjoint for some cylindrical PVC flutes I’m making to correct for the flat upper 2nd octave. I know I can use a wedge and have had limited success with it in some of the flutes I’ve made but I also want to explore this option.
Is there any kind of guide out there on the ratio of taper and how much of a parabolic curve it it needs to have? I’m having little luck finding any information about this. I fully plan to experiment to get the right shape but want to have some sort of guide so I don’t waste too many print runs.
Interesting question! If you’re interested in how tapers work, Arthur Benade did a lot of work on that. The most intuitive model is his W Curves (which paddler has previously mentioned in C+F posts):
There’s a classic paper by Benade and French that I took some data from and formatted in musical units (cents) rather than physicist units (Hz) that will give you some idea of the effect of the Boehm taper:
Benade notes that the effect on pitch of a small change in the bore is proportional to the ratio of the before-and-after cross sectional areas. So if you know the overall effect of a particular taper that might give you a rough guide of how much to tweak it…
As many others have noted, the tapers aren’t really parabolic. The relationship between the taper and the affected notes is not a simple one (I really wonder how Boehm developed it!).
One thing to keep in mind is that Boehm flutes have very large, uniform holes. I think flutes with smaller, finger-covered holes will probably need more correction/taper.
If you have a wedge design that seems to work well, maybe you could convert that into a taper design. You would want a taper that removes an equivalent cross-sectional area to the wedge at each point. Or, you could start with a Boehm-like taper, then experiment by reducing the cross-sectional area along the taper by, say, 5% and see where your 2nd octave ends up. Just some ideas…
As Terry and Rob have pointed out, the head tapers are not actually straight tapers or parabolic, and they tend to vary from one manufacturer to another.
I have a Rudall Carte Radcliff flute in my collection, which is a cylindrical body, tapered head flute similar to a Boehm, but with different fingering. I have profiled the head bore. The data at the link below might be useful to you. You’ll see that there is a definite step in the head bore. I did not expect this, but I have measured it carefully several times and its definitely there. Also, this flute is made of ebonite, so unlikely to have shrunk or warped much.
One important issue that I think is worth mentioning is that the point at which the embouchure hole is located along the taper is critical. You should experiment with this a bit once you have chosen a taper and made a reamer. The insertion depth of the reamer relative to a pre-drilled embouchure hole will make a difference.
Hmmm, the bore graphs in the article I posted make no sense. They are not Boehm head bores. Something went wrong somewhere, I need to work out where. Sorry about that. But the graph Paddler links to in his response is the kind of info I had in mind.
I’ve read some analyses (can’t immediately recall which ones) that suggest the taper is only significant to the center of the embouchure hole, and the remaining area up to the cork acts as a Helmholtz resonator where only the total volume is significant. I’d be interested to know if anyone has read anything that suggests otherwise…
The units are mm. The length of the head is 220 mm. In the spreadsheet data the last 0 got truncated, so the length measurements there are cm. Sorry about that. It is unnecessarily confusing doing something like that, but I never originally intended to share it with anyone.
Anyway, you can see the overall length is 22 cm which is 220 mm, so the data does make sense when you take that into account. The location of the embouchure hole is in mm from the 0 point in the table. The end cap is not included, nor relevant.
Whoops, sorry, silly me. There is nothing wrong with my bore graphs, they are just presented R to L, rather than in the conventional way L to R. I did that as I wanted all the far end of the tuning slides to line up to make it easier to compare the tapers.
On page 10 of this paper on Acoustic Impedances of Classical and Modern Flutes it make the following statement, which I think is consistent with your statement above:
Further, the chimney tube plus the closed end of the flute tube together constitute a Helmholtz resonator, this is in parallel with the rest of the instrument. Its impedance rises with frequency, and it has a resonance at several kHz, which then dominates the Z(f) curve. It also decreases the frequencies at which the minima occur.
Yeah I was able to infer that the X axis was in cm and the Y axis was in mm so the chart made sense to me, though I’m wondering if the diameter measurements are of the outside diameter or the bore? If it’s the outside diameter, what’s the wall thickness?
Both Paddler’s and my graphs show bore diameters. The body bore (and therefore the tuning slides) around 19mm (3/4”), the heads tapering down to around 17mm around the embouchure. Generally I might expect outside diameters around the embouchure area to be about 27mm, giving a wall thickness around 5mm, but I haven’t checked that assumption.
Yep, thanks Paddler, that makes perfect sense. A typical head for a conical flute would be 27mm outside diameter and 19mm or thereabouts inside diameter, giving a 4mm embouchure hole depth. But since the bore diameter at the embouchure for Boehm bore heads is around 17mm, the OD can afford to come down a mm or two as well.
A 26mm OD on a 17mm bore would give a 4.5mm embouchure depth. So a bit deeper than the earlier flutes (better tone) and a slightly smaller OD (fits better under the lip). A win-win!
Of course, we can find many exceptions to the 27mm OD. My Geo Ruddall Willis Fecit is 29.3mm and I have a Nicholson at 26.5mm. I’m sure I’ve seen some even thinner, maybe 26mm? I wouldn’t like to go over 27mm.
You can see too why I came up with my Eccentric bore idea. Push the bore down a little below the centre of the head and you can have individual control of embouchure depth and OD! Turns out you don’t need to go too far in this.
On a poly tubing flute you could achieve that by building up the OD of the head at the embouchure hole by gluing some larger tubing around it. Then sanding down some or even all of that protrusion on the underside of the head to get the OD that you feel best about.
But maybe I’m getting ahead of the immediate aims!