# Scaling an object to change its key

I used a 3D Printer to make a kid's Recorder for my son. It plays surprisingly well. However, it's not tuned to match the common recorder used in schools. Using a phone-app tuner, the base mouth-piece section of the print blows a (very high) A, while a Yamaha school recorder blows an A♭. Sadly, my son lost the rest of the print and I only have the mouth-piece section now.

One of the nice things about 3D printers is you can scale your models. I'd like to scale this model so it will be tuned closer to what we see widely for sale. It doesn't need to be perfect — these are elementary students and great intonation is never gonna happen — but I want it close enough to avoid really bad dissonance.

The question is in two parts:

1. Is this a reasonable thing to do (will scaling the whole model result in something playable, where things like finger holes are still in the right places)? Alternatively, would I do better to add some length to the tube, and if so should it be before, after, or in between the fingering holes?

2. What ratio (in percent) should I scale the part up to tune down from the A to the A♭?

• @Tim The ratios on string instruments are simply the frequency ratios, so each half-tone step is 2^(1/12) but I believe wind instruments are different, because scaling them up will increase not only the length but also the width en height of the air column inside them, so you'd get a squared or cubic ratio. Anyway, see e.g. astro.pas.rochester.edu/~aquillen/phy103/Labs/FluteLab.pdf Jun 15 '19 at 18:03
• @YourUncleBob I believe it's the length only that determines the frequency. The diameter of the bore affects the tone and perhaps the relative ease of blowing overtones. If you look at the photo at moeck.com/en/service/faq/the-recorder-family.html, you'll see that the tenor and bass are just about twice the size of the soprano and alto, respectively. Jun 15 '19 at 19:20
• It's a 19 hour print, and the pieces are narrow but tall, which makes them failure-prone. I'm up for some trial-and-error, but only if I have some good math to back up the attempts and converge quickly on a reasonable solution. Jun 15 '19 at 19:37
• I have absolutely no experience in building flutes. Why am I allowed to vote on answers. ;) Jun 15 '19 at 19:51
• as far as I know recorders can be tuned - almost a halftone - by adjusting the mouthpiece. you could at least try to make only the piece between a little longer to test it. Jun 15 '19 at 20:23

You certainly need to make the complete instrument before you try to scale it, as another answer said. Unless the design is an accurate copy of the Yamaha recorder, the pitch of the head joint on its own could be anything. In particular, it will depend on the design of the joint between the head and the rest of the instrument.

To scale an instrument "properly" is a very complicated task. For example to make an instrument that plays accurately in tune, the position of the finger holes depends on their diameter, and professional instrument makers tweak such things by applying single coats of varnish to change the shape of the hole, not by changing the shape to the precision of a 3D printer!

Note that in the pictures of the printed recorder, the finger holes are not all the same size. That is not an accident - they need to be different sizes to make the instrument play in tune. But whether you can print the different sizes accurately enough is another question.

So there is little point worrying about such details, and you might as well scale everything by the same amount. To lower the pitch by one semitone, increase the size by a factor of 1.059. The finger holes should be repositioned in the same proportion.

If you want to be slightly more ambitious, you could apply the same scaling rule that is used to make organ pipes have the same timbre over a wide range of pitches. To achieve that, the high pitched pipes have to be relatively fatter than the low pitched ones. A typical scale factor would increase the diameter (both internal and external) by 1.044 per semitone, compared with 1.059 for the length.

This sounds like a fun project, but you are never going to produce a professional-quality instrument this way, so don't set your standards unrealistically high!

If you wanted to scale the design by a larger amount, for example to make an alto or tenor recorder instead of a soprano, then simple scaling is not feasible. If nothing else, you will need to move the finger holes so the player can reach them, and adjust the intonation by changing their diameter as mentioned earlier. There is no real alterative to either practical trial and error, or very sophisticated computer modelling - and the modelling usually only gives a starting point for trial and error, in any case!

• There are online calculators for simple flute designs now (see comment under the question) but turning the calculations into a perfectly-tuned instrument is probably far from straightforward. Jun 15 '19 at 22:55
• @YourUncleBob but these links all only apply to Querflöte (flute), not Flöte (recorder), so it’s going to be more tricky. I guess this instrument is less loved than it deserves. Jun 16 '19 at 18:18
• Exactly -- don't assume head joints need to be any particular pitch, especially in the world of Thingiverse :-) Jun 17 '19 at 15:08

Have a look at the picture at https://www.moeck.com/en/service/faq/the-recorder-family.html. The instruments are indeed (roughly, at least) scaled proportionally in all three dimensions. I would do that, especially since you're talking about a small change of scale rather than printing an alto or tenor recorder from plans for a soprano.

I would try to measure the frequency of your recorder playing an A. Then I would divide that by 440 and use the result as the scaling factor to increase all three dimensions of the model to be printed.

Alternatively, you could try doing the same with just the head joint, measuring the frequency both of the one you printed and of the Yamaha to calculate the scaling factor.

I would try to be fairly precise with this measurement, since the last thing you want is an instrument that can only get close to 440 Hz.

After reading the page you linked to in the question, I have a couple of additional thoughts:

The pitch of the head joint by itself may not be so important. Two well-made recorders that both play in concert pitch could conceivably have a different pitch coming from the head joint because of differences in design. The length of the head joint might be slightly longer in one than the other, even though the assembled instruments have the same length. I would assemble the final instrument and do my best to tune it before deciding that the solution is to change the scale of the model.

• Except the recorders at school may be tuned to 432 Hz (or something else slightly different from 440 Hz). 432 Hz isn’t that uncommon, but old music (usually 415 Hz) tuning, or anything far off 440 Hz, is normally only seen in professional instruments, though. Jun 16 '19 at 18:20
• @mirabilos of course. One could also measure the actual pitch of the instruments at school. But I suppose it's probably 440 Hz, since one might want to play them with a piano from time to time. Jun 16 '19 at 19:44
• I'd be very surprised to hear that any school (other than in a baroque program) is using recorders tuned at anything other than 440. Jun 17 '19 at 10:13

You should scale the length and the hole spacing should be scaled along with the length. You might want to have the hole sizes scale up also. You should probably not scale it up in all three dimensions, only the length. And you should make sure that the finger holes don’t lose their circular shape when scaled.

I think you want to make it 6% longer (1.0564 times its original length).

• I'm no mathematician, but I really think that finding the scaling factor in all 3 dimensions will be more controllable as a modelling/maths/music issue. Jun 15 '19 at 18:15
• @Tetsujin part of the problem will be the thickness of the instrument's walls (and therefore the length of the finger holes). That is, the outer diameter may be scaled up by a different factor than the inner diameter. What about the diameters of the finger holes? The length of a tenor recorder is twice that of the soprano, but is the fipple twice as wide? Fortunately, these considerations will be less important when scaling an instrument by a relatively small factor. Jun 15 '19 at 19:30
• you scale it in one dimension & you're going to have to correct for oval holes... not so simple. Jun 15 '19 at 19:40
• @Tetsujin I can segment it and stretch at the front, end, or in between fingering holes Jun 15 '19 at 19:52
• @Tetsujin Scaling it in more than one dimension will almost certainly have unexpected and unintended consequences to pitch, timbre, and playability, although at only 6% scaling it might work out fine. Jun 15 '19 at 20:09

I recently ran into this problem when scaling my F bass dimensions up to make a wooden contra bass in F. I went with my gut feeling that the length needed to be twice as long, and the bore size and holes needed to be twice the area. This produced a bottom note that was 7% sharp. There is definitely something theoretical about the relationship of bore size between the instruments that I do not fully understand.

Contrary to some of the earlier answers, bore size most definitely does affect the tuning. Try temporarily placing a peanut sized lump of plasticene inside your Yamaha opposite hole 1. Play a C (3rd space on the stave) - it will now be seriously out of tune. My Moeck bass seems to have a "lumpy" bore taper that allows grouping some holes closer together. Measuring my 6 recorders (soprano through to bass) dispels the theory that the bore size increases linearly.

I presume that you have not tapered the head bore - it should be parallel sided. The body bore should taper steadily down to just past the bottom two half holes, then run parallel towards the bell with a small flare at the bell.

So getting back to your original problem, you need to find out why the head joint is pitched differently to the Yamaha. I would check the following:

• Is the head joint bore exactly the same size? You need to measure it with calipers.
• Are the labium dimensions identical? (especially the block to edge dimension)
• Are all the surfaces smooth? 3D printers tend to leave quite a rough surface.
• Is the socket diameter identical? It is acting as part of the bore when the body is unplugged.

Another thing to watch is the bottom two half holes on a soprano are acoustically in the wrong place so the your right hand pinky can reach them. They are bored at a steep angle towards the bell. You can safely explore them with a wooden toothpick.

If you get seriously into recorder making, Adrian Brown's book on recorder maintenance is invaluable https://www.dolce-edition.com/workshop.html. It covers the basics of tuning and altering the bore to correct common tuning problems.

You can certainly scale it. But build the whole thing before you decide it's wrong. It doesn't matter what note the first section alone plays - you never WILL play the first section alone. Judge it when all three sections are assembled. I suspect the problem will go away.

However, I note one comment on the Thingiverse site:

"I printed this. It looks, sounds, and plays beautifully. However, it's tuned to a few tick above C#. Any geniuses out there know of a decent way to either tune the design, or else tune the printed instrument via modification? Even with all the joints just barely hanging onto each other, it's still just a tick or two above C."

User error or poor design? I'm not saying it's the case here, but it's not unknown for this sort of project to be designed by an engineer with no musical background. He gets SOME aspects of the design right. But it doesn't occur to him to test it musically.