ChatGPT Flute Making Advice

[shrines]

I've been struggling to correct the tuning of the notes in my flute's third octave for a month or so. I built my a mathematical flute model in MathCad based on all the traditional sources (Nederveen, Lefebvre, et al) and, over time I have amended my model to incorporate guidance from the theses from Paul Dickens and Yang Lan.

My model is is quite good (I am within 10 cents on all notes in the first two octaves) but the third octave notes are 30-60 cents sharp. I have been hunting the Internet for definitive guidance for a month or so with no luck. So yesterday, in act of frustration, I succumbed to all the hype on the Internet and main stream media I submitted a query to ChatGPT.

I asked ChatGPT:

What flute design characteristics influence the tuning of the third octave notes?

Here's what ChatGPT replied:

The tuning of third octave notes on a flute is influenced by a number of design characteristics. Here are some of the most important ones:

Bore Diameter: The bore diameter of a flute affects the tuning of the high notes, including those in the third octave. A larger bore diameter will produce sharper high notes, while a smaller bore diameter will produce flatter high notes.

Taper: The taper of the flute refers to the way in which the diameter of the bore changes from the headjoint to the footjoint. A more gradual taper will produce flatter high notes, while a steeper taper will produce sharper high notes.

Embouchure Hole Shape: The shape of the embouchure hole can also affect the tuning of the third octave. A more oval-shaped embouchure hole can help to sharpen the high notes, while a more circular-shaped embouchure hole can help to flatten the high notes.

Tone Hole Placement: The placement of the tone holes on a flute can also affect the tuning of the third octave. If the tone holes are too high or too low on the flute, it can cause the third octave to be out of tune. Therefore, the placement of the tone holes must be carefully considered in the design of the flute.

Material: The type of material used to make the flute can also affect the tuning of the third octave. Different materials have different densities and properties that can affect the acoustics of the flute, including the tuning of the high notes.

Key Design: The design of the keys can also affect the tuning of the third octave. If the keys are too long or too short, it can cause the third octave to be out of tune.

All of these design characteristics must be carefully considered and optimized to achieve good tuning of the third octave on a flute. A skilled flute maker will take all of these factors into account when designing and building a flute.

---
Okay, it's not definitive guidance, but it is interesting; especially regarding bore taper. I have not found any mathematical formulae for calculating the proper bore taper; I have only found anecdotal taper ratios. In my mathematical model I used the same taper as measured on my Casey Burns 5-key flute.

What do you think of the "AI" response above? Do any of you know mathematically how taper length and ratio relates to flute accuracy?

Best,
--Steve

[Terry McGee]

ChatGPT confirms the basis of a quip I sometimes make: "Flute design isn't rocket science. It's much harder!" Essentially everything effects everything else, which is why most makers work from historical models, making improvements empirically rather than mathematically. But one feels that breakthroughs are not that far away, so good on you for trying!

Have you looked at WID - wind instrument designer? If you Google that term you'll find quite a bit in the way of resources.

I'd be interested in an example of a note that's bothering you just in case that triggers any thoughts. The fingering you're expecting it to use, the note you get on the actual flute and the note as predicted in your current model.

Oh, and to state the probable obvious - are you putting the stopper in the right position (both in the model and the real flute!) for playing up into the third octave. It might be down around 17mm from middle of embouchure.

Interesting that ChatGPT didn't mention stopper position. I guess nobody's perfect - we're all human...

[shrines]

Hey Terry, thanks for the reply.

I am ignoring stopper position. I am designing a vertical flute. The embouchure hole is located in the center of a transverse head that is joined to the body in a T formation.

The width and diameter of the transverse head form a Helmholtz resonator tuned to D7. I'm in territory uncharted in the standard mathematical models.

Overall, my math works using the same libraries as the Woodwind Instrument software, with modifications to work with the T design.

The tone and range of my flute are quite good. I just need to figure out that pesky 3rd octave.

[paddler]

I am ignoring stopper position. I am designing a vertical flute. The embouchure hole is located in the center of a transverse head that is joined to the body in a T formation.

The width and diameter of the transverse head form a Helmholtz resonator tuned to D7.

I wonder if the sizing of your T head is causing the problem with third octave tuning. If it is tuned to D7, as you say, that is 2349 hz. According to this page the sizing of the upstream space (between embouchure and cork) on Boehm flutes is such that its resonance is centered around 5k hz. See the section titled "The cork and the 'upstream space'". Apparently, the sizing of this space matters because it introduces an impedance whose magnitude varies with frequency in a way that is helpful for compensating for the frequency-dependent end effects that would otherwise throw out octaves out of tune. So if you get the size of this space wrong (i.e., such as by putting the cork in the wrong place in a conventional flute, or making the same too big or too small in your design) your octaves will not be in tune.

So, I would be curious to see the effect of varying the size of that space either side of where you have it now. Just a thought.

[Terry McGee]

Indeed, for the experimental model, you might need two stoppers!

I imagine you're familiar with Señor Giorgi's flute? http://www.mcgee-flutes.com/Giorgi-flute-patent.htm

He didn't appear to have any width to the top of his T. I guess since we're talking volumetric (Helmholtz) resonance, a starting point would be half the usual stopper gap on each side? Which is about the bore width!

[shrines]

I started with equal volumes on each side, using a depth equal to the bore diameter. I experimented for a couple of months with mixed results, and finally decided to try to understand the problem mathematically.

The space between the cork and the embouchure hole on a normal flute is a closed tube, the resonance of which would be calculated as a quarter wave transmission line. A horizontal tube with the embouchure hole in its "side" defines a Helmholtz resonator. I initially modeled the horizontal tube as a pair of opposing transmission lines to model the impact on the flute model, but that approach did not account for the significant tuning variations I saw from minor variations in the embouchure hole size. It is, however, the behavior one would expect from a Helmholtz resonator: fᵣₑ= c/2π × √(A₀/(V × L₀)), where the A the area of the embouchure hole, V is the volume of the cavity and Lo is the wall height of the embouchure hole.

I built a test headjoint that accepted two stoppers, one at each side, so I could experiment with various volumes and offsets. If the horizontal head was acting as a pair of transmission lines then making one side larger than the other for the same internal volume would affect the tuning. It did not.

The logical approach at this point would be to build an impedance tube that I could use to measure the impedance of each note. Armed with those test results I can vary one parameter and another to see which combination gives the maximum harmonics to facilitate third octave note production and which parameters influence the third octave tuning and cross-fingering.

I have already built a recorder style headjoint that I will use for testing to get a consistent embouchure rather than relying on lip position. That make my only playing variable air pressure.

I was just hoping someone could offer a less complicated solution.

[paddler]

Intuitively, I would have started with equal volumes on each side, using a depth equal to half the bore diameter. The reason being that in your flute there are two connected chambers facing each other, which one might expect to behave like a single larger chamber, hence to achieve the same overall chamber volume to that of a standard Boehm flute you would need each of your chambers to be half the volume of the one above the embouchure in a Boehm flute. Again, intuitively I would have expected a chamber of equal volume, albeit a different shape, to resonate at roughly the same frequency, and hence compensate for the end effects in the same way. I would also expect that by using a depth equal to bore diameter on both sides that you would have created a chamber of roughly double the volume of that in a Boehm flute, and that such a chamber would resonate at roughly half the frequency (which it seems to, given what you said earlier). But all of that would have been simple guesses to get started. Regardless of how the two opposing chambers behave relative to a single chamber, I would have initially aimed for a resonance centered around 5k hz, as in a Boehm flute.

Of course, underlying my "guesses" stated above is the assumption that you are using a cylindrical bore of approximately the same diameter of a standard Boehm flute, and hence require similar end-effect compensation. Conical bore flutes tend to require a different stopper position in order to balance their octaves, at least in my practical experience.

Also, I am not surprised that the embouchure hole size has a very big effect. I've played quite a few vertical, end-blown flutes, and one thing that stands out is how drastically one can manipulate the tuning with slight changes to embouchure and coverage. But in all the flutes I tried there was no helmholtz resonator built in (i.e., no horizontal head section). In some ways this sensitivity to embouchure negates the need for one, since the player can compensate to bring notes into tune more readily than with a transverse flute.

Anyway, it sounds like you are already deep into exploring these questions, so maybe you have already looked at the above configuration. I'm also quite unsure as to whether having a helmholtz resonator that is too large would sharpen or flatten your third octave notes, so it might be the case that by cutting its volume in half you exacerbate the problem. It does sound like your current configuration is over compensating for the frequency dependent end effects though.

Sounds like an interesting modeling project.

[Tjones]

"All of these design characteristics must be carefully considered and optimized to achieve good tuning of the third octave on a flute. A skilled flute maker will take all of these factors into account when designing and building a flute.

---
Okay, it's not definitive guidance, but it is interesting; especially regarding bore taper. I have not found any mathematical formulae for calculating the proper bore taper; I have only found anecdotal taper ratios. In my mathematical model I used the same taper as measured on my Casey Burns 5-key flute"

Steve, If I understand you correctly, the 3rd octave is very important to you and you've done your modeling using your Burns flute. It's my understanding that Casey worked on optimizing the first two octaves, and not dealing that much in refining the third octave. With my Burns 6 Key flute it's much more difficult to play into the third octave, than my antique Blackman flute.

I'm not sure if this matters that much in what you're trying to accomplish, but if the measurements are incorrect to begin with, then it might effect your outcome.

[Geoffrey Ellis]

Bore Diameter: The bore diameter of a flute affects the tuning of the high notes, including those in the third octave. A larger bore diameter will produce sharper high notes, while a smaller bore diameter will produce flatter high notes.

Interesting. This conflicts with my own experience. Using a larger bore for a given key most certainly does NOT sharpen the high notes--quite the opposite. In fact, if I have my facts straight that is the very reason why Boehm opted for a 19mm bore instead of the 20mm bore. The larger bore gave a fuller, sweeter tone that he liked, but the higher notes (especially the third octave) suffered as a result. I'd have to re-read that portion of his book to confirm that "flatness" was the issue, but I believe that was the case (though I may speak under correction).

I make a lot of end blown flutes (mind you, none of them feature the T head like Giorgi's flute) with both cylindrical and tapered bores. The physics behind the tapered bores is frankly out of my reach, and I rely entirely on trial and error (or flagrant copying of existing, proven designs) to hit the mark. But I've done a lot of experimentation with cylindrical, end blown flutes, and getting the aspect ratio right (the length to bore diameter relationship) makes a big difference on the tuning balance across the octaves. Bigger bore = flatter second octave. As a rule of thumb, about a 30:1 aspect ratio is the target, and even then an optimal design is going to have largish finger holes and thinner walls.

I've found that trying to deliberately engineer the third octave tuning is also above my pay grade. I only make open-hole (keyless) flutes, and generally speaking I've found that when I get the first and second octaves in good tune, the third octave tends to fall within acceptable parameters. Fussing with the bore, the tone hole placements and other design factors in an attempt to really dial in the third octave notes inevitably leads me into trouble because you can't really impact the third octave without simultaneously tweaking the first and second, and not necessarily in a desirable way. Yang Lan really did a deep dive on this with his optimized xiao thesis, and I am (as far as I know) probably the only maker who attempted to replicate his findings by making a cast-bore xiao using the data he produced. In short, he had to really think outside the box with his design but the results did not deliver a substantially better tuned instrument (though in fairness my replication might not have been perfect--impossible to measure it without some very sophisticated gear). I think there was some minor improvement in both tuning and (arguably) in timbre, but considering how extreme the process was I seriously question the real world value of the exercise.

[waltsweet]

Steve,
You mean you want to tune the 3rd octave on a classic ("Irish") flute, 6 or 8 key? Here are some ideas:
The E-hole (#6) is very small. Siccama had the right idea.
The D# key can prove very useful (each new key doubles the possible fingering combinations). Historically, there are 1-key flutes; I think the key helped find the right combo, even if haphazardly.
I have three Peterson analog Strobotuners. I strung them together (all live in real time), the top one being more modern with a +2 range. Let's say I play F# in the flute's first octave. I set the first strobe to F# (±0 cents) and the lowest traces will appear as F#4; Second strobe is set to C# (2 cents sharp), lowest traces C#6; Third strobe A# (+2 range, 14 cents flat), lowest A#6. These scan the 1st, 3rd and 5th natural harmonics of F#. If you play F# and the harmonics are fighting each other, then your bore is not supporting the basic note, and the upper partials will always be pulling in an undesirable direction. Oh, sure; you can hunt around to find fingerings to put the high note on pitch, but I have great doubts about clarity, responsiveness, stability, balance, etc. if the harmonics are not cooperating. In my experience, you must explore the harmonic cooperation of the 1st two octaves before expecting to have much of an impact on the third.
One model of the tonehole lattice compares the lattice to the fretboard of a guitar, where the low end of the flute corresponds to the guitar's nut, and the top of the lattice corresponds (conceptually) to the 12th fret. I call this the "0-12 bore". Well, I've been focusing on the "12-24 bore," i.e., the 1/4 of the resonator above the C# hole. As we ascend the scale (0-12), there's a flow station (displacement antinode) at the highest open hole; in simultaneous correspondence, there's a pressure station (pressure antinode) in the 12-24. The bore in 12-24 is twice as critical as 0-12. It will shift your octave relationship. While it's true that the endstopper will stretch/compress octaves over the whole compass, the taper of 0-12 will adjust mostly octaves 1:2, while the bore in 12-24 will change octaves 1&2 in relation to 3. For a given headbore, you'll be looking for that right combination of hole size, endstopper position, length of the 0-12 and the 12-24, and taper rates for each. It's all about Rayleigh's Rules.
I hope you're comfortable breaking away from the historical constraints. What do they say about doing the same old thing, yet expecting different results? I think there's considerable room for technical improvements while maintaining traditional playability.
Walt Sweet

[shrines]

@Tjones, I agree. My Casey Burns 5-key has the most beautiful tone and is well-tuned for the first two octaves and it is mostly in time in the third octave. It is that beautiful tone that I wanted to recreate with my own flute, so I started my adventure by copying Casey's dimensions. That was about ten years ago. Now, well more than a hundred flutes later, My flute *almost* captures the tone and tuning of Casey's design, but all my dimensions have changed; all except the dimensions of the position and taper of the bore.

It may be that I have been trying to solve the wrong problem trying by adjusting headjoint-stopper volume. Even small changes in volume and embouchure size/shape have a huge impact on third octave tuning so it's been difficult to get the solution dialed in. Perhaps changing bore dimensions could accomplish the same thing without being so finicky. --Yang Lan in his master's thesis varied the bore dimensions to optimize the tuning of his Xiao flute design and redefine the third octave fingering.

[shrines]

@waltsweet, thank you! I was modeling my flute and planning on building my impedance tube test setup specifically with the intent to capture the harmonic artifacts of each note. My goal was to learn the very relationships you describe in your post.

[shrines]

Geoffrey, I agree with your conclusions regarding bore diameter as it affects first and second octave tuning. What originally struck me as interesting about the ChatGPT response was the statement that a longer taper flattens the higher notes. I guess the only way I'll find out is to extend the taper and see if it makes a difference. Now I have to figure out whether to extend the taper toward the headjoint or toward the foot.

[Geoffrey Ellis]

Yang Lan in his master's thesis varied the bore dimensions to optimize the tuning of his Xiao flute design and redefine the third octave fingering.

He did, but as far as I've been able to determine it didn't actually work in practice. I just now tested his alternate fingerings on the copy I made of his optimized design. In fact, in his thesis he indicates that it is possible to play a wider range of third octave notes that go beyond the traditional four notes that one can usually get. None of those theoretical notes were obtainable when I tried them, and the only notes available at all were the usual ones (D, E, F# and G). And these were obtained using the usual fingerings and they were not actually in better tune than on an ordinary cylindrical bore xiao. Forgive the digression--this just happens to be related to my favorite flute (the xiao).

But the point being that his carefully engineered bore design that was aiming for improved tuning and harmonics was only marginally successful in my view. And the hassle of creating this design of his is monumental.

As a side note, I also made a copy of one of his PVC prototype designs/experiments, and the tuning improvements over the traditional design were nil. I'm not saying that he isn't ultimately on the right track, but I kept encountering this phenomena of theoretical versus real-world disconnects. In short, what looks good in an equation (or in a computer model) does not work the same in real life.

[shrines]

@paddler, I agree that the volume of the space between embouchure and stopper is typically tuned to about 5kHz in Boehm flutes. It is my understanding that the stopper space provides an increase or decrease in specific harmonics that encourage high note production. I recognize that my initial tuning of the stopper space to D7 was less than optimum. It is my understanding that I should be tuning the stopper space to a different note altogether, since third octave notes are all based on cross-fingerings that depend on second or third harmonics of multiple open holes. I was building my mathematical model and planning to build the impedance tube for testing to rapidly modify individual parameters and retest in search of the right stopper space and associated fingering.