A new understanding on flat foot flute physics?

[Terry McGee]

We’ve had much discussion over the years about the relative merits of flutes that are “well tuned” – i.e. just blow them and you get the notes you need, vs the “old-style flutes” which tend to have very flat low end notes. The enthusiasts for these old-style flutes (either actually old flutes or in the form of modern replicas) maintain they get better results by having to deal with the tuning by special blowing. So far no-one has managed to support this claim in terms of physics. I’ve been putting in a lot of work (almost to the point of detriment of health and fortune!) on flute acoustics over the last few months and now believe I have an angle on it. Whether I can spell it out well enough to make sense to others is still to be determined. We need to start with a short flute physics refresher…

When we softly play low D (D4) on a “well-tuned flute”, a pressure wavefront runs down the flute from one end to the other, reflects at the first open hole and returns to kick the jet out of the embouchure hole. The turnaround time is precisely what is needed to generate D4.

When we push a bit harder, some of the energy goes only halfway down the flute, before turning back. Because it takes only half as long to get back, the frequency and therefore the pitch is double – we’ve added some 2nd harmonic content, D5.

Push a bit harder and some of the energy goes only 1/3rd of the way down the flute, we’ve added some third harmonic, known also as A5.

It doesn’t stop there – there will be harmonics up to about 5KHz. But we’ll stop there – it’s enough for our story to consider the first three.

Because this is a “well-tuned flute”, the three energy waves arrive back at the embouchure at the same time, reinforcing each other as they kick the jet back out of the embouchure hole. The now harmonically-enriched D4 sounds focussed, crisp and hard-edged. A Fast Fourier Transform (FFT) of the resultant sound would show the three harmonics. The D4 note is easy to play.

When we blow softly in a “old-style flute”, D4 comes back flat, probably between 20 and 60 cents flat. But when its skilful player starts pushing into the note, the perceived pitch rises as the note becomes harder, as the harmonic content rises. We’ve assumed in the past that the skilful player is “lipping up” the note, closer to the D4 we want. I now believe that is not what is happening.

(Aside: In the real world we can expect a range of things to happen, and lipping might be part of it. But not, in this submission, the main part.)

Instead, I believe the skillful player is getting the D4 in tune by eliminating it from the mix of harmonics. Or, at least, by minimising the proportion of D4 in the mix. Let’s see how that would work.

The skilful player introduces significant offset between jet and edge, typically by blowing downwards towards the centre of the flute. Fletcher has shown in both experiment and theory that this offset reduces the fundamental content(D4 in our example) and increases the 2nd and 3rd harmonic content (D5 and A5 in our example).

The mix of wave energy returning to kick the jet out of the embouchure hole no longer has appreciable amounts of D4 in it. So the timing of the resonance is no longer dependant upon it, but on the timing of the remaining mix of upper harmonics. In the old style flute, second octave notes tend to be reasonably well tuned, so that’s got to help.

The ear still hears the missing D4, as the brain assigns pitch based on the distance between harmonics, not just the pitch of the lowest harmonic. This can be readily demonstrated by filtering the fundamental out of a triangular wave – the listener can readily appreciate the change in timbre but will not hear it jump an octave.

So our skilful player has taken an old-style flute with a very flat D4 and, by dint of selective blowing, made the flute sound more in tune, not by lipping up the defective note, but by minimising the contribution of the defective note. Further, the tone has been enhanced by two different and equally important mechanisms. Firstly, the combination of harmonics gives a hard-edged sound that pleases the traditional player. Secondly, eliminating or minimising the discordant contributor improves the focus of the sound, as the remaining in-tune harmonics reinforce at the jet. The flute will be more resonant and more in tune than if the flat D4 were permitted to remain in the mix.

This would explain why it took about 100 years to eliminate the flute's flat low end - there was not a great need to do so. This seems more likely than the alternative explanations, such as "they liked it flat" (why?) or "they didn't realise it was flat" (I bet the piano player did!) or "they couldn't find a solution" (and yet they made such lovely flutes in all the other respects).

It’s often felt by skilful players that the “old-style flute” is better than the “well-tuned flute” as it permits the harder toned low notes. Nothing in the theory above supports that view – the “well-tuned flute” should be capable of being played in the same way, producing the same results. It's just less necessary. Rephrasing slightly, we can say ‘it is only possible to play the “old-style flute” in tune if we use a selective blowing method that reduces the impact of the discordant fundamental’. It may be that the greater, indeed absolute need to push the “old-style flute” provides more incentive to the skilful player, subsequently providing great satisfaction when achieved.

A few other things become evident:

We are not all skilful players – not everyone will be able to achieve satisfactory results on an “old-style flute”.

If new makers want to make “old-style flutes” they need to advise their customers that they will need to use a selective blowing technique.

New “old style flutes” really should be marked so that 2nd-hand purchasers know what they are getting.

Those makers wishing to build “well-tuned flutes” need to consider the relationship of their harmonics to ensure that a range of tonal colours can be educed without unwanted apparent pitch changes. A combination of this new understanding and real-time measurement techniques (RTTA) should be able to deliver flutes that are capable of great results across a wide range of blowing approaches.

The usual wisdom of placing the stopper at around 19mm will need to be reconsidered. Soft blowers may find better results at greater stopper distances; selective (hard-edged) blowers may benefit from reduced stopper distances, even in the 15 to 17mm range. This weakens the fundamentals (who cares, if you are trying to kill them off anyway!) but improves the tuning and focus of the harmonics, where such blowers are intent upon placing their energy.

It would be good to hear from selective blowers as to what works best for them. Interestingly, players using Rudall & Rose’s Patent Head are probably already there. This supports my view that the 19th century English tonal ideal is not dissimilar to the 21st century Irish ideal.

There are quite a few more ramifications of this new understanding, but they’ll have to wait their turn.

(Exhausted, he slumped back. Foolishly, he allowed himself to lapse into a deep sleep before donning his flak jacket....)

Terry

[Akiba]

Your description of the sound produced through selective blowing makes me think of Paul McGratten's playing, in particular his set of reels on track 5 of "Music at Matt Molloy's". He has the "reediest" tone I've heard that is also very focused. His hard D sometimes sounds like D5 and other times more like D4; regardless, his D's have a complex, honking tone.

Then I think about John Wynn playing his Murray, I believe a flat footed flute. His D4's sound full and rich (as he plays with a more rounded, less harmonically complex tone) and I wonder how he gets that note in tune....

Thanks for all your work and insights, Mr. McGee. I'll be doing some experimentin' myself tomorrow, that's for sure.

Jason

[flutefry]

Every class has an idiot. There's a reason I went into biology....

If the fundamental (distance from embouchure to first open hole) is flat, why is the first harmonic in tune if the energy wave goes half the distance from the open hole to the embouchure? Shouldn't it also be flat? Or is it 50% less flat than the fundamental?

Thanks in advance,
Hugh

[Terry McGee]

If the fundamental (distance from embouchure to first open hole) is flat, why is the first harmonic in tune if the energy wave goes half the distance from the open hole to the embouchure? Shouldn't it also be flat? Or is it 50% less flat than the fundamental?

Hmmm, good question. No, it's not 50% less flat than the fundamental. If I'm right, once the fundamental has been annihilated (by selective blowing), it's out of the picture. If it has only been minimalised, it will continue to have an influence, but that influence will be proportional to how much it remains in the picture. The better it is annihilated, the less influence it has.

[Aside: The interesting thing about the 19th century flutes (and therefore their accurate copies) is that the 2nd and 3rd octaves comply far better with what we expect than does the first octave (the fundamentals). Before I hatched the current theory, that bothered me. If the makers couldn't get both the lowest two octaves in tune, why would they choose the 2nd one over the first? Why did they let the bottom one droop? If we go with my current theory, that solves itself. It's because the bottom one matters less - by selective blowing you can safely ignore it. You can't ignore the second octave - it's right there, smack bang in the middle of the range (which was by then 2 & 1/2 to 3 octaves, for classical music). You couldn't do anything about it. It had to be right. Better to get it right and let the low octave go hang.]

The weakness in my description (but fortunately not in my thesis) was my use of the term half-way. As you've illustrated, it would seem that if it takes too long to go the full way (ending up with D4 being flat) it will take too long to go half way (making D5 flat). But indeed, the flute maker has separate control over how long it takes for the pressure wave to propagate the full length of the borelength or half the borelength, and that control is exercised by the profile of the bore. Part of the bore (at the head end) is cylindrical; the rest is tapered. And not just simply tapered, but cunningly complexed. The tapered section "slows down" the fundamental compared to its octave. We tweak the bore profile to achieve the desired octave relationship. We can then fiddle lengths and hole diameters to get the actual notes right.

To take some extremes, in a plain cylindrical bore, such as a flute made from plumbing pipe, the lower octave full-length propagation time is too quick compared to the 2nd octave's half length, and the second octave ends up distressingly flat by comparison. In "the old style flute", they seem to have gone too far and it's the first octave that ends up flat. In the to-be-desired "well-tuned flute", they end up equal.

I hope that's explained it well enough - this is not easy stuff and I know I'm going to have to get better at explaining it! But thanks for trying!

Terry

[Carey]

Great work Terry. and a nice description of your findings/theory/revelation. Interesting how we hear notes too. I guess it is all pattern matching in the end, and missing out this bit or that of the pattern matters not.

In my mental model of what goes on in the flute, I imagine a positive pulse of pressure running down the barrel much like a "thrown" hump in rope runs down along it (or a wave along a flag.) When this pulse reaches the end I envision it blasting the air at the end of the bore out into the room. This air has momentum and carries on a little longer than end of the flute, creating a lower pressure region at the end of the flute. This negative pulse goes back up the flute and pulls the stream in the embouchure to start over. That may not be completely correct, and I'm all about hearing more about it. But my question is this, given that mental model, what causes the harmonics to decide to go back early? I imagined there were some bits of the process that had more energy that others and they got there at a different time, and reinforcing happens only for the lucky few frequencies. So they all must reach the end in order to start back up.

Once in steady state, the picture I have is of pulses of the first harmonic bouncing off both ends and each other in the middle. The second having three sections and so on. (I spent a lot of time with a Slinky when I was a kid .) So in my universe the harmonics travel the full distance but do it twice/thrice as fast. But then there would be that pesky instant where the positive pulse on the way down meets the negative pulse on the way back. Hmmm..

From physical observation it might just appear that the waves bounce off each other, and perhaps they pass right through each other. I'm not sure it really matters. (There's an energy/matter pun in there somewhere.) I guess I can use either mental model and it doesn't really matter once the tone is established, but how does it decide to get started in the first place? (Pesky transition states. ) But using this model I can see where putting more (different?) energy into the embouchure would favor the harmonics over the root.

Thanks for allowing/causing/making me think about that.

[peeplj]

A couple of questions:

First, do you know when "flat-foot" started--did the late Baroque traverso have a flat low D? What about when the foot keys were first added? What I'm wondering here is if there would be a corresponding change in the cut of the embouchure hole--if you could find a starting point for when flutes were started to be blown in this way, and there was a change in the embouchure cut at the same time, it seems like it could go a long way towards supporting the theory.

Second, can the sound of a skillful player be analyzed and shown to be missing the fundamental content?

What you've come up with here sounds more likely to me than have previous attempts at an explanation.

Nice work!

--James

[brotherwind]

Your description of the sound produced through selective blowing makes me think of Paul McGratten's playing, in particular his set of reels on track 5 of "Music at Matt Molloy's". He has the "reediest" tone I've heard that is also very focused. His hard D sometimes sounds like D5 and other times more like D4; regardless, his D's have a complex, honking tone.

Jason

When I tried to learn that tune ("Rolling in the Ryegrass") from the very recording, it was one major obstacle to decide wether it was D4 or D5, actually. Great playing in deed. Convinced me to take up the flute at all!

Moritz

[Terry McGee]

Hi Carey

Yes, the operation of the flute takes a bit of work to get into the mind and keep in the mind. I really need to come up with a full but concise explanation of it, but in the meantime let me refer you to Joe Wolfe's very good explanation at http://www.phys.unsw.edu.au/jw/fluteacoustics.html.

But let me address your substantive question: "what causes the harmonics to decide to go back early?" You do, by favouring them with your jet speed. A length of pipe has a number of natural resonances, eg D4, D5, A5, D6, etc. The exact pitch depends on the length of the tube and the speed of sound in air (which is why warm days we go flat).

But the jet also has a resonant frequency, dependant upon the length of the jet and the speed of the air in it (note it's the speed of air in the jet case, but the speed of sound in the tube case. That explains how two very different lengths can have the same resonant frequency). The harmonic we select out of those available in the tube depends on which resonances match best.

The diameter to length ratio of the tube is also very significant. If it's very fat, the fundamental will be by far the strongest resonance. If it's very thin, you may not be able to even play the fundamental. The French horn is an example of a long thin tube where you play on the upper harmonics.

When you play D4 softly, you use a relatively low jet speed, which has its own resonance around that pitch. When you play D5 you increase the jet speed (either by blowing harder, or shortening the jet length, or some of both). But if you do the "blowing at the centre" offset trick (see http://www.mcgee-flutes.com/Gunn%20on%20Tone.htm), you can force the harmonic content (D5) up without losing the fundamental (D4).

In fact, try this trick. Finger D4 (xxx xxx) but blow D5 by blowing fast, directly at the edge. Now, without changing the jet speed, lower the jet towards the centre of the flute by pushing out the top lip and/or withdrawing the lower lip. The flute will drop back to D4, and you will have to blow really hard to convince it to play D5. And the D4 will be rich with harmonics and hard. It's my surmise that not only does the jet offset enrich the harmonic content, but it makes it less likely to overblow into the next harmonic, even though you are using a jet speed consistent with the harmonic. Tricky stuff, eh?

There's a lot of mixed thoughts in the above (it's still early here!), but hopefully it helps! Now, breakfast!

Terry

[Terry McGee]

A couple of questions:

First, do you know when "flat-foot" started--did the late Baroque traverso have a flat low D? What about when the foot keys were first added? What I'm wondering here is if there would be a corresponding change in the cut of the embouchure hole--if you could find a starting point for when flutes were started to be blown in this way, and there was a change in the embouchure cut at the same time, it seems like it could go a long way towards supporting the theory.

I believe things got violently worse around the time they added the long C foot. The embouchure hole was probably going from round to elliptical at about the same time. I don't know whether the offset blowing style was used in the baroque (I'd like to know if anyone sees a reference). The earliest reference to it I've found so far is Gunn (about 1793).

Second, can the sound of a skillful player be analyzed and shown to be missing the fundamental content?
--James

Easily done if anyone would like to send me a .mp3 of some low notes.

(And remember, it doesn't have to be missing the fundamental, just significantly reduced so that the harmonics are calling the shots.)

Terry

[akfluter]

I have pretty much the same question as flutefry, I think: Are you saying that with a flat foot, D4's fundamental frequency is flat, but its harmonics are not? I assumed that the harmonics of a note played on the flute, like on most instruments, were pretty close to integer multiples of the fundamental (at least they weren't 20 to 60 cents away from being integer multiples). It would be interesting to see some measurements of the frequencies of the harmonics.

Ben

[Terry McGee]

The harmonics are in tune when you play D4, so that if D4 is say 30 cents flat, so will its harmonics D5, A5, etc. This is because the jet switching in and out of the embouchure hole resynchronises them at each cycle. (The real A5, D5 etc might well be different as they use different fingerings.)

But if you play the harmonics individually (but continuing to use the xxx xxx fingering), you may find that they are or are not in tune. On my old Metzler, which has about 30 cents flat foot, the harmonics D5, A5 and D6 are 20 to 30 cents sharper than the D4. (This is a bit open to question too, as to play each harmonic I have to change the jet speed or length or both, but that's the best I can do.) But it seems to make sense that by increasing the content of the sharper harmonics should help pull the D4 up, which is what happens when I use the downblowing embouchure approach.

Terry

[jemtheflute]

A couple of questions:

First, do you know when "flat-foot" started--did the late Baroque traverso have a flat low D? What about when the foot keys were first added? What I'm wondering here is if there would be a corresponding change in the cut of the embouchure hole--if you could find a starting point for when flutes were started to be blown in this way, and there was a change in the embouchure cut at the same time, it seems like it could go a long way towards supporting the theory.

I believe things got violently worse around the time they added the long C foot. The embouchure hole was probably going from round to elliptical at about the same time. I don't know whether the offset blowing style was used in the baroque (I'd like to know if anyone sees a reference). The earliest reference to it I've found so far is Gunn (about 1793).

I have a provincially made (Dorset!) one-key boxwood flute by Norton of Shaftesbury that is, apparently, c1820s, though I haven't been able to find out much about it/Norton. It has an elliptical embouchure and fairly Rudall-ish typical tone-holes, but even with a short D foot (it's the instrument in my avatar pic) it has a distinctly flat D4. FWIW, most of the Baroque fingerings work quite tolerably on it, though forked Fnat has to be lipped way down. Its intonation generally does improve with a Gunn-type embouchure approach. Visually at least, it seems very typical of the baroque-type flute adapted into the early romantic era that e.g. Boehm himself refers to having started his career on. If I can find time, I'll try to take measurements and do a sound clip Terry can RTTA...... assuming that would be any use to anyone.

I wonder if there is any element of an attempt to help to flatten the forked F natural by extending the tube that contributed to the flattening of the fundamental D, whether on short or C feet? One would perhaps expect it then to be present in at least some true baroque flutes, and especially as the bore, embouchure and toneholes were widened to attempt to produce greater power........

Second, can the sound of a skillful player be analyzed and shown to be missing the fundamental content?
--James

Easily done if anyone would like to send me a .mp3 of some low notes.

(And remember, it doesn't have to be missing the fundamental, just significantly reduced so that the harmonics are calling the shots.)

Terry

I would think that the ideal candidate here would be Conal O'Grada's Top of Coom CD. Conal has absolutely fantastic "hard" Ds. In fact, he makes such an emphatic point of hitting them at every opportunity that I find they detract somewhat from the listenability of his playing - that is, he overdoes 'em for my taste. It's a bit like Finnegan's fancy tonguing - the obtrusive emphasis on the technical virtuosity is somewhat at the expense of interpretation/musicality: wish I could hit 'em like that, though! He uses a R&R flute-body with a Fentum head on that recording and boy can he honk it!

[Jack Bradshaw]

Hi Terry....just passing through....

One point to add at the moment.....bellmakers were familiar w/ the "missing note" technique for their tuning for a very long time.......someone may be able to comment further on that....

Jack

[Aanvil]

He uses a R&R flute-body with a Fentum head on that recording and boy can he honk it!

Really? Damn it... now I haven't an excuse.