Embouchure Physics by Benade via Hammy Hamilton's blog.

Irish Flute
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There is a relatively dense but interesting post at Hammy Hamilton’s blog about the physics of the flute embouchure by Arthur Benade

http://hammy-flutemaker.blogspot.com/2019/08/benade-and-embouchure.html

The Benade article discusses which parameters of embouchure cutting are more or less influential or sensitive to the tone of the flute.

I’ve seen endless discussions here on the effect of flute material on the sound, when it seems apparent that material is a second-order effect. I also see plenty of discussion about what flutes might be easiest for a beginner, but these are pretty much vague and qualitative. It seems to me the embouchure shape should be paramount when it comes to flute tone and ease of playing.

In my relatively young experience with flute, I have played one with a generous and easy embouchure (Solen Lesouef), and another that requires a precise and focussed breath (Firth, Pond & Co). Both are great flutes, but certainly different.

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Fascinating stuff. I had no idea Benade had written anything of that detail on flute embouchure holes.

Interesting that Benade puts very little weight on the length-to-width ratio, much more on the under-cut angle of the splitting edge, and emphasizes that the undercutting be flat, not curved.

I was puzzled by Hammy’s rant:

None of the calculators I’ve seen make any claim to do more than lay out the holes to get the notes in tune … that’s math. Questions about tone, embouchure hole shape, even undercutting of the toneholes, … that’s wisdom. They are beyond the state of the calculator art, and likely to remain so for a long time to come. For wisdom, you have to look to the masters, and people like Benade.

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I think a lot of stuff people know doesn’t get published because it would be of interest to so few readers. I found the same with Prof Neville Fletcher. I’d make some remark on a pretty esoteric topic, and he’d launch into a pile of discussion. But it doesn’t seem to appear among his publications.

Interesting that Benade puts very little weight on the length-to-width ratio, much more on the under-cut angle of the splitting edge, and emphasizes that the undercutting be flat, not curved.

Something that interests me here is how should we take the angle, when we Irish players tend to blow more down into our flutes than was fashionable among the players of the kinds of flutes Benade was dealing with? I freely admit I eyeball this one!

On the curved vs flat issue, I can certainly concur. I’ve transformed some flutes by identifying and removing a seemingly tiny convex bulge in the surface of the chimney under the edge. Hard to imagine what’s actually happening under those conditions. But I’d guess it must muck with the jet-switching operation. I imagine it’s an aerodynamic issue rather than acoustical. I’ve often wondered what a concavity would do, but by the time you’ve made a really nice head, you are disinclined to put it at risk!

I was puzzled by Hammy’s rant:

[quote=“Hammy”]
there is a special level of hell reserved for the creators of [flute design calculator] software
[/quote]

None of the calculators I’ve seen make any claim to do more than lay out the holes to get the notes in tune … that’s math. Questions about tone, embouchure hole shape, even undercutting of the toneholes, … that’s wisdom. They are beyond the state of the calculator art, and likely to remain so for a long time to come. For wisdom, you have to look to the masters, and people like Benade.

I think that’s probably what Hammy was saying. He’s probably not had experience with serious attempts at modelling such as Paul Dicken’s work or WID, but is referring to interpolaters such as the Flutomatic. On the few occasions I’ve felt moved to make something in poly tube, I’ve found them pretty approximate.

There’s a bio and a listing of Benade’s papers at https://ccrma.stanford.edu/marl/Benade/BenadeBio.html

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I just read FoMRHI’s Comm 2070, where Arthur Benade advises on Cutting of the flute’s embouchure.

This is real info, otherwise very rare. Boehm says 7° undercut, but almost nothing else. Jonathon Landell knows his stuff, and can train you in one of his flutemaking courses. To the machinist: holding the flute between centers above the milling table, the X-axis extends left-to-right; the Y-axis extends front-to-back; the Z-axis runs up and down.

The flatness (YZ plane) is advocated almost everywhere in the flutemaking world. Terry knows that this surface is where the airjet strikes (I call it the “angle of impingement”). It affects how the instrument supports high notes versus low notes. Too much, and the flute doesn’t seem to want to get out of the low notes (strange harmonics). To play higher notes, the airjet must take a different angle of impingement, and the player finds himself searching for a strikepoint that is above the edge of the chimney wall, out of the hole: The tone disappears! I know that flat is best; it seems that a curved surface on the blowing wall (YZ plane), combined with the all-important function of jet-switching, will lead to unpredictable behavior (manifesting as a loss of control).

Benade establishes an azimuth (XY plane) of ±60° (=120° included angle). The blowing wall must be flat (YZ plane) along this entire arc (XY plane). He also points out that the undercut needs to be held in relation to the central axis of the blowhole itself (all around the Z axis, not a simple Y offset as shown). I have dovetail-shaped cutters that give me the 7° undercut as specified. It is ironic that Benade’s drawing implies a method that would NOT give this 7° undercut at the ends of the 120° arc (should the correct angle be scraped by hand to increase it?). The drawing shows the hole being drilled on a line that does NOT pass thru the central axis of the flute bore. This Y-offset handily effects the 7° undercut, but only at 0° azimuth (while at the ±60° ends, it’s less). If the cutter had 7° per side, we’d align the axes and we’d be all set.

I’m inclining to think that the general shape (ellipse, stretched semicircle, rounded rectangle) means less in relation to the other parameters.

Outside the 120° blowing edge (i.e., from 4:00, leading CCW to 8:00), the walls should not be parallel. I’ve had good results when the wall is convex (when it bulges into the blowhole cavity). Another cutter does this job.

The last significant parameter is the extent of sweeping undercut on the back wall and at the back corners (10:30 and 1:30). The sides of the blowhole (along the bore) do not have such extreme sweeps. The finished blowhole can be seen as an ellipse when viewed from the outside. If we were to cut the flute open, the back wall and back corners would more closely resemble half a rounded rectangle.

I don’t want to say the tuning (accurate musical scale) is unimportant. However, tone and playability are king. Players “light up” when playing and tone improve. The software does nothing to improve these latter factors, as far as I know.

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Excellent posts. I’ve always wondered what overall principles are at work in a good embouchure cut, but up to now not a lot has been said, making it more of a mystery than anything else. This brings some clarity to it. Thanks!

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I did not see this claimed anywhere in the article. All I saw was that the undercut angle of around 7 degrees should occur at the due east point,
and that the transition to whatever undercut angles one has at the points 90 and 270 degrees around from that needs to be done carefully
to maintain a flat surface in the YZ plane in the area where the players breath hits the embouchure wall.

Despite the unusually high level of precision in his writing, I think it was still underspecified in terms of how to actually do this, other than to
use a stubby, straight-edge, scraper rather than a file .

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Right, underspecified. Very annoying to someone who just wants to do what others have done!
Here’s the excerpt that talks about the undercut at ±60°.

Transitions between Undercut Angles
. . . . . .
First and foremost we consider azimuths lying in the region one blows against, running counterclockwise from about θ = 300° (= —60°) and θ = +60°.
In this region the embouchure wall must have a profile such that at every angle θ the intersection of the hole surface with a plane containing the hole’s own axis is a straight line.

Note: using the offset drilling (shown in the diagram), the undercut may well be 7° undercut relative to the XZ plane (offset Y), but not as measured from all azimuths around the Z axis.

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I agree, but is it still possible for the intersection of the hole surface with a plane containing the hole’s axis to still be a straight line at all angles?

One thing that is underspecified is what he means by “the hole’s axis”. Does he mean the axis on which it was drilled or is he assuming some
rotation has taken place after an offset drilling in order to make the hole’s axis now pass through the centerline of the bore?

Another issue that gave me pause for thought, and which relates to my first question above, was why it was so important for the “straight line” to be defined only in the vertical plane?
My own observations of flutes that are widely regarded to be easy to blow, shows that they tend to have an almost flat plane region on the blowing wall, within the critical region
defined by Benade. Such a shape satisfies Benade’s constraint (I think – at least in so far as I have interpreted his intent), but not while also maintaining a 7 degree angle across the
whole region with respect to the hole’s axis. The transition regions at the edge of such flat regions are also problematic with respect to his precise definition.

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Well, Prof. Benade is throwing a lot of good terms around, but I still don’t think the method is completely described. I, of course, am interpreting based on my experience. The references to choice of plane would help, but I’m not sure he was a machinist. When he says straight line, my understanding is that the blowing wall should be straight (not convex) from the OD to the ID, at any azimuth. I believe the line should make a 7° included angle with “the hole’s axis”. You raise a good question about where that shall be measured. I once used the drilling offset shown (as in the 2nd figure). However, it is more common in flutemaking literature to set a line across the top of the blowhole from front to back (the path of the airjet, simplified, as in the first figure). There are souvenir gauges to measure the 97° angle from “horizontal” (7° from vertical, figure 1). The back wall will be undercut anyway, so we don’t measure its angle of the initial cut. If you insert the gauge in the hole at 0° azimuth, then twist it to 60° azimuth, the gauge should still show the 7° undercut (I understand from Benade’s description). If you simply had slid the gauge toward the cork (sideways, to the end of the blowing wall), the gauge won’t fit the same. I’ve said the wall should have straight line top to bottom, at 7°; the blowing wall itself will not be a cylindrical surface (or other standard geometric solid). To illustrate, make a plane that rotates around the Z axis (hold a miniature pencil plumb and attach a tiny sticky note). Now rotate the pencil around Z; this is the azimuth angle. Go to 60° azimuth. Where the plane cuts the blowing wall, you will see a straight line segment, top to bottom, making a 7° included angle with the axis of rotation. Now go to 0° azimuth. On the plane you will see a parallel line, closer to the axis of rotation (but still 7°). One way to make this cut is with a cutter ground 7° undercut per side (like a dovetail cutter). I know more than one maker who cuts this way, as in fig 1. In figure 2, the offset shown in the diagram will give 7° undercut at 0° azimuth, but nowhere else. To create a uniform undercut at 60° azimuth, the offset would need to be more (it boggles my imagination). I believe Benade intended the undercut to be increased by hand at the ends of the blowing wall. The “straight line” from top to bottom of the blowing wall at 7° relates directly to the angle of impingement, a fundamental factor for control of the airjet. At the ends, I scrape the surface elsewhere around the blowhole (from +60° azimuth CCW, and from -60° azimuth CW). I use a cutter with an hourglass profile. Benade doesn’t say how to blend. N and S have a different undercut; he doesn’t say how he gets this job done. He shows the back wall as convex. Done with files?

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My reading of the article is that he did intend for the blowing wall to be finished by hand, regardless of whether the 7 degrees was to be maintained only at the 0 degree azimuth angle, or at all angles within the target region (+/-60 degrees), and that the tool to use for doing this was a straight edged scraper .. .stubby and with a large handle. He seemed down on the idea of using a fraise, although didn’t really explain why, as far as I can see.

The more I look into the details, the more ambiguity I see in the article. Even if we focus just on the supposedly critical angle α, we see that he only defines it to be at azimuth 0, and that some of the text contradicts the diagrams. For example, he says that as the size of the embouchure hole increases, that results in a larger value for α. However, in his diagrams he defines α to be the angle measured between the blowing wall and the vertical axis of the hole. This doesn’t make sense, because if α was defined that way then the size of the embouchure hole size would not change the value of α. So in my reading of the article I assumed that what he really meant to do was to define α as the angle between the blowing wall and the tangent to the outer surface of the flute at the point where the hole’s edge intersects a vertical plane at azimuth 0. In other words, α defines how sharp the blowing edge is … the angle between the blowing wall and the external surface of the flute, at the splitting edge. This interpretation would match with his statement that changing embouchure hole size changes α, but it would contradict his diagrams.

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I just applied my best interpretation of Benade’s instructions, with outstanding results

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To be clear, I think the diagram showing the opposite wall as convex was only to illustrate what he meant by convex, not to suggest what the hole should look like.

Congratulations on your successful experiment. Good to know.

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I gave it a whirl as well and it worked great. A big time saver on the embouchure cutting and it gets the angle just right. Still a respectable amount of hand work to do, but a substantial reduction overall.