tone hole maths for undercutting

[hans]

Can the undercutting of a tone hole (finger hole) be calculated for the effect on tone (frequency) in respect to first and second octave? I would like to quantify amounts of undercutting, to be able to calculate the relative differences for frequency changes in the upper octave compared to the lower octave. ideally I would like to integrate some formula into a flute making spreadsheet (I have been using an adaptation of Koesel's flutomat with good success in designing cylindrical flutes).

If there are no general formulas, what kind of undercutting would be best for maximum raising of second octave frequencies, but little or no rise of first octave frequencies for a tube with 3mm wall thickness?

Thanks for your interest and help!
~Hans

[Feadoggie]

Can the undercutting of a tone hole (finger hole) be calculated for the effect on tone (frequency) in respect to first and second octave?

I am sure it can. But the math wouldn't be particularly straightforward. You may want to check out this site for some background.

http://www.flutephysics.com/

The author of that site is a member here. I had the pleasure of meeting Nelson at Augusta Irish Week year before last. He's a charming man.

The undercutting subject has recently been discussed on the flutemakers board as well.

http://launch.groups.yahoo.com/group/flutemakers/

Feadoggie

[chas]

I found the discussion on the flutemakers list quite enlightening in that the answer isn't nearly as cut-and-dried as I'd thought. One flutemaker I've talked to, when I asked him about undercutting just kind of shrugged and said he'd do it till it looked right, then till it played right.

[Thomas-Hastay]

Undercutting is a form of Bore Perturbation. Because a tonehole is located at an "anti-node" or place of greatest vibration, the undercut hole will flatten the pitch. Because of this fattening, the tonehole can then be slightly moved up the bore to correct for increased air column velocity in the upper registers.The amount of undercutting is directly proportional to the rise in pitch. So Undercutting at the anti-node flattens the pitch, but the increased size of the hole sharpens it. This is verydifficult to explain without mathematics.

There is some phase shifting of the nodes in a bore due to increased air column velocity. This shifting changes the ideal placement of toneholes in the upper registers. Recorders deal with this by using a tapered bore that corrects for nodal shifting. Undercutting also helps with this shifting by providing a larger inner hole to align this nodal phase shift by slight changes in the "tonehole lattice structure"under increased velocity.

The <b>secret</b> to calculation is to average the tonehole diameter(this is, in fact, a trucated cone).

<b>(R1 + R2)/2 = D-avg</b>

This value is used in the playing hole formula when solving for tonehole placement. Once averaged, you can use the resulting value in the Flutomat Calculator. I suggest the Craft manuals located here...

http://www.shakuhachi.com/TOC-CM.html

The Manuals of Lew Paxton Price and Bart Hopkin are best for simplified mathematics concerning tapered playing holes and simplified flute acoustics.
(post edited for proper content)

[srt19170]

The <b>secret</b> to calculation is to average the tonehole diameter(this is, in fact, a truncated cone).

<b>(R1 + R2)/2 = D-avg</b>

You're saying this for the contribution of the tonehole column, I presume? It terms of venting, it doesn't seem likely to me that an undercut hole acts as a hole of the average size.

In the flutemakers thread, I challenged Terry McGee to do some experiments with Flutini to see what measurable impacts undercutting has on tuning in the various octaves. He hasn't responded yet with any results. My own results with undercutting on PVC flutes suggest that the effect is minimal (probably due entirely to the tonehole column contribution).

[Thomas-Hastay]

The reason undercut toneholes are more accurate in the upper registers, has to do with aerodynamics/pneumatics. If the bottom of the tonehole has a sharp edge, the aircolum will pass most of the open hole before the pressure pushes it up and out. The aircolumn cycles into/out of the tonehole every frequency cycle, so tapering the top helps the tonehole as well. The "Ideal" situation is a hole that has the shape of the inside of a Doughnut so the air will flow in/out without "tripping" on any sharp edges.

By means of "surface tension", the air follows the curve of the undercut and most of the pressure/energy finds it's way out the hole. The "nodes and anti-nodes" also shift farther down the bore with the increased velocity in the upper registers and this aggravates the problem of "blow-by" that reduces acoustic energy at the tonehole.

http://en.wikipedia.org/wiki/Surface_tension

I'm sure Terry is an itelligent man, but the field of Acoustics does not always provide answers. Using the combined knowledge of Acoustics/Hydraulics/Aerodynamics/Pneumatics and Applied Physics will help the field of "Woodwind Acoustics" tremendously. I find that Mathematics is the "Traders Language" that ties all these physical sciences together and provides me with epiphany.

[Terry McGee]

I'm sure Terry is an itelligent man...

I think I can see the weakness in your argument ....

... but the field of Acoustics does not always provide answers. Using the combined knowledge of Acoustics/Hydraulics/Aerodynamics/Pneumatics and Applied Physics will help the field of "Woodwind Acoustics" tremendously. I find that Mathematics is the "Traders Language" that ties all these physical sciences together and provides me with epiphany.

I feel confident we can rely on the field of Acoustics to provide us reliable answers to acoustics questions, such as this one. Aerodynamics is also important to the flute, but in questions relating to efficiency, attendant noise, etc rather than tuning.

Even without turning to other sciences, the mathematics of flute acoustics are incredibly complex. I believe I have on my desk computer the most advanced computer model yet developed of the flute (if there's a better one, the developer is lying very low) and even it has limitations. If I make any small change in the design in that model, it takes the computer (an average hot recently purchased machine) over a minute to recalculate the effect on tuning of the change. The changes are often surprising in their location - eg make a change in the head taper of a Boehm flute and the most effected note is low E. So, I'm afraid we can discount mathematics excepting for the most straight-forward of first approximations - it would just take forever. Indeed, that's why we are working on this computer model.

Going back to Hans' original query:

If there are no general formulas, what kind of undercutting would be best for maximum raising of second octave frequencies, but little or no rise of first octave frequencies for a tube with 3mm wall thickness?

I can't immediately think of any kind of undercutting that would produce no or little change in the low octave but would sharpen the second. Any increase in hole diameter will do both. But undercutting down hill (away from the embouchure) will sharpen the 2nd octave more than the first, and you may be able to counter the sharpening of the low octave by pulling out the slide more, giving you a net sharpening of the second octave.

But beware interconnectedness! Sharpening a note will sharpen the note above it by improving venting. Increasing undercutting may also flatten notes below it, by increasing effective bore diameter. And there are others (which is why my poor computer gets so hammered calculating them). So, make change slowly, measure often.

Terry

[hans]

Thanks everyone for your replies!

I have now rejoined the yahoo flutemakers groups, and read througfh the thread about undercutting, which is very informative.

I have been going through both Lew Paxton's books and Nelson McAvoy's online article, to check the maths offered for flute acoustics. I also dug into the javascript of Kosel's flutomat, to see what formulas he used.

There is nowhere any attempt to deal with undercutting. The best I can try on the theoretical side is perhaps to develop the flutomat javascript so it can take into account input for undercuts, i.e. as a first simplified approach use an averaged tone hole diameter as mentioned above. Then I can try to compare it with actual results from my making of PVC flutes.

I have been using the traditonal empirical wisdom about undercutting Terry points out with very promising result. I made some cylindrical flutes with decent tuning which need only a little lip adjustment, or a bit of extra power in the high octave, to get the octave balance right.

Thanks Terry for the pointing out that undercutting may also flatten the note below by widening the bore. To perceive a flute as an oscillator made up of a series of inductors (cylinders of air) is a great way to create a simple mathematical model. What I see now is that this model cannot accommodate undercutting easily, as an undercut changes both the tone hole induction and the bore induction. Correct?

I try to understand exactly why the second octave notes are flat, and to make the javascript show these flattening compared to the first (base) octave notes. Am I right or wrong to think that an idealised flute with no wall thickness at the tone holes would not produce these flattened second octave notes (since there are no tone hole impedances)? And what is the effect of overcutting tone holes, by reducing wall thickness from the outside, making dips for the fingers? Will that be more beneficial in reducing second octave flattening?

Hans

[Terry McGee]

Second octave notes are flat in the plain cylinder flute because we push forward our lips more to reduce the jet length when we go to the second octave. That covers the embouchure hole more, flattening the flute. Doesn't happen with the whistle because the jet length is determined by the length (along the whistle) of the "window". In the whistle, we get to the second octave by speeding up the jet using more air pressure. In the flute we get to the second octave with a mix of speeding up and shortening. It's what gives us more control (top notes can be quiet, low notes can be loud). Making one end or other tapered offsets that effect.

The holes are not always inductors - they are inductive when open but act as shunt capacitance when closed. (Acousticians call them compliances rather than capacitors, but you get the idea.) Just because they can have several effects doesn't make it impossible to calculate for them, but it sure makes it more complex! In our computor model, every section of the flute is represented as an impedance (combining both capacitive and inductive aspects and permitting them to be summed at whatever frequencies we are interested in).

We are not just interested in hole 4's direct determination of low G, but all its harmonics too, as their correlation determines how responsive the note will be, how clean its tone and of course the tuning of the notes based on it - 2nd octave G, third D, and into the third and fourth octaves as needed. And as I said, the hole affects the pitch of notes above (by venting) and, when closed, of notes below. And I'm still giving you the simplified picture. Add in murky end effects like radiation resistance and face impedance, shadowing of the tone holes by the pads of open keys, etc etc and no wonder my poor computer uses so much aspirin.

Unfortunately I can't be of much help to you on plain cylinder flutes - it's an area I haven't delved into. But perhaps something in the above will give you some ideas.

Actually I did make a cylindrical flute once - a copy of the Rafi in Brussels. I just wanted to confirm that such a big cylindrical (renaissance) flute would be hard to play, and hard to play in tune. I was right!

Terry

[srt19170]

Then I can try to compare it with actual results from my making of PVC flutes.

I'll be interested to see your results, either here or on the flutemaker's list. In my own experiments on PVC flutes I could not reliably get a significant change in tuning from undercutting. On the flutemaker's list it was suggested that this was because PVC flutes are too thin to show much undercutting effect.

One thing I'll suggest is that you do your tuning measurements very carefully using something like Flutini. When you're looking for a tuning change of a few cents, it is far too easy on the flute to introduce bias, or to see bias on a typical electronic tuner.

Maybe Terry has a better notion of how to measure, but the routine I settled on to measure a note on a particular hole was to use Flutini and blow the neighboring upper hole to a set measurement, and then cover the upper hole with no embouchure changes to get a measurement for the hole being undercut.