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Reed math, a question?


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This question is directed at those who make traditional instruments. I need to find the equation for the shape of reeds in a concertina. Most importantly for the curve/taper that is found in the edges of concertina reeds. I am not currently interested in making traditional instruments though I have the ability to cut shoes now. What my interest is the formula; as I am doing some work on another member of the free reed family (harmonicas) and wish to further the testing I am doing with tapered reeds in harmonicas. As this allows for a longer reed for a given note and in the upper end of the register would make a reed that was much easier to sound.

 

I thank all for any help in this and hope everyone is having a great year.

 

Michael

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Michael,

 

there are no equations, everyone I have talked to started with copying a successful old instrument and then modifying according to hypothesis and results. There is not the volume of reeds made to justify someone coming up with money to pay for the research to lay out a prescriptive equation.

 

Not sure about your theory on longer reeds starting more easily; in the higher register I would have thought wider reeds had an advantage.

 

regs

 

Chris

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Chris,

 

I hate to disagree but there is math for it. There is formula for parallel edged reeds and their frequency before shaping (which has a funky ratio for including the tip mass versus total mass). There is known Math in the size and shape of reed chambers. There is theory being kicked about for the materials which the chambers are made of. Even though I speak Italian fluently, getting them to giv up th math they use for accordion reeds is no go over a phone. Need to go down there and arrange to meet some folks in the near future.

 

I was just hoping Dana, Wim', or someone with their experience might know a formula that works for the amount of reduction in the width from tip to base does to the end frequency. I all ready know that on a reed of 14mm L x 2mm W with a root thickness of .6 to .7 mm and a tip thickness of approx' .5mm; reducing the tip by .25 mm uniformly raises the given frequency by approx' 1 to 1.25 tones. What I am trying to find is the math to further test without doing a lot of blind testing. Cutting custom reed plates and shaping the reeds takes a good bit of time. It is also very difficult to control imperfection I might put into the testing which can really drive you nuts.

 

So again anyone who has experience and might know the formula please. I am not interested in building traditional concertinas (even with the opening of the oriental market to concertinas, not enough time in the day). I might at some time be willing to cut shoes for folks but that would be about it.

 

Michael

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There is formula for parallel edged reeds and their frequency before shaping (which has a funky ratio for including the tip mass versus total mass). There is known Math in the size and shape of reed chambers. There is theory being kicked about for the materials which the chambers are made of.

 

Michael,

could you please indicate to me references about this theory ?

(I'm a scientist and not afraid of maths)

Thanks

David

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First,

 

Chris my unreserved apologies, currently in the concertina world there are not these formulas. There these formulas in the Accordion world though and currently figuring the math as both my wife and an Italian friend pointed out that I should be able to reverse engineer the math from the reeds in my possession.

 

David and Larry,

 

The Formula for parallel edged free reeds can be reached at http://talkingreeds.com/reed-construction/

 

The "K" part refers to the 1/3 of the tip of the moving section of the reed which is not the overall reed though you take that into account in the ratio. I am still working on getting it right all the time (don't always remember to do the functions of the equation in the right order, you sure this isn't alchemy :huh: )

 

What I am looking for will start with the basis that is used in accordion reeds, which is based on a trapezoidal shape structure rather than a parallelogram. At least now I have physical model to wrap my head around.

 

Special thanks to Dana for pointing that the math is possible just not yet done.

 

Michael

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The theory on materials does not deal directly with the concertina bur another free reed instrument which is the harmonica. The latest research which object and subject components to it, seems to say yes it does affect the sound you get, but few if any are able to notice this difference. The Equipment picked up and documented the difference but those listening did not hear any difference. It is a start and needs more research and a bit of refining in the way the study was done. It is a start.

 

With Concertinas it would be interesting to see in a balanced and studied fashion what the differences in design and material do to an instruments voice. We know that the reed and the chambers involved are coupled in the production of the frequencies we get out of an instrument. It is also know that many different parts play an effect upon the final sound. It would be interesting to document as best as possible these differences and their effects. Though I don't see the ASA or the ICA being able to afford to support that level of research. That is no disrespect on either group they have done and continue to do a lot in the field and research.

 

Michael

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Are you looking for a "design formula" that will define the shape of the taper based on other parameters? I can see why manufacturers wouldn't want to give up a trade secret (possibly empirically derived) like that.

 

If you're looking to get the formula for natural frequency based on a given reed profile, it's going to be a modified version of that formula at talkingreeds.com including an integral over the length of the reed of the profile (thickness as a function of length).

 

I would start by backtracking to the derivation of the formula you have. I've seen free reed frequency formulas before, but that funky mass ratio term is a new one on me. Also, I thought the "breadth" term (X) ended up having no bearing on frequency, only amplitude, starting force, etc., but maybe that's only in a reed of uniform thickness.

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but maybe that's only in a reed of uniform thickness.

 

The Talking Reeds page describes the formula in these terms:

 

The natural frequency of a parallel sectioned reed is found from

 

I take this to mean the reed is has the same width and thickness throughout its length

 

Its a long time since I studied this, but I suspect that you will find this formula in standard mechanical engineering textbooks where it would be used to describe the natural frequency of a cantilever beam.

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Similar but not the same Theo, the one you are referring to is this: f=[kn/2pi][sqrt(EI/wL^4)] (ref: Formulas for Stress and Strain, 5th edition by Raymond J. Roark and Warren C. Young) or can be expressed as f=1/2pi times the square root of the fractional equation of 8EI/pAl^4. Where the following is used:

 

f = frequency

E = Young's modulus for the material

I = moment of inertia which is dependent on cross section

p = Density of material

A = Cross section of the material

L = Length of the beam (I assume that refers to free end, though I could be wrong and it accounts for total beam length)

 

This Formula does differ from the one given on the referred web site.

 

Thanks for the input.

 

Michael

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Similar but not the same Theo, the one you are referring to is this: f=[kn/2pi][sqrt(EI/wL^4)] (ref: Formulas for Stress and Strain, 5th edition by Raymond J. Roark and Warren C. Young) or can be expressed as f=1/2pi times the square root of the fractional equation of 8EI/pAl^4. Where the following is used:

 

f = frequency

E = Young's modulus for the material

I = moment of inertia which is dependent on cross section

p = Density of material

A = Cross section of the material

L = Length of the beam (I assume that refers to free end, though I could be wrong and it accounts for total beam length)

 

This Formula does differ from the one given on the referred web site.

 

Thanks for the input.

 

Michael

I spent some time puzzling over this sort of thing fifteen years ago since it seemed such a logical way to go, but found it had very little application to actual instruments. Factors such as reed loudness, reed lengths, stiffness requirements and overall instrument balance made things instantly too complicated. Ironically , much of the work to find good solutions was already done by people making reed sets over the last century or more by trial and adjustment. Unfortunately none of that ended up in any kind of handy formula. As you can see from the one above, even the formula for the simplest form of reed ( parallel cantilever beam ) Isn't all that simple though if that was all you needed, a simple spread sheet would take care of it in an instant. When faced with a real instrument of 3 octave range, that simple reed covers too much length spread, only has appropriate stiffness for a small range on notes ( your choice which ) and has terrible loudness balance between the high and low notes.

To remedy all this, long reeds were shortened, short reeds lengthened, changed in thickness and sometimes in whether reeds were parallel sided or slightly wider at the base than the tip. Reed widths were adjusted along with the length to produce better volume balance. On top of all this to make the reeds all playable at the same general air pressure, the reeds had a thickness profile ground into them that changed the weight distribution needed to keep the reeds in pitch at their new lengths, and importantly to adjust each reeds stiffness in the more or less neutral area of the reed. There are an infinite number of ways ( albeit in a limited range ) the thickness can be varied and reach the same result. Some provide even well distributed reed bending, others don't. Some produce better tonal qualities than others, but that becomes a subjective, not mathematical choice. All of this could be boiled down into equations if you could mathematically define your ideal tone ( if that weren't enough, the reed is only one part of the tone equation ) and make your choice about what you wanted. This is even more complicated than it seems since the reed's overall harmonic spectrum is not the result of multiple vibrational modes of the reed, but the deviation from sinusoidal motion caused by the reed's interaction with it's surroundings. All of a sudden a lot of other poorly understood factors enter the equation feeding back into the reed profile from outside the reed itself.

Trial and adjustment seems like a poor way to go about things in this modern era of ultra powerful computers sitting on our desktops or laps, but you can learn a lot very quickly and do it while taking into account the full environment of the reed.

We have been fooled over the last hundred years of rapid advancement in science and technology into viewing the world through the lens of science, engineering and mathematics that does a marvelous job at handling small numbers of variables and things isolated from their surroundings. We build relatively simple structures compared to nature and reduce physics problems to levels of complication that are manageable and then hope they bear some resemblance to the real world, often deluding ourselves in the process thinking we have understood the problem. Mind you we have done very well at this, but sometimes simple processes done over a few times with an attentive mind can narrow down a solution that is connected to the actual world we live in, not a highly limited approximation of it. Our minds were built for this type of work, and in many cases, this is the most powerful tool we have. Math and engineering will continue to have an important place, but if we let our powers of observation and intuition atrophy instead of developing them to their fullest, we'll find ourselves growing more and more separate from the world around us in all it's complexity.

Sorry to spout off so much, but empiricism has gotten short shrift and needs to be brought back into the picture.

Dana

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Dana,

 

I agree with you in all you wrote. The reason for searching for and boiling down the math for the reed base shape is the building of a foundation to do all the complicated stuff that builds upon it. Which is so long a list with just the chromatic harmonica let alone the concertina to on occasion drive to a fondness for strong drink.

 

My primary interest is building that foundation with which to than build a scaling or in reality a set of scalings with which from there be able to set up a Chromatic that does not suffer so horribly from the piercing bite of the upper octave due to the shape and shortness of the reed. With Chromatic Harmonicas the very thickness of the reed plate and the shape of the reed channel have effect (as they do in concertinas but to a different extent). So really the formula is a set of starting blocks with which to build from there. I am making progress on this though and the reverse engineering is making sense and hopefully will bear fruit from real world testing (cause it the boat don't float what use is it).

 

I tend to flavor Tesla's view on research and how one should look at thinking.

 

“The scientists of today think deeply instead of clearly. One must be sane to think clearly, but one can think deeply and be quite insane.”

 

and the second is:

 

“Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.”

 

So I work with the math to set a foundation upon which reality will impose itself and make the math bend as needed.

 

Michael

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Dana,

 

I agree with you in all you wrote. The reason for searching for and boiling down the math for the reed base shape is the building of a foundation to do all the complicated stuff that builds upon it. Which is so long a list with just the chromatic harmonica let alone the concertina to on occasion drive to a fondness for strong drink.

 

My primary interest is building that foundation with which to than build a scaling or in reality a set of scalings with which from there be able to set up a Chromatic that does not suffer so horribly from the piercing bite of the upper octave due to the shape and shortness of the reed. With Chromatic Harmonicas the very thickness of the reed plate and the shape of the reed channel have effect (as they do in concertinas but to a different extent). So really the formula is a set of starting blocks with which to build from there. I am making progress on this though and the reverse engineering is making sense and hopefully will bear fruit from real world testing (cause it the boat don't float what use is it).

 

I tend to flavor Tesla's view on research and how one should look at thinking.

 

“The scientists of today think deeply instead of clearly. One must be sane to think clearly, but one can think deeply and be quite insane.”

 

and the second is:

 

“Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.”

 

So I work with the math to set a foundation upon which reality will impose itself and make the math bend as needed.

 

Michael

I can sympathize with your approach and am glad you are making progress. While there are many ways to arrive at pitch /length solutions ( as an example, Rich had his profiles constructed of three different sections with a different formula for each ) you can impose constraints you feel might be desirable like evenly distributed bending or length width proportion constraints, or stiffness constraints that will give definite results. For your problem of the shortness of the high reeds, you are limited by the allowable stiffness. The primary reason the reeds are so short to begin with is that it is the only way to keep them in the same stiffness range as the lower reeds. There is a limit to how thin you can make the tip especially since stiffness effects are much greater per unit of thickness than the weight effects.

Regardless, I'm sure you will learn a great amount and reach your goal one way or another as you master the math. I found my own efforts illuminating even though I eventually abandoned that path myself. It certainly gave me a greater understanding of the reed as a whole.

Dana

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Dana,

 

You are most definitively telling the truth there. So in tuning and shaping reeds both parallel and trapezoidal/elliptic sided, I have learned that you must preserve the curve of the belly of the reed in relation to the tip and root. That gapping while it can change tuning is more important for response and setting the quickness a reed responds to pressure than in tuning (though that does not dismiss the effect there) overall mass must be accounted for in reed shape and thickness as you have pointed out due to the implied rigidity issues. The list gets longer from there and really don't want to go down that list right now. I am currently finishing the tuning on a Suzuki SCT 128 and IF I ever work on another one it will be costly to the person who wants me to do so. Sixteen pairs of reeds tuned to beat and than having to be balanced against one another when there is duplication is a bit nuts, than add in that the duplicated notes are on different length reeds and it really gets fun.

 

On the positive side have finished laying out my own fret work and will be cutting the test panels once the client work is done and hope to be seriously putting time in on the concertinas this week. Besides mapping the reeds. Thanks again Dana the information yo have given is very useful and I am very thankful for it. I miss Richard, though never meet the man, we had some wonderful discussions and was for a bit going to look into cutting reed shoes for him.

 

 

Michael

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This is even more complicated than it seems since the reed's overall harmonic spectrum is not the result of multiple vibrational modes of the reed, but the deviation from sinusoidal motion caused by the reed's interaction with it's surroundings.

 

 

Hi Dana,

 

Thanks for sharing your extensive experience and understanding on free reed operation. I do, however, believe you make an error with the above statement, which, upon close examination, is self contradictory. Assuming I'm not misunderstanding your statement, any deviation from sinusoidal motion of the reed tongue would necessarily be caused by the excitation of multiple vibrational modes of the tongue. Cottingham at Coe College has measured the time dependence of the vibrating reed tongue and found it to be virtually sinusoidal, but with a very small, albeit measurable, departure, due primarily to the first overtone. I doubt any researcher would attribute the rich harmonic structure of the musical tone to this very slight departure from sinusoidal motion. Thus, the first part of your statement does agree with scientific consensus, with only the second part being in error. The rich harmonic structure of the musical tone is rather due primarily to the sharp pressure pulses caused by the tongue passing back and forth through the slot, with significant modification of these pulses by acoustic interaction with surrounding structures - most significantly the reed cavity. In other words, a perfectly sinusoidal motion of the reed tongue is entirely capable of producing a musical tone with the rich harmonic structure we hear.

 

Best regards,

Tom

www.bluesbox.biz

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This is even more complicated than it seems since the reed's overall harmonic spectrum is not the result of multiple vibrational modes of the reed, but the deviation from sinusoidal motion caused by the reed's interaction with it's surroundings.

 

 

Hi Dana,

 

Thanks for sharing your extensive experience and understanding on free reed operation. I do, however, believe you make an error with the above statement, which, upon close examination, is self contradictory. Assuming I'm not misunderstanding your statement, any deviation from sinusoidal motion of the reed tongue would necessarily be caused by the excitation of multiple vibrational modes of the tongue. Cottingham at Coe College has measured the time dependence of the vibrating reed tongue and found it to be virtually sinusoidal, but with a very small, albeit measurable, departure, due primarily to the first overtone. I doubt any researcher would attribute the rich harmonic structure of the musical tone to this very slight departure from sinusoidal motion. Thus, the first part of your statement does agree with scientific consensus, with only the second part being in error. The rich harmonic structure of the musical tone is rather due primarily to the sharp pressure pulses caused by the tongue passing back and forth through the slot, with significant modification of these pulses by acoustic interaction with surrounding structures - most significantly the reed cavity. In other words, a perfectly sinusoidal motion of the reed tongue is entirely capable of producing a musical tone with the rich harmonic structure we hear.

 

Best regards,

Tom

www.bluesbox.biz

I am happy to assume what you say is correct regarding the pressure pulses. To be sure the reed is only a moderate part of the influence on the harmonics. I do wonder however given it's mass and the nature of the forces created by the air on the reed especially at the narrowest part of the window ( in a concertina and most of the window depth in a reed with vertical window sides ) something you can feel pretty dramatically when playing, whether the effect on the air stream by the regular deviation of the reed from the true sinusoid isn't amplified. You don't hear the reed anyway, only it's effect on the air. Also since the reed is moving at varying velocities with respect to the air flow, sometimes with sometimes against, I would think the reed's influence on the nature of the pressure pulses is not insignificant. I haven't seen anything definitive about the nature of the downstream side of the system ( chamber, wall thickness, dimensions valve port/ position of the pad hole ect. ) and the only useful stuff I've found has been from empirical tests, not from theory) I'll presume the reed tests you mention were air driven. The key to the harmonic structure is the interaction of all the influences on a regular (sinusoidal or not ) signal in a regular non sinusoidal way. What is actually in the mix is still as far as I am concerned up for grabs.

Dana

Edited by Dana Johnson
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