The greater the neck angle the taller the bridge.

[pefjr]

I am curious after reading this thread Single foot Loar-style bridge how the neck angle effects sound. Can we hear from some builders? Thanks in advance.

[billhay4]

I can't say precisely about sound, but it does affect the downward pressure of the strings on the top. The higher the bridge, the sharper the strings break over the bridge and the greater downward pressure they exert.
Bill

[Fretbear]

Jimmy Gaudreau used to have a '25 Gibson Fern that he played for years that had the neck angle shimmed up by 1/8" to increase the break angle and response.

[fscotte]

I think a stiffer top would need a greater down pressure at the bridge, and vice versa. But neck angle isn't going to change, so all you can do is raise or lower the bridge, which has its own separate issues in relation to playability.

[sunburst]

The pressure on the top from the strings breaking over the bridge is a static pressure. In other words, the strings are just lying there in the bridge slots, unmoving, and the tension on the strings presses the bridge down against the top. The forces on the bridge that make sounds are the dynamic forces from the moving strings, and they are the same regardless of static downward pressure, so more break-over angle does not necessarily increase response or loudness or anything else sound-related. Too much pressure on the bridge may even "choke" the top, or dampen top movement in response to plucking the strings.
There is a "school of thought" that says less downward pressure is better, and that just enough to keep the bridge in position is optimal for sound.
I assume that there is an optimal downward pressure for each instrument and each player and listener, and no absolutes can be stated.

[pefjr]

I was thinking about the difference in the height of the bridge. The distance from the sound board to the strings. Does the neck angle mean less/more volume with a higher angle or maybe that is tap tuned? I have no idea, I just know some mandolins have a higher angle than others. I happen to have one that is high, it might be right at the critical point of being too high, and it has superior tone and volume. I know there are many factors to tone, but is neck angle another one in the magic formula?

[Mando-Mauler]

John Duffy always rejigged his mandolin necks so they had a greater height above the top-plate at the bridge. He maintained it increased volume and made things "stiffer". In one interview, he said that was one of Gibson's main faults...the neck angle was too low.I forget where I read it, but it's out there somewhere.But remember, this mando God belted a mandolin harder than anyone else on Earth.Years ago, Roger Buckmaster (first class maker) and myself, being of inquisitive nature, designed & built a strengthened top F5 with a REALLY sharp neck angle. It put so much pressure on the strings it used to snap the vertical metal tuning posts on the bridge...ended up we had to stick a Gibson guitar bridge on because the posts were thick enough to withstand the pressure. We messed around with this thing for about 25 years & got some weird sounds out of it; some people liked it some of it , other folks ran away screaming. After all those years of rejigging, adding & subtracting things & finally decreasing the neck angle, it plays and sounds beautifully. Only one moral here - don't go overboard. From memory, which is a bit fragile now, the best angle is up or down from a base 17 degrees. What does Steve Gilchrist use?:mandosmiley:

[sunburst]

I was thinking about the difference in the height of the bridge. The distance from the sound board to the strings.

Yes, but depending on other aspects of construction the string break-over angle can be different with the same bridge height. In other words, a higher or lower bridge on the same mandolin might make some difference in the sound (though the mass of the bridge is more likely to make a difference) but one mandolin with a 5/8" bridge might have the same break-over angle as another with a 7/8" bridge, so bridge height is not a good indicator of anything. Same thing with neck angle. If the neck overstand is different and/or if the arch height is different, different neck angles result in different string break-over angles on different mandolins.

[Big Joe]

As a single element in total construction, neck angle is not as critical as numerous other issues. That being said, if the neck angle is too much... or too little... it will be detrimental to tone and volume. There is a pretty wide range that seems to be very workable. The most important issue is to not get it too high or too low. Each instrument responds differently to so many things that it really is the total build and not one single issue that is important. I think the shape and size of the bridge can have more effect than the normal neck angles we see.

If the neck angle is so low the bridge cannot function properly, then it is probably too low. If it is so high that it is hard to keep the bridge in place, then it is likely too high.

[Bernie Daniel]

As a single element in total construction, neck angle is not as critical as numerous other issues. That being said, if the neck angle is too much... or too little... it will be detrimental to tone and volume. There is a pretty wide range that seems to be very workable. The most important issue is to not get it too high or too low. Each instrument responds differently to so many things that it really is the total build and not one single issue that is important. I think the shape and size of the bridge can have more effect than the normal neck angles we see.

If the neck angle is so low the bridge cannot function properly, then it is probably too low. If it is so high that it is hard to keep the bridge in place, then it is likely too high.

Not totally on topic but how do you accurately measure the break angle? I've thought about doing it just so I could make a precise estimation of the down pressure vector on the top plate.

But its an awkward thing to do -- not being able to slip a protractor through the bridge for example. Best I have been able to do is measure the bridge height and then measure the string height two inches on either side of the bridge. I then use those those measurements to draw a simulation of the angle on paper to measure. But that estimate is not too accurate because the top plate is not flat.

[sunburst]

...how do you accurately measure the break angle?

I would use a bevel gauge, set it to match the strings over the bridge and then measure the angle of the gauge.

[Bernie Daniel]

I would use a bevel gauge, set it to match the strings over the bridge and then measure the angle of the gauge.

Of course! Thanks!

Sad to think I hadn't thought of that myself!

[Willie]

Back to Johb Duffey, ....When he built his first "Duck" he said he increased the neck angle over what Gibson was using, I forget exactly to what degree he said he did that but as Mando-Maker stated, John believed that it made the mandolin sound louder and better...He also tried different things with the points by having some filled in and some not....I don`t know if he ever said what he finally decided to do with all of that info when he built the second "duck", I don`t believe #2 sounded exactly like #1 though and it wasn`t exactly the same shape, some variation in size and arch of the top....

Willie

[AlanN]

I had an early Duff where the neck angle was cocked way back, bridge was sky high. Don't really remember how it sounded.

[Darryl Wolfe]

My observations: All else being equal, extreme neck angle and bridge height equal stiff, super heavy guage string feel. Most likely equivalent to what I think John posted as being "bound up"

[Chris Baird]

Neck angle can affect the string's breakover angle at the bridge, which can affect the downward force on the top. The amount of downward force affects the modal frequencies of the instrument and their relationship to one another. This can be easily seen with an FFT program and various degrees of "tuned-up"... If one does a "bonk test" with just enough string tension to hold the bridge on, and then further bonks tests while incrementally tuning the instrument up, you will see that varying downward tension affects the modes.

[bluegrasser78]

Well I Know Duffeys' early F-7 conversion has "IT" and I'd put that mando up against anything out there..It has all the Great things we all look for in a mandolin..And it has the steep neck angle,I believe 1 inch from the top of the mandolin to the top of the bridge..:grin:

[Willie]

Duffey was always tinkering around with his mandolin and most of the time I saw it and heard it it sounded different each time but always sounded good, I am talking about his F-12 conversion, he did play an F-7 earlier but as far as I know that was not his
so I don`t know if did any conversions on it or not....When we are talking about a high bridge with a neck angle being raised the strings can still be close to the fret board but Duffey liked them even high with the steep neck angle... and it surely didn`t kill his mandolin....Or maybe it did because I never heard it as a stock mandolin....

Willie

[arigato]

Chris:

For an elastic system, the initial force does not affect the model properties of the system. The model properties are totally based on the properties of the instrument and not to the load you apply. So are you saying an instrument behaves non-linearly (P-delta affects) while remaining elastic?

[Chris Baird]

Arigato,

I'm not sure what causes the effect I described, however, it can easily be tested as I've said. Try it and see. The forces that the string tension exerts throughout the mandolin affect the modes. The modes of a mandolin with bridge and strings on, but with only the slightest string tension, are significantly different from the same mandolin fully tensioned (tuned up). I assume the combinations of relatively extreme tensions, compressions, and torsions affect the stiffness of the woods??? I don't know.

[Dave Cohen]

I haven't observed what Chris is describing. Early on in Tom Rossing's lab, I did see the (1,0) or sideways rocking mode split right at 440 Hz. After a little puzzling about that, followed by a forehead slap, I realized that the undamped open A strings were stealing energy from the sideways rocking mode, thanks to the overlap in frequency and the non-negligible mass of the strings. Damped the strings after that w/ a piece of foam between the strings and the fingerboard. Never saw that effect again, even though I did all my interferometry w/ the strings at pitch.

More recently, I was doing some interferometry in Thom Moore's lab at Rollins. He wondered out loud about the possible perturbations from the strings. After running through the modes, I removed the strings and did the interferometry again. Within error, the modes were the same and their modal frequencies were nearly the same with no strings as they were with strings at pitch. That is, they were up to about 1.5 kHz. At higher frequencies like that, things started to look a bit different. The main events for plucked string instruments are all well below that, though. So I agree w/ Arigato; the normal modes are s'pozed to be the normal modes, regardless of how you excite them, and that is what I have seen. If Chris is looking at audio spectra (his "bonk testing"), he is seeing a whole lotta stuff besides the body modes all scrambled together. Can't really be sure from audio spectra which peaks correspond to normal mode peaks. I have even had trouble with that despite the hindsight of having just done the interferometry.

http://www.Cohenmando.com

[Chris Baird]

I actually use an accelerometer and a small pcb force transducer hammer to gather frequency response functions. I'll post some data when I string up the next batch of mandolins. I also damp the strings. I'm pretty sure I'm just measuring modes and not audio junk.

I've found really large differences between a mandolin sans bridge and strings vs. fully strung and tuned up. Which I find predictable considering the strings and bridge add considerable weight, if nothing else, to the equation.

Anyway, as I've said, I'll post my data when I finish the next batch.

[Fretbear]

I should know better than to post amongst these big dogs, but string tension, weight and response is something I have spent alot of time putting my attention on. I have a guitar that strung up to pitch is nothing to write home about. If you tune it a half-step down, it is better, but still nothing exceptional. If you tune it down a full-step and increase the string gauges very slightly (to adjust the tension and feel) you start to get that kind of flat-top sound that makes one turn their head. Same guitar, different circumstances.
My mandolin has a full 1" distance from bridge-top to soundboard, necessitating the use of a Tall-Boy bridge and again I use slightly heavier than standard gauges on it. I am not going to brag on my mandolin, but let's just say not only will I be heard in any acoustic situation, but it will also sound sweet as well. I tried to use some of those same .011.5" E strings that I use on my brother's MK mandolin, and all it did was bind it up, shut it down, whatever you want to call that phenomenon when there is too much tension to let the instrument speak loudly and clearly. I don't have a specific point to make, other than that I know from direct experience that there is definitely something to finding the particular tension/action where an individual instrument sounds optimal, even if it involves increasing or decreasing the overall string tension and this will not always be found at "pitch'" or by using "standard" measurements for the neck angle, bridge height, etc.

[pefjr]

I should know better than to post amongst these big dogs, but string tension, weight and response is something I have spent alot of time putting my attention on. I have a guitar that strung up to pitch is nothing to write home about. If you tune it a half-step down, it is better, but still nothing exceptional. If you tune it down a full-step and increase the string gauges very slightly (to adjust the tension and feel) you start to get that kind of flat-top sound that makes one turn their head. Same guitar, different circumstances.
My mandolin has a full 1" distance from bridge-top to soundboard, necessitating the use of a Tall-Boy bridge and again I use slightly heavier than standard gauges on it. I am not going to brag on my mandolin, but let's just say not only will I be heard in any acoustic situation, but it will also sound sweet as well. I tried to use some of those same .011.5" E strings that I use on my brother's MK mandolin, and all it did was bind it up, shut it down, whatever you want to call that phenomenon when there is too much tension to let the instrument speak loudly and clearly. I don't have a specific point to make, other than that I know from direct experience that there is definitely something to finding the particular tension/action where an individual instrument sounds optimal, even if it involves increasing or decreasing the overall string tension and this will not always be found at "pitch'" or by using "standard" measurements for the neck angle, bridge height, etc.Interesting. I had a Martin Guitar for 20 years and thought it sounded great. My son put on a heavier string gauge than I had ever had on it, just before I put it up for sale. I sold it recently to a Hawaiian, but before he bought it, he tried it out and he tuned it to what he called 'slack tuning'. Open 'G' I think. That guitar put out a beautiful sound I had never heard before. Too late to back out, I was already committed to sell it.

[Dave Cohen]

Rossing and I put some accelerance spectra in our 2003 paper inAcoustical Science & Technology, and we identified some of the spectral peaks w/ body mode peak frequencies. We had the benefit of hindsight from just having done the interferometry and knowing the modal frequencies, and even there, it was difficult. With accelerometry, you don't observe the air modes (even though they are there), but there is still a lot of stuff to deal with. I'm assuming that your accelerance spectra are OK, but I think you may be over interpreting your data, trying to say things that you can't really say. Even with interferometry, it is difficult to do the interpretation. There is lots of frequency overlap between the various modes, mode mixing, etc. All of the modes have different bandwidths, Q values, etc., and there is lots of difference between individual mandolins. Even mandolins that look very much alike and were built very similarly can have surprisingly different modal frequencies and Q values for each of the modes.

http://www.Cohenmando.com

[Ed Goist]

Why do double basses have such an extreme neck angles and such tall bridges?

[sunburst]

Violin family instruments, in general, have relatively taller bridges than mandolins. They are designed to be played with a bow (though bass often is plucked), they have a bass bar and a sound post, and the mechanisms involved in sound production are somewhat different from mandolins. Likewise with flat top guitars and other things not directly related to carved top mandolins and their bridges that keep showing up in this thread, things work somewhat differently.

[Chris Baird]

Dave,

I'm not really sure what you mean when you infer that I'm interpreting my data wrong. I'm careful to take my measurements the same way, and, as you know, frequency response functions help produce more consistent results over simple FFTs.

The main differences I see between varying degrees of tension are in the frequencies of the modes. I agree that the higher modes overlap so much that it's hard to get consistent results and interpretations even with the same setup. However, I'm seeing differences from the fundamental body modes on up.

And, like I say, I will post my data on the next batch and we can discuss. I may be wrong and it wouldn't be anything unusual.

[Dave Cohen]

The main problem is that I have done experiments in which there were not significant differences in modal frequencies between a mandolin with eight strings at tension and the same mandolin with no strings at all. That is in direct contradiction with your results.

For an experiment to be "science", it has to be (i) repeatable, (ii) verifiable, meaning that someone else can repeat the experiment and get the same result, (iii) falsifiable, meaning that a future experiment can come along and contradict it. That is, it is not "belief", but rather maintains the potential to be overturned. It may never be overturned, but it has to be designed and expressed such that it potentially could be overturned. Now, I know that I can repeat my experiments with the same results, and I am confident that someone else could use a comparable interferometer and get the same results. If someone should repeat my experiments and get different results, my results would be falsified. And, there is one more thing: Not seeing significantly different lower modal frequencies between a strung mandoln and an unstrung mandolin is consistent with what is known about normal modes of motion in elastic solids. The eigenfrequencies of a mandolin are characteristic of the mandolin itself and should not be significantly affected by a perpendicular static down force (i.e., from the string tension), and that is exactly what I have observed. Since you have observed the opposite, you have some tall explaining to do. It doesn't have to do with frequency response functions vs audio spectra or how you transform your data into the frquency domain. It also doesn't have to do with the very real sound differences people are describing between a guitar at pitch and a slack-key guitar. It has to do with normal modes of vibrational motion, since you originally made a claim about mormal mode eigenfrequencies.

http://www.Cohenmando.com

[Chris Baird]

How is it that a string's modes can change with tension where as a mandolin's body modes can't?

Like I said, in any event, I'll gather some more data and we can discuss.

[Dave Cohen]

"How is it that a string's modes can change with tension whereas a mandolin's body modes can't?"

First, a string is under tension, whereas the mandolin top plate, as you are describing it, is not. It does have a static down force on it from the string tension acting on the bridge, but that is not "tension". The tension in a string is parallel to its' length. The string itself is not very stiff, at least not stiff enough to sustain a normal mode of motion. The tension along its' length effectively stiffens it, until it is effectively stiff enough to sustain a normal vibrational mode. The more tension, the more effectively stiff it gets, with no increase in mass. As it gets stiffer w/ no increase in mass, the modal frequencies have to increase. So much for the string.

The next stage in an analogy would be a banjo head. It is a membrane, and also not very stiff, so it is stretched over the pot until it is in tension and is effectively stiff enough. Increasing the tension does increase the modal frequencies in a banjo head, b/c the increasing tension is effectively making the head stiffer, just as with the string. You can read about that in the chapter just before mine in the Rossing book.

Finally, to a mandolin top plate. The static down force from the string tension does not stiffen the top plate at all, since its' direction is normal to the surface of the plate. The total down force for a set of J74 strings w/ 13.875" scale length is about 50 pounds (+/- 5 lb), assuming standard pitch and a 16 degree breakover angle. The strings do deform the plate a bit, but not very much. In fact, not enough to significantly effect the modal frequencies. Unlike the string and the banjo head, the mandolin top plate is already quite stiff. The amount of static down force it would take to deform the plate enough to significantly increase its' stiffness would be more than enough to cause catastrophic failure, i.e., to crack the plate. Hence, the plate's modal frequencies should not change appreciably under the 50 lb of static down force, which is exactly what I have observed. That doesn't mean that there aren't changes in sound associated with changes in total string tension, but the "sound" is the product of a lot more than just the body modes.

There are a lot of things that can be confusing about modal analysis. Rigidly clamp the instrument, and you get one set of eigenfrequencies. Simply support it, and you allow the neck + body whole-instrument bending motions to perturb (some of) the body modes, which leads to a different set of eigenfrequencies.

http://www.Cohenmando.com

[tburcham]

:popcorn: I'm loving this exchange!

[arigato]

Just as some back ground, I myself am a structural engineer that did a phd in earthquake engineering (non-linear effects). I am a humble guy who thinks that I just went to school longer than someone else, who is more than willing to share some info on what I have learned.

Chris, I think your talking about the response of the instrument (measuring sound waves and such) with the strings on and changing the tension in them, which certainly changes the sound quality emitted from the instrument. What I, and Dave are reffering to is the instrument itself where the strings (the input mechanism) dont add any mass or stiffnes to the system (instrument in this case). If the wood crushes, top plate buckles, etc. the instrument has gone into the non-linear behavior which is not the norm (we cant play anymore now!) so, the instrument under normal conditions remains "elastic" with small displacements expected. Meaning we play the mandolin, put it away, and its essentially the same mandolin we had before we played it.

In that realm, the instrument will not change (modes of the instrument, not of the response) with string height alone. It will change with the bridge because you are adding mass to the system and some energy will be spent deforming the bridge. Playing it standing, or sitting, will change the modes of the instrument (and sound quality) because the boundary conditions of the instrument have changed. We are also assuming that the strings have not appreicable changed the shape of the instrument itself appreciably. If it does, it does change its modal/structural characteristics.

Im still not sure all that schooling I did was worth it, but its helps a ton when trying to undertsand how our little mandolin does what it does! (and im still learning)

[Chris Baird]

As I've said, I'm not trying to explain anything, only report that I've seen significant changes to the modes with changes in string tension. I've seen it many times with both microphone gathered FFTs and hammer/accelerometer FRFs. I may be doing something wrong, but, it is hard to see how. And, I don't think it is worth discussing further until I get some data and method to report.

[arigato]

I look forward to seeing what you got when you get around to it.

[Dave Cohen]

Chris has several times made a distinction between the "fft" (fast fourier trasnform) and the "frf" (frequency response function) which is incorrect and, I think, misleading. I'd like to clear that up fyi.

The Fourier transform is an integral function that effects the transformation from a function in one domain (commonly the time domain) to the domain of a conjugate variable (commonly the frequency domain). It is not specific to sound spectra. The acronym "fft" stands for fast Fourier transform, which takes some computational shortcuts to make calculating the Fourier transform easier and faster for a computer. An frf can be any function that depends on frequency, including accelerance (=F/m), mobility, compliance, stiffness, Impedance, and others.

Whenever data is acquired in the time domain (i.e., some function versus time), some kind of transformation has to be performed in order to get into the frequency domain, i.e., to get an frf. There is more than one kind of "transform" that will do that. There are Fourier transforms, including the fft, Hadamard transforms, and something that was trendy for a while, the "wavelet transform". Most of those transforms are (or at least can be) some kind of integral function. Chris noted that he used an accelerometer and a force hammer. The significance of the force hammer is that he was acquiring data in the time domain. That means that he had to perform some kind of transform in order to get an "frf". Now, chances are it was a Fourier transform or an fft. There are other transforms that offer some advantages, but they are largely used by the folks who are looking at frequency ranges up to and in the megaHerz region. The acoustical region extends up to ~20 kHz as an upper limit. Musical instrument studies typically don't go above 5 kHz, for which the fft is good enough.

http://www.Cohenmando.com

[Dave Cohen]

Forgot to add: Accelerance does not necessarily produce more consistent results than the audio spectra which Chris referred to as "FFT"s. An accelerometer is a motion dector, which means that if you stick it on an instrument top plate, it will respond to the motion of that top plate. It will not respond to the air motion in the cavity or the soundhole, or the motion of other parts of the instrument,or the strings, except inasmuch as they affect the motion of the top plate. So, the accelerometer has the advantage of discriminating against some motions, thus simplifying the resultant spectrum. The catch is that those other motions do have an effect on the top plate motion, or at least they can. So it is not a matter of "better", more like a matter of what you want to observe.

http://www.Cohenmando.com

[Chris Baird]

An FRF is more consistent and reproducible than an FFT because the input is also a factor in the FRF. Which means that you can bonk at different levels of force and come out with nearly the same FRF. And, yes, an FRF uses the input and output FFTs. FRFs can be done with microphones and impact hammers as well. But, an FFT by itself gives less useful data because the input is unknown. Each bonk not only has different levels of force, but, also a different frequency input. For the sake of consistency from one measurement to another, an FRF created from the input FFT (the hammer) and the output FFT(from the accelerometer or microphone) gives much more meaningful data.

[Dave Cohen]

Geez, I try to help and it gets worse. Chris, you keep coming up with wacky uses of "fft". An fft is nothing but a mathematical operation that transforms from one conjugate variable to another. What you are calling "FFT"s are not ffts at all. There is no such thing as an "FFT by itself giving less useful data", nor are there "input ffts" and "output ffts" as you are using the term. An fft doesn't give any data. To get data, you have to excite some physical system and then measure the response with some kind of transducer. If you do an fft on some function of time, you get anther function of a conjugate variable. You don't then do a 2nd fft on the transformed function. Do an fft on a function and follow it with an inverse fft, and you are back to where you started, in the domain of whatever variable you started with, e.g., time. I can only guess at what you are trying to say; maybe you mean "signal"?

Chris does have one minor point mixed in with the misconceptions. If one wants to compare amplitudes of audio spectra or other frequency response functions, you need to be very careful about how you excite the system. However, whether you use an impact hammer, an acrylic ball, or your knuckle to excite the body of a mandolin, in every case you are applying an impulse during a short time interval ("dt") which has many frequency components and will excite many or even most of the normal modes of the system. And I stress, the normal modes of a system are characteristic of the system, and not the excitation. They have characteristic modal frequencies ("eigenfrequencies"), which do not change with differing excitations. An impact hammer will have more high frequency components b/c it effects an impulse over a shorter time interval. A knuckle has fewer high frequency components b/c it is "squishy" and the duration of the impulse is longer. But in neither case will the modal frequencies of a given instrument be different than what they are. The relative amplitudes will certainly be different, but not the frequencies. I have done countless bonks of plates and assembled mandolins and guitars and transformed to get the spectra. In every case, every bonk yielded spectra with the very same peak frequencies for a given instrument.

http://www.Cohenmando.com

[Tavy]

Chris, can you say all that again without the acronyms? :(

Apologies, but I'm having difficulty understanding what you're actually doing - as Dave says there's simply no such thing as an "input FFT".

Dave: what you're saying matches up with my limited experience - how you bonk, and where you bonk effects which modes get excited and how strongly they're excited, but not their frequencies - no doubt if you were clever enough the positional differences would tell you useful information as well. I haven't compared strung up with not strung up to see if there are changes, but I might just rush off and do so..... or not.... ;)

[Chris Baird]

Ok, yes, I am using "FFT" to refer to data obtained from a time domain signal transformed into frequency domain using an FFT. I should have used the term "spectrum", or FFT results, or frequency components of an audio signal, etc.

The results of a frequency response function (FRF) can be calculated in a variety of ways, but, it always uses data from the impulse (the hammer) and data from the response (the accelerometer or microphone).

And Dave is right, there is no difference between the frequency peaks of a measurement that doesn't use the impulse. But, when you measure the impulse you can then calculate damping, bandwidth, relative amplitudes of all the modes, and also create a matrix that can be used to model the modal shapes of the mandolin using curve fitting etc.

Joseph Curtin has an article that is very well written on calculating the frequency response of a violin using an impact hammer and microphone. I don't have this nice a setup, however, Mike Kemnitzer aka"Nugget" does.

http://www.josephcurtinstudios.com/images/violinsoundradiation.pdf