Why do double basses have such an extreme neck angles and such tall bridges?
Why do double basses have such an extreme neck angles and such tall bridges?
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.
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.
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.
"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.
I'm loving this exchange!
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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)
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.
I look forward to seeing what you got when you get around to it.
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.
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.
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.
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.
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....
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.