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So this is something I've never understood but I'm curious about.

Specifically as a radius gets flatter or rounder how does the tone differ?

A tighter radius creates a stiffer top accentuating the higher frequencies as the mass doesn't appreciably increase in relation to the increased stiffness of the top due to the radius. How much the highs are accentuated is difficult to say, since like everything else in a guitar it depends on a lot of other factors taken as a whole, it's a system.

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Jim Watts
http://jameswattsguitars.com

Although the stiffness increase as you radius the top is real, I question how much of an effect it has in practice. Certainly it's true that carved arched tops gain usefully in stiffness, but the usual 25' radius or so that is used on flat tops would have an effect that could easily be swamped by other things, many of which could be hard to quantify. Given that it's probably impossible to make 'identical' guitars that sound the same, it would be hard to have any confidence in the small added stiffness making a consistently audible difference in itself.

Most of the effect on archtop guitars is in the low frequency range; the lowest 'main top' mode is shifted upward in pitch as compared with a flat top. As you go up in frequency, and the vibrating areas of the top get smaller, the frequency effect becomes less and less, since the amount of curve is decreased relative to the size of the vibrating area.

Arching the top of a 'flat top' guitar does seem to help with longevity. For one thing, as the top shrinks with drops in humidity the arch goes flatter. The top is unlikely to crack until it gets flat, so having a bit of arch built in gives it someplace to go before then. Here in New England, where we see wide swings in R.H., and particularly low values during the heating season, it helps.

A physicist and or an engineer will argue with mathematics that a top doesn't get stiffer if you arch it. Stiffness is an inherent property of a material that has to do with its Young's Modulus and it's volumetric measurement. What arching does do though is make it stronger. When you see those old roman arched aqueducts it illustrates the point. So like the forces on the bridge of an archtop guitar for example, the arched top is stronger against the downward force of the bridge.

It seems to me that it does something to the tone though. I have built many of the same model guitar in both true flat top and arched. They sound different but then all guitars do too so it's hard to tell whats real. I mostly am building with arched tops now just because it's a good idea to build in that safety net for RH fluctuations.

It's a great question and one I tried to answer myself by building true flat tops and I have not really concluded anything.

so many theories I have heard after 20 years of doing this is that often if you can hear the difference you can also wag your tail. I have seen and done many difference bracing schemes. I think in the long ans short of things , this is what I learned
A it will most likely sound like a guitar
B glues stick wood together. If your joinery is poor don't expect it to give you good result glue isn't a filler
C hide glue is better than tite bond and thats better than this glue. fact is they all can work
All glues have their uses I like HHG and Fish glue for ease of repair again joinery is more

D things I do think are worth working on
Brace location and angle of the X braces
E do not over scallop
F JOINERY and this is so important
G neck angle and total string height at the bridge. I find 1/2 in height is a good target for energy exchange. Too high can over torque the top
H Woods amazingly you can make great sounding guitars out of lower grade wood.
I don't believe everything you read in forums
J To make a good sounding guitar is easier than making a good looking one

to sum things up and I hope my other pro friends can add to this
Fit and finish it hard to achieve so take your time do not force a joint , use proper gluing techniques. The more I do this the less I use CA.
I like
fish and hhg for bracing
fish to glue the top and back
I like dovetails but any secure neck joint will do the job.

fretting is very important the better the fret work the better the guitar will play

tite bond is great of fingerboards I stopped epoxy for that

again it is to be fun and we all hear things differently the perfect guitar has yet to be made so lets keep building till we learn how to make that perfect guitar.

I have met so many great people in my years in this profession and I am proud to call many friends
stay well we may disagree but we don't have to be disagreeable

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John Hall
blues creek guitars
Authorized CF Martin Repair
Co President of ASIA
You Don't know what you don't know until you know it

jfmckenna wrote:
"A physicist and or an engineer will argue with mathematics that a top doesn't get stiffer if you arch it. Stiffness is an inherent property of a material that has to do with its Young's Modulus and it's volumetric measurement."

Ever try to bend a piece of corrugated metal? It's easier one way than the other.

Why do they make magnetic drive speakers conical instead of flat?

Material properties are certainly important in determining the stiffness of a structure, but section properties are as well.

An arched top deflects less under the down load of the bridge: it's stiffer than a flat top of the same material and thickness under that load. The material of the top is not stronger, though, as you might find if you drop it.

I certainly could be wrong but it's my understanding that the stiffness of a plate or a beam is dependent on it's material property Youngs Modulus and it's width, length and height.

If you measure YM then that will never change. It's an inherent property of the material. So the only way to change the rest of the equation and make a plate stiffer is to make it thicker or shorten it.

Does arching it make it stiffer? According to my (possibly wrong) understanding of it no, but an arced structure has a bunch of other physics acting on it. Maybe it's just semantics I don't know.

Carry this analysis to an extreme. Arch a beam like the St. Louis arch. It will deflect less under a given load than a straight beam of the same dimensions, simply because more of the load is compressing it axially.
Any experimentation involving altering the arching of the top will reveal the differences in sound attributable to the increase in stiffness with more arch. It is one of the reasons Gibsons don't sound like Martins.

Sent from my SM-G950U1 using Tapatalk

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John

" it's my understanding that the stiffness of a plate or a beam is dependent on it's material property Youngs Modulus and it's width, length and height."

That's correct as far as it goes.

Take a a narrow strip of paper, say 1/2" wide. Hold it flat on the table, with the surface parallel to the ground, and pick up one end: the free end bends down, of course. It doesn't take much force because even though the Young's modulus of paper is rather high, it's also very thin, and the bending stiffness goes as the cube of the thickness. Now bend it to form a flange on either side by turning up 1/8" of paper and leaving it perpendicular to the surface. Now when you pick up one end it doesn't droop because you've made the height of the 'beam' very much greater out at the edge. You get much the same effect if you crease the paper along it's length with a fold down the center. It's harder to get paper to form a nice arch, but you can see that the effect should be similar. I've done that with wood. I will note that with the paper it's stiffer with the flanges or the 'V' facing up: when the free edges are in compression there's a tendency for the thin paper to buckle in compression, and the stiffening effect is lost.

Of course, in the case of a fold or a cylindrical arch the stiffening effect only happens along one axis. The Howe-Orme guitar company used a cylindrical arch on their instruments to stiffen the top against the torque of the bridge without having to make the top thicker and heavier or adding braces along the grain.

Many makers these days use a radius dish when gluing in shaped braces, so that the top and back plates are nominally spherical, except perhaps above the sound hole, where they are often flat. Normally a radius in the range of 15'-30' is used. Classical guitars have traditionally been made with the top rim flat around the edge, but with some degree of arching in the lower bout, which can be somewhat arbitrary in shape. One method advocates having a dome between the bridge and the waist, with the top flat behind the bridge, to begin with. Bridge torque from the static string load pushes the arch down in front and pulls it up behind the bridge, resulting in a more or less uniform arch of the lower bout. However you do it, this adds stiffness along both axes.

A top that has been glued down all around the edges has to bend in both directions at the same time, if it is to bend at all. Even a cylindrical arch probably adds some cross stiffness in effect on something like a Howe-Orme, although I have not measured it.

Violin or archtop guitar plates use a more complex arch, with recurves around the edges, and those classical guitar tops do something similar to a much lesser extent. The exact configuration of the arch has effects on the way the plate vibrates at various frequencies, and produces somewhat different 'voices' depending. A violin or arch top guitar maker can use this, along with the way they graduate the thickness in different places, as a way to control the sound. In some sense the shape of the arch serves a similar purpose as the bracing on a flat top guitar. Careful measurements of 'old master' violins suggest that certain graduation and arching schemes go together.

Neapolitan mandolins and similar instrument use a 'cranked' top; creased just behind the bridge location and folded down to allow for the string break angle over a low bridge, and add stiffness to withstand the force. The use of a nice, big cross brace right there helps, too.

Carved arches are used to withstand the down load on the top imposed by the break angle over the bridge when a tail piece is used, in the same way an arch or dome is used in architecture. Viols da gamba are often made with a top in three or five staves. The center piece is left fairly thick and bent to a longitudinal arch, to which similarly bent staves can be glued on, at a bit of an angle, to get more width (think of a barrel). The outer segments are solid. The whole thing is then carved to the desired contour. It's also possible to heat form arches with other than cylindrical shapes: the Australian maker Smallman uses a combination of bending and carving to form BRW backs that are about 5 mm thick. We've all seen laminated plywood archtop guitars and basses.

Sometimes it's hard to stop typing: this is an interesting subject with lots of branches one could follow up. The point is, again as with the dome of a building, the geometry of an arch or dome can add stiffness, both in supporting a load and in vibration. The stone in Brunelleschi's dome is no stronger than the stuff in the flat wall, but the dome can support a load over a much larger span than a stone beam because the dome is all in compression. Stacked stone is no good in tension. I really must stop....

I look at this whole subject in a way that it appears hasn't been mentioned in this thread...and that's the fact that on a "flat top" acoustic guitar the "dome" is created by gluing curved braces to a flat plate of wood, usually using a radius dish to "force" things in place until the glue sets.

so, OK, fine then...what's been done is to force a flat plate of wood (most likely 2 pieces joined together) into a curve...this is not the natural state of that plate of wood...it puts stress on it, and therefor changes the way it's going to react as compared to a plate that's not stressed, rather simply reinforced via the braces.

from my experiences in cold bending woods to conform with some situation the following occurs: at first things move (bend) fairly easily...the further you push the wood the more force is required (probably an exponential aspect as opposed to logarithmic)...enough force and the wood fails (snaps in two, splinters but still holds together...whatever)...

the point being is that by pushing a flat plate of wood into an arch/dome/cylinder/whatever you've for all purposes made it stiffer than when it was in it's natural state of being "flat"

I guess if some curious soul has time on their hands they could build a flat top guitar, record it's data, then rebuild the guitar changing only the braces and convert it to a dome topped guitar ...of course that endeavor is flawed in that new wood will be used for the replacement braces and that of course will cause lots of arguments about any perceived results now won't it?

Arching does make a plate stiffer. There is another property in play other than Young's modulus, it's the section modulus. Different shapes use different formulas.
The stiffness of a structure is E*I. E= Young's modulus and I = the section modulus.
I've attached a couple of images from finite element runs showing the effects of doming.
The first image shows a flat 16 inch disk constrained at the edges with a 10 N load in the center, it deflects .25 mm. The second is the same set up only with a 25FT radius (somewhat typical for a guitar), it deflects .17 mm.
I also looked at the natural frequency of the plates and the doming raised the first mode about 20%. I'm not saying a guitar will increase 20% due to doming, just that doming impacts the resonance of the plate.
I hope this helpful to some. Oh, sorry about the large images

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Jim Watts
http://jameswattsguitars.com

I really like seeing things like this, Jim. I've always said: there are three kinds of people, those who are good with math and those who aren't.
John Greven once gave a talk about two reasons why today's guitars don't have the pre-war Martin sound. First, the red spruce we get now is not like the old stuff. He claims Lutz is whats needed. More importantly, he claims the arched soundboards most are using pushes the fundamental pitch higher than the old flattops. As far as the spruce goes, maybe there is something to it, but raising the pitch of the top would be something that we could actually hear.

This is an interesting discussion. Opinions/theories are coming from a variety of disciplines.
Two points that I think are salient:
If you arch the soundboard you raise its resonant frequency. Just try it with an unbraced soundboard. Hold it against your chest with minimal contact and no compression across the grain. Listen to the tap tone (This sounds like a Spinal Tap song). Then, bend it into a slight cylindrical arch and tap again. The pitch goes up.
The second point is the visual examples of domes and Roman arches are not relevant, IMO. The forces on domes, etc, are downward forces. An equivalent analogy of designing a dome on a soundboard in the same fashion would dome the soundboard inward so that the force is pulling the dome into compression. While it is true the soundboard experiences some downward compression in front of the bridge due to rotational force, it is safe to assume most of the tension on a soundboard is pulling the soundboard up in relation to the edges. Here's another visual example: The Hoover dam is curved inward towards the impounded lake, which is the same method as the aforementioned domes of conveying compression to the periphery. If the Hoover Dam were curved the same way a guitar soundboard is arched the curve would be in the opposite direction. This would rely on the tensile strength of the concrete dam rather than the compressive strength.

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Stay with the happy people.
--Reynolds Large

Stone and concrete arches are curved the direction they are so that the material is in compression. Stone and cement are lousy in tension.

I looked at the effect of arch height on 'free' plate vibrations in my series of articles on plate tuning by Chladni patterns in American Lutherie. See, in particular, AL # 28, Winter '91, pp 29 (text) and 31 (diagram), or the appropriate 'Big Red Book'. Of the lowest five resonant modes the only one that changes pitch as the arch goes up is the 'ring' mode. On a flat plate the restoring forces are provided by the Young's modulus in bending the plates (which is different along and across the grain in wood) and the Poisson's ratios of the material. On an arched plate the edge has to get longer or shorter in the ring mode so the Young's modulus also enters in direct tension and compression of the edge, raising the mode pitch.

I don't think any 'flat top' guitar top is actually flat once you put the strings on. Deformation under string load certainly does slightly affect the pitches of the higher resonant modes of the assembled guitar top, with the effect size and sign (higher or lower pitch) mostly depending on how the local curve impacts the particular bending mode.

Some years back Joshua Gordis, of the Naval Post-Graduate Research Institute, pointed out to me that when you load a column the pitch of its fundamental bending mode drops, going to zero at the buckling load. He wondered if there was a change in the pitches of the lower resonant modes of a top under string tension. Aside from small changes, as cited, mostly to higher order modes, there is no major effect on the lower 'signature' modes of a flat top. The 'main top' resonant mode on an archtop drops in pitch as the string tension rises, as one would expect under the column loading model. You can see the same effect if you cut a disc of foam plastic and check the 'ring' mode frequency with a tight rubber band on it, as compared with the same bad cut so that it is slack, and taped on with double stick tape. On a flat top the plate in in tension behind the bridge, and compression in front, and it seems to more or less cancel out.

I do note that the 'main back' mode can (but does not always) rise in pitch with string tension, because the back is being stretched....

Old guitars seem to 'lose it' IMO when the soundboard buckles in front of the bridge, though. That may explain why the most successful brace patterns, 'fan' bracing on classicals and 'X' bracing on steel strings, tend to concentrate wood in the area between the bridge and the sound hole.

The ca 1780 English guittar (cittern) that one of my students restored used X bracing to form a thin, flat top into a tall dome, both to provide the necessary break angle over the bridge with the strings tied off at the tail block, and to withstand the consequent down load. He took it from basket case that had been written off by the museum to a very nice performing instrument, learning a lot of repair techniques along the way. Whether you call it an 'arch' or a 'dome' is somewhat a matter of degree, with no clear dividing line IMO.

Old growth red spruce is available today, if you spend enough money.

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John

For those into the mathematics of this, in 1955 Reissner published a paper entitled "On the axi-symmetrical vibrations of shallow spherical shells" and included a formula for predicting the vibrational frequencies of the lower modes. Fig. 1.6-10 in the book charts the results. In a nutshell, his formula predicts that a 10 foot radius dome is required to double the frequency of the first mode on a shell of guitar sized dimensions compared with a flat plate. The relationship between doming and frequency is not linear though.

With respect to the sound, doming adds stiffness as other have said, and consequently you get the sound of a stiffer guitar.

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

Ok so I learned something, I think. My thinking was sort of like this... That a gram of water is a gram of water regardless if it is spread out into it's gaseous state of steam or condensed into ice. That it is an inherent property of the substance. That is what I was thinking of what the term 'stiffness' meant. So stiffness can be changed not just by making a plate thicker or thinner, longer or shorter, but also by bending it into an arch? My thoughts were that it was a constant and a constant can be taken to Mars and it would still be the same thing.

JF - yes, you were missing the section modulus piece of it, another way to put that is it's geometric stiffness. As you state though the material properties are constant regardless of the shape. Different shapes have different section modulus formulas, not everything is treated as a beam or flat plate.

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Jim Watts
http://jameswattsguitars.com

Young's modulus is the material property; a measure of how much effort it takes to stretch a given size piece by a certain amount, if I have that right. When you bend a piece of wood most of the restoring force that is resisting the bending comes from stretching and compressing the wood at the surface. If you think about a wood beam making it deeper adds to the stiffness: a 2x4 up on it's side is stiffer than it is on it's face. Stiffness is the resistance to bending, which varies depending on the size and shape, as well as the material properties. We do tend to mix those things up: 'Young's modulus' and 'density' are material properties, while 'stiffness' and 'weight' depend on the size and shape of the object. You can't talk about the 'stiffness to weight ratio' of a piece of wood unless you specify the size and shape, but given those, and the Young's modulus and density, you can calculate stiffness and weight.

I brought this topic up on my personal facebook page and tagged a bunch of engineers and a few phd physicist that I know and boy howdy did that bring up a storm

They all do agree though that the top of the guitar will be stiffer if arched. Did you know that a tube can be stiffer then a solid rod of the same diameter?

https://physics.stackexchange.com/quest ... bsbv57vTDs

About as counter-intuitive as you can get. That’s why we need people good at math.

I did not read it that way. What I got was:
1) given a rod and a tube of the same outside diameter and material, the rod would be somewhat stiffer, assuming it had sufficient wall thickness to avoid buckling or denting under the imposed load.
2) given a rod and a tube of the same length and material, and the same mass, the tube would be larger in diameter and stiffer in bending, however,
3) the tube would have a higher frequency in bending, since the mass is less relative to the stiffness. The tube might thus withstand a higher load in axial compression than the rod before buckling, so long as the wall thickness was sufficient to withstand local failure.

The way to think about this is as analogous to an I-beam. The stiffness of a round rod is proportional to the Young's modulus and the fourth power of the diameter, assuming the same length. To a first approximation you could calculate the stiffness of a rod of the appropriate O.D., and then subtract the stiffness of the rod that you drill out to make the hole down the middle of the tube. A 1" diameter tube with 1/4" wall thickness should be as stiff as a 1" diameter rod minus the stiffness of a 1/2" diameter rod. If I did the math right it's .9375 as stiff as the solid rod, and weighs about 31% less? A tube with the same mass as the 1" rod would be larger in diameter and stiffer.

It is stiffer when weight is factored in, but if weight is not a factor then an equal diameter hollow tube is less stiff.

Ah, got it. I’m leaning back to trusting my intuition more now.

My head is spinning after reading most of this. I dome my tops with the expectation that they will be slightly more resistant to deformation due to string tension... and tap tune the tops to my target frequencies. On my jumbo's I do a 30' radius on the tops and on the parlor and dread, 40' radius. Backs are 15' on all three.

Make what sounds good. If it doesn't sound good, do some research or experimentation to figure out why. I find that 80% research and 20% experimentation seems about right.