Graduated thickness of top plates

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Graduated thickness of top plates

Postby John Clifford » Tue Jun 19, 2018 3:43 pm

I’m thinking about the graduation of the top plate in an archtop guitar. The conventional wisdom (Benedetto, et al.) is that the plate should be thicker in the center (along the longitudinal axis), and thinner toward the edges, with the recurve being the thinnest point. The three rationales commonly given for this are (1) that’s how the Cremonese masters carved their violins and cellos, (2) this will allow the top plate to move in and out like a speaker cone, and (3) it is structurally necessary.

However, there is at least some research suggesting that, in fact, this is NOT how the Cremonese masters did it most of the time:

http://www.liutaiomottola.com/myth/varch.htm

And other research seems to have proven, rather conclusively, that guitar tops (regardless of how carved) do NOT in fact move in and out like speaker cones, but rather ripple with standing waves traveling in and out from the bridge to nodal lines inside the perimeter. (See, e.g., T. Gore)

And the structural argument doesn’t make sense to me either. I’m not an engineer, but it seems to me that if you were designing an arch to withstand downward pressure, you would make it thinner at the top and thicker toward the base, like say this one:

https://en.wikipedia.org/wiki/Gateway_A ... ropped.jpg

Also, there is the fact that “flattop” guitars use plates of mostly uniform thickness (sometimes thinned at the edges to push the nodal lines as far out as possible). So I see no reason why the structural needs of an archtop guitar cannot be satisfied with adequate bracing.

So what am I missing? Is there really any reason the top plate shouldn’t be carved uniformly thin?
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Re: Graduated thickness of top plates

Postby Brian Evans » Wed Jun 20, 2018 8:04 am

The top plate is under two different loads - one is approximately 35 lbs of down force on the bridge and the other is around 170 lbs of longitudinal force from string tension between the neck and tail blocks. Of the two, the string tension load is what will most likely eventually cause a neck reset if the body is made too weakly. You can "fix" either situation with bracing, and Benedetto recommends different graduations for X braced vs parallel braced tops for this reason. You can adjust the download with changes to bridge string break angle (we had a recent thread discussing that). Reading the Benedetto book he warns against carving too thin, saying "it's a common misconception that if the top is thinly carved it will vibrate more and produce a better sounding guitar. Nothing could be farther from the truth." One can only assume that for him to say that he did an experiment or two to see what happened, which is something that anyone could do.

Many archtops made by the masters of the 1930's and 1940's used quite different bracing techniques to the now accepted X and Parallel standards. Flat top guitars are such a different beast that I don't think there is any value at all in looking there for inspiration. I even think that an archtop with no bracing at all would be a great experiment, sound-wise (my 1946 Epiphone Zephyr, being a laminated body electric archtop, has no bracing at all). I also know that adjusting the recurve results in significant changes to the tone of the box, when doing the final adjustments. The St Louis arch is an visual art piece, not the best example of a structurally efficient arch. Maybe the simplest example of a structurally strong arch is a simple roof truss - very thin at the edges. I have a 65 foot span truss system in my big shed that is made of 2X8's with a web of 2X4's. It's 10" thick at the edge, where it meets the wall, and is 12 feet deep at the center where it peaks.

Just some of the thoughts I've had while considering the same questions you raised! Good questions...

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Re: Graduated thickness of top plates

Postby John Clifford » Wed Jun 20, 2018 10:13 am

Of course I realize the longitudinal string tension is the greatest stress, but it seems to me that a carved arch that tapers in thickness from side to side cannot be the most efficient or easily calibrated structure for handling that stress. Nor would something like your roof truss. Here you are simply trying to keep the neck and tail block apart (not the direction of stress that an arch or truss is designed to handle), so some sort of beam like a brace would seem to be the obvious solution.

I know Benedetto (and everybody else) says that an arch of graduated thickness, tapering toward the edges, is necessary for the guitar to sound right, I'm just wondering why that would be. Depending on what research you believe, the old violin makers may or may not have followed that method. You're certainly right that adjusting the recurve makes a big difference, but so do many other things, including thinning the entire top.

Yes there are vast differences between flattops and archtops, but they both sound like guitars because they both produce sound in fundamentally the same way - a vibrating plate with one or more holes in it, affixed to an enclosed box, and excited by plucked strings. As I understand it, the mechanics of plate vibration work essentially the same way whether the plate is flat or arched. To me, that implies that the same factors of surface area, stiffness and mass should control the sound of both.

Anyway, not really arguing with you, just pondering the mysteries! I guess the only thing to do at this point is keep experimenting!
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Re: Graduated thickness of top plates

Postby Brian Evans » Wed Jun 20, 2018 12:53 pm

I've put a 1/2" maple dowel between the tail and neck blocks on two guitars that have curly redwood tops. The redwood seems a lot lighter and more flimsy than spruce, so they got the extra brace (which is under the top). That takes much of the distorting load of the string tension. I might use than on a tradition X braced spruce top too, next time I make one. I didn't use it on the parallel braced spruce top I am jut now finishing. Consider the point load of the bridge - does the thicker top directly under the bridge act like a sort of brace to support and distribute that specific load? It does, obviously, but does it have to? To make other comparisons, for me an archtop is very much closer to a pure monocoque in structure, like an aircraft or an egg, while a flat top is much more like a frame structure with a stressed skin. When I look at the bracing of a flat top I am reminded of flying buttresses keeping church walls from collapsing, and WWI aircraft made of triangulated braces covered with a stressed skin of varnished cloth. The top has little inherent strength so you add braces to make it strong only where needed, while an arched top is kind of strong everywhere. Consider the F-holes - located precisely where they interrupt the load flow across the top from the bridge to the sides. Why does that work? I've started making archtops without F-holes to see what happens, so far I am very happy with the result. Changes the nature of the sound a lot, for me in a good way. You're right - many many questions have I...
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Re: Graduated thickness of top plates

Postby John Clifford » Wed Jun 20, 2018 1:46 pm

The guitar I'm working on now is going to have a single sound hole in the upper bout, something like this drawing. I'm leaning toward some version of x bracing, possibly with one or more additional cross bars. I'm going to try carving the top closer to a uniform thickness in the vicinity of 3/16", but still thinning to around 1/8" in the recurve area.

It's tempting to conceive of the recurve as functioning something like the rubber suspension ring around a loudspeaker, but this seems to be illusory. Rather I suspect (but don't know) that it has the effect of extending the wave form closer to the outer edge - or stated another way, moving the nodal line outwards, thereby increasing the in-phase surface area of the vibrating plate and lowering its resonance. Also of course you are removing mass in an area that is structurally supported by the sides.

Interesting stuff. Thanks for your comments.
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Re: Graduated thickness of top plates

Postby Brian Evans » Wed Jun 20, 2018 3:18 pm

Here is my latest... It sounds really quite good, exceptional low tones - the "air resonance" sings around low F, but does not create a deadish tone. I tune it to D-D, one step down, to take advantage of the bass response being so good. 16" lower bout, 2 3/4" deep sides, redwood over birdseye maple, X-braced with an additional transverse brace below the bridge, and that longitudinal brace below the top..
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Re: Graduated thickness of top plates

Postby Brian Evans » Wed Jun 20, 2018 4:12 pm

If you like conspiracy theories, here are facts to ponder. The holes in the side/top of my guitar add up to 10 square inches of area and 23.5" of perimeter length. The area of a 3.6" diameter hole is very close to 10 square inches, but has only 11.3" of perimeter length. Be interesting to compare to F-holes, which are quite variable but I think trend towards 23 - 30 inches of perimeter and 10" - 15" of area. I wonder what that means...
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Re: Graduated thickness of top plates

Postby Beate Ritzert » Wed Jun 20, 2018 4:14 pm

Brian Evans wrote: I've started making archtops without F-holes to see what happens, so far I am very happy with the result. Changes the nature of the sound a lot, for me in a good way.


May i ask in which way (leering a bit to the archtop bass diskussion...)?
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Re: Graduated thickness of top plates

Postby John Clifford » Wed Jun 20, 2018 5:09 pm

Very nice work, Brian! I like the woods and modern shape. Looks like we are both fans of celluloid binding. What made you decide to include the transverse brace below the bridge? The one I'm working on now will have a western red cedar top, which is a lot floppier than spruce, and I'm planning to carve it thinner in the center, so I may need something like that.

I wish we could have an archtop makers meet-up so we could see and play each others' instruments. It's a rare occasion when I get to try anything besides a factory guitar, so I have few points of reference for what I'm trying to accomplish.
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Re: Graduated thickness of top plates

Postby Brian Evans » Wed Jun 20, 2018 8:24 pm

Beate Ritzert wrote:
Brian Evans wrote: I've started making archtops without F-holes to see what happens, so far I am very happy with the result. Changes the nature of the sound a lot, for me in a good way.


May i ask in which way (leering a bit to the archtop bass diskussion...)?


Beate, in a very good way. My previous guitars had suffered, as do many if not most guitars I have played, from dead notes, and from inharmonicity when playing the bass strings in the upper frets. This guitar sounds, to me, very much like a good archtop amplified with a good pickup and amplifier - a rounded, balanced tone, mid rangy, but with overtones and nuances. Odd to say an acoustic guitar sounds like an electric guitar, but there is that jazz tone that is fixed in my ear. Volume is very balanced across the octaves. It's possible to play a mid A on the fifth string 12th fret and third string second fret and they sound the same - all you have to do is move the pick a bit closer to the bridge for the fifth string A. Similarly a D sixth and fourth string. I've never heard that before, usually a string played full or almost full length has more richness and sounds more in tune with itself. Trebles are round and balanced. But the bass is tremendous, due I am convinced to the helmholtz resonance around low F. Low F isn't dead, and the notes around it, to low D, are huge. I've only been playing it for two or three weeks, and I haven't heard anyone else play it, but my wife says it sounds very big and loud.
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Re: Graduated thickness of top plates

Postby Alan Carruth » Thu Jun 21, 2018 2:54 pm

I tried out the 'reverse graduation' on fiddle tops as soon as I heard about it, and it seemed to work fine. Some of the literature I've seen suggests that making the top uniform in thickness, except for a small, slightly thinner, area along the center line near the bridge was used in conjunction with 'curtate cycloid' cross arches, so that's what I've been doing. I have also used that system on several archtops guitars, with good results. I have not made so many that I've fully explored all of the ins and outs of it by any means, though.

Back when I was working out how to make arch top Classical guitars I thought that making the top thin and the arch high would be the way to go. The object was to make a light top that would have good trebles, so it seemed logical. The ones I made that way didn't sound very good as Classicals: I put Thomastic 'S' series Classical strings, which use steel rope cores wound with either plastic or metal, on one, and it had a good sound...for Blues. Ah well. At that point I got a copy of John Schelling's article on 'The Violin as a Circuit'. In it (in a footnote) he says that if dynamic similarity is desired arch height should scale with plate thickness, rather than something like body length. The next arch Classical I made had a 'reverse graduated' top 3mm thick, with the same arch height I'd use on a violin with that thickness, 15mm as measured from the bottom of the plate to the top of the arch at the bridge. It worked great.

Generally on the reverse graduated arch top steel strings I've used a plate thicknesses ranging from 3.5mm to 4.5mm OA, with a central area about .5mm thinner tapering out over an area about 5 x 15 cm. Arch heights range from 18-21mm as measured over the top. On my most recent arch top, which was based more closely on a Gibson 'Lloyd Loar', I went with graduations on the top that were similar to the original: 5mm in the center tapering out to 3.5 outside of the F-holes and into the upper bout. That one also has 'parallel' bracing, instead of my more usual X brace.

Arch top guitar tops vibrate very similarly to flat tops in many respects. The two main differences that I see are that:
1) resonant modes on a flat top stop at the upper transverse brace, whereas they can run all the way up to the neck block on an arch top, and,
2) the lower order modes on an arch top tend to be at higher frequencies, due to the stiffening effect of the arch height.
One of the rationals behind making arch top Classical guitars was to get more top area moving to produce more sound. It seemed to work. Of course, there's a lot of stuff to balance to get that to happen!

It's important to keep in mind that using the resonant modes as a way of describing the movement of a top is misleading in some ways. The modes on a guitar top are not sharply defined in pitch the way they are on strings. Your low A string will only vibrate easily at more or less exact multiples of 110 Hz: it's a lot harder to move it at 111, or even 110.2. That's why it's a usable signal generator for music making. Plates have wider band widths: the resonant pitch of the loudspeaker-like 'main top' mode might be close to 200 Hz, but you have to go out about 5-10 Hz on either side before the amplitude only falls off to 1/2 of the peak value. Also, since that 'main top' mode is by far the most effective it's likely to be the main sound producer all the way out to 1000 Hz; it has a 'long tail'.
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Re: Graduated thickness of top plates

Postby John Clifford » Thu Jun 21, 2018 3:59 pm

Thanks for the information, Alan! I'm not sure that entirely clears things up for me, but at least it reassures me that it's not foolish to try something other than the standard "Benedetto" thickness profile. I know James D'Aquisto felt that a flatter arch yielded better acoustic tone (whatever that is). That kind of makes sense to me when considered along with the "standing wave" model of plate vibration. That is, if you conceive of the top plate rippling in a way somewhat analogous to what happens on the surface of a pond when you toss in a stone, rather than moving in and out all at once like a loudspeaker, then it also makes intuitive sense that a flatter, thinner profile would be more conducive. Is this path ultimately leading to turning the archtop into nothing but a really hard way of making a flattop? Who knows, maybe.
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Re: Graduated thickness of top plates

Postby Alan Carruth » Fri Jun 22, 2018 12:06 pm

All resonances are 'standing waves'. There are a large number of them in any guitar top. The lowest frequency one is normally the 'loudspeaker' mode, where the whole top moves in the same direction at any given time. then come a couple of 'dipole' modes, where the vibrating area is in two parts going in opposite directions. On most guitars these will be a 'cross' dipole, where the treble side goes down while the bass side goes up, ad a 'long' dipole, where the dividing line runs across the top, usually near the bridge, and the area below goes down while the area above goes up. As you go up in pitch you get more and more areas that get smaller and smaller.

The 'dipoles' illustrate one important point: at resonance adjacent areas will always be out of phase: always moving in opposite directions. There's a 'node' line in between that doesn't move, like the fulcrum on a see-saw. What happens is that both sides produce sound, but since they're going in opposite directions some of it gets cancelled out. In essence, the air would rather slosh back and forth across the top than actually produce sound. The situation sorts itself out once you get a wavelength or so away from the top (five or six feet for the cross dipole, usually) and you will hear some sound at that pitch if you're off to the left or right of the center line of the box.

For the higher order modes, at higher frequencies, things are more complicated. Basically you tend to get 'beams' of sound going off in different directions. One of the hallmarks of 'good' guitars seems to be that they tend to beam the sound out toward the audience instead of all around. I've seen guitars that were 'cannons' in the back of the hall and barely audible to the player.

It's a good thing you don't have to be able to calculate all of this stuff in order to make a good guitar. Most of us would be out of luck: I sure would be. Still, I think it helps to be able to visualize it to some extent.
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Re: Graduated thickness of top plates

Postby John Clifford » Fri Jun 22, 2018 5:55 pm

Alan Carruth wrote:All resonances are 'standing waves'. There are a large number of them in any guitar top. The lowest frequency one is normally the 'loudspeaker' mode, where the whole top moves in the same direction at any given time. then come a couple of 'dipole' modes, where the vibrating area is in two parts going in opposite directions. On most guitars these will be a 'cross' dipole, where the treble side goes down while the bass side goes up, ad a 'long' dipole, where the dividing line runs across the top, usually near the bridge, and the area below goes down while the area above goes up. As you go up in pitch you get more and more areas that get smaller and smaller.


Alan, I'm sure you know more about this than I do, but from what I've been able to glean from the literature, plate vibrations work much like string vibrations, but in more dimensions. That would mean (to me) that although the whole top appears to be moving in the same direction at once in the monopole mode - just as a whole string appears to be moving up and down at once in the first harmonic mode - that is not what actually happens. In fact, there are traveling waves that move out in all available directions from the source of displacement (the plucking point for the string, and the bridge for the top), travel through the medium, and are reflected back from the fixed ends of the string or the sides of the guitar. The overall "envelope" of these waves conforms to the familiar visual picture we have of a vibrating string or plate, but that is an optical illusion. This is different from the primary functioning of a loudspeaker cone, which is constructed so that the whole cone does in fact move in and out as one (although I understand there may also be traveling waves present in speaker cones, which contribute to the distortion we guitarists love to hear from our tube amps). Now maybe I have this all wrong, in which case please correct me. In any event, thanks for your insights.
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Re: Graduated thickness of top plates

Postby Alan Carruth » Sat Jun 23, 2018 6:19 pm

Actually, when driven at it's 'main top' resonant pitch the whole top (or lower bout, depending) does move in phase. It's not the piston-like motion of a speaker with a soft edge, with the plate the center moves more than 'average' and the edges don't move at all. This has been confirmed with holography, by instrumentation such as accelerometers, and using microphones, just as a few examples. This also obtains for a top when the box is open at the back, such as an open-back banjo skin. Momentum is conserved, of course, so as the top moves up and down anything attached to it, like the sides, has to move down and up, and that's been measured as well. Gore talks about this, and the consequences of it, in his book.

For that matter, if you drive a string at it's fundamental resonance it also moves in just the way the texts books show. You can do this by passing an alternating current along the string at the right pitch, and placing a horseshoe magnet with it's poles on either side of the string in the middle. The string vibrates in and out of the 'U', across the line drawn between the poles. This is a bit tricky: as the string vibrates it warms up, and the current probably helps with that. This changes the pitch of the string, so it can be hard to track. You might be able to do something with a UST and a feedback amplifier; with a signal generator you have keep adjusting it manually.

Plucked strings don't look like that. There are on line animations of what happens, and I drew out some diagrams in an article called 'String Theory' (I couldn't resist) that you can get on my web site as a .pdf. Suffice to say that a plucked string starts out with varying amounts of energy at the frequencies of some set of it's resonances, all added up in various phase relationships, depending on where you plucked it. It sounds more complicated than it is, in some ways. It really helps if you learn to think of it in the 'time domain': how the string looks just before you release it, and at various times afterward. We usually see diagrams in the 'frequency domain' . That's nice if what you're interested in is the frequency content of the signal, but it's hard to visualize how the string is actually driving the bridge that way. In the 'time domain' you can see that more easily. They're totally equivalent in terms of describing what the string does, but each approach has strengths and weaknesses.

"Everything should be made as simple as possible, but no simpler"
Einstein
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Re: Graduated thickness of top plates

Postby John Clifford » Sat Jun 23, 2018 8:32 pm

Well, I'm out of my depth here because I can't do the math. I have Gore's book, where he explains how traveling waves work in strings, giving the illusion that the string is simply moving up and down, when actually this effect is created by a wave moving back and forth along the string in both directions. You point out the same phenomenon in your "string theory" article (which I have read multiple times!). Gore asserts that plates with fixed edges vibrate somewhat like two dimensional strings, with traveling waves moving across the plate and reflecting off the edges. In Section 4.5.1 of his book he says:

"The monopole plate mode can almost be regarded as an optical illusion formed by waves washing across the soundboard and being reflected, just the same as the optical illusion of standing waves in strings . . . This optical illusion is responsible for the speaker cone analogy, because it looks as if the soundboard in the monopole mode is just pumping up and down."

I'm not trying to win an argument here, I just want to understand how the top plate of a guitar actually needs to flex in order to produce sound. My immediate goal is to coax a little more low frequency sound out of my archtop plates than I am currently getting. That means I'm focusing on the monopole mode and associated air resonance. I know from Gore's equations that I need to reduce the stiffness of the plate (in relation to its mass) in order to lower the resonant frequency, but the question is: where? It seems to me there might be different answers depending on what is actually happening in the vibrating plate. If the plate were acting like a speaker cone, then the answer would clearly be: only around the edges. If it were just moving up and down in the middle, with no movement at the edges, then maybe the answer would be: only in the middle. But if the plate is actually rippling with traveling waves moving across it from edge to edge, then the answer might be: everywhere.

I appreciate all the time you've taken to respond to my musings. Maybe someday I will be in a position to help others find their way through this maze!
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Re: Graduated thickness of top plates

Postby Bob Gramann » Sun Jun 24, 2018 12:19 am

I can tell you how I think of it. The vibration of the string needs to move as much of the plate as possible. So any floppiness around the bridge allows the string energy to be lost by moving only a little of top instead of a lot of the top. So, the purpose of the gradation and the bracing is to control where it is stiff and where it isn’t. The art is figuring out where you want it to be stiff. I want mine stiff all around the bridge.
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Re: Graduated thickness of top plates

Postby John Clifford » Sun Jun 24, 2018 12:36 pm

OK, the visual aides on the UNSW physics website finally managed to clear up some of my confusion about the relationship between traveling waves, standing waves and modes of vibration:

http://www.animations.physics.unsw.edu. ... x.html#1.6
http://newt.phys.unsw.edu.au/jw/strings.html#travelling

I was confused by Gore's references to "optical illusions." The key point I was missing is that the traveling waves are added together with their sum being the standing wave and its associated modes of vibration, so that the medium does actually move in the familiar pattern, once the standing wave is established. It's not just an illusion. But it still seems that a traveling wave would have to make at least one complete trip from the point of displacement to the edge and back before a standing wave could be established. So the medium would at least have to be capable of bending in the shape of the traveling wave. Oh well, enough already.

Bob, thanks. Yes I agree you need a minimum level of stiffness, both for structural reasons and to avoid the damping effect you describe. So as you say, the challenge is figuring out where it's overbuilt and correcting that without going too far.
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Re: Graduated thickness of top plates

Postby Alan Carruth » Mon Jun 25, 2018 5:41 pm

Part of the issue here is that everybody's right, at least some. In it's fundamental mode the top or string does move most in the center, and the edge/end is fixed, but that motion is the result of the addition of waves that reflect from the edges. Note that it's not the 'piston-like' motion of a loudspeaker, moving the same distance everywhere within the rim: the maximum displacement profile for the fundamental mode of a string is one hump of a sine curve. For a top it can be, and probably usually is, different, except maybe on banjos. In the real world you probably will never see this displacement profile, either for the string or the top. There's always more than one frequency involved, and the displacements add up in complex ways.

The objective here is not to move the top per se, but to move air, of course. Evan Davis gave a modeling paper at an ASA meeting a few years back which showed the best way to move air is to thin the top in the center. This is essentially what scalloped bracing a la Martin does, and accounts for the strong low end fundamentals and 'punch' of something like an HD-28. Thinning around the edges is not nearly as effective in that respect. This would imply that 'reverse graduation' might be the way to go to get more bass, but one has to wonder how much of an effect that would have. I think that using a lower arch, with a thinner graduation, however you like to do it, would work better, but that's just my opinion.

I've long felt that what flat top guitars do with bracing is done with arching shape and graduation on arch tops. If 'free' plate tuning results are to be believed the top bracing on arch tops and fiddles pretty much simply replaces the stiffness that you lose when you cut the holes. Some of the recent work I know of on violins suggests that there were several different schemes of arching and graduation used at various times and places, without a lot of mixing and matching. 'Curtate cycloid' arches seem to go with 'reverse gaduation' on the top; other arch schemes call for other sorts of graduation to work well.

One of the demo experiments I did when I was writing my plate tuning series for American Lutherie (almost thirty years ago!) was to make a couple of sets of small, elliptical plates with uniform thickness and heights but different arch shapes. One used a more or less circular arch, one was flatter in the middle and came down fast at the edge, and one had a broad recurve all around. If you take the circular arch as 'average' in some sense, then either of the other two gave sightly lower #2 mode pitches, while the 'O' mode was higher on the one with the recurve, and lower on the one that was flatter in the center. In other words, the stiffness distribution is partly a function of the arch shape. You'd need to graduate them differently to get the same stiffness distribution.

FWIW, my most recent arch top has the best basses of any steel string one I've made. It has a relatively flat center and wide recurves, and is graduated so that it's thicker in the center and thinner at the edges, in keeping with the Gibson original we were trying to emulate. It also has smaller F-holes, which makes a big difference and 'parallel' bracing rather than my usual 'X'. With so many changes who can say which one is responsible for the tone?
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Re: Graduated thickness of top plates

Postby John Clifford » Mon Jun 25, 2018 10:39 pm

As always Alan, I appreciate your willingness to share your wealth of experience with the rest of us. Everything you've said here makes sense to me.
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