I was talking to my BIL, who used to play guitar with a local band. he maintains that the classic guitar body shapes do have some effect upon the sounds produced.
As far as I can see, the electric guitar is just an array for strings with transducers. There should be no difference-unlike an acoustic guitar, where the shape of the sound box definitely affects the sounds produced.
Am I right?
No.
Sort of. The shape of the body doesn’t affect it as much as the transducers and electronics will. But how those transducers are mounted, and the material they’re mounted on will affect the guitar’s sound greatly.
Probably some. But there are guitars with through necks, very solid. The block of wood attached to the through neck has minimal effect, the neck wood is all that matters. It runs the length of the guitar.
What other posters have said - shape is far down the list of factors compared to basic guitar design, materials used, electronics, etc.
Eddie Van Halen famously had an Ibanez Destroyer (their Gibson Explorer ripoff from the 70’s) that he used for non-whammy parts on the first album. In between that album and…which one? Women and Children First? the guitar pictured on the back cover - he took some big-toothed saw and ripped a chunk out of the body, resulting in more of a Star shape. He claimed it never sounded the same after that - he felt because he took too much wood out close to the bridge…
I don’t think the OP is talking about differences between a hollow body electric versus a semi-hollow body. That’s what I consider construction. I think he’s asking about the shape. I don’t think a double cutaway LP is going to sound any different than a single cutaway. Sawing off the lower (treble side) horn of a Strat isn’t going to effect the sound either. How the pickups and bridge are mounted can effect the sonic qualities. I guess, on the other hand, by altering the shape of a solid body you’re changing the mass of the body and that might effect the natural sustain.
I think this may be the key point of the OP’s question. The classic “squashed vase” shape as commonly adapted to solid body guitars (example) from acoustics (example 2) doesn’t add any inherent tonal difference to a solid-body guitar because of its geometry. This shape evolved particular to acoustic hollow guitars to best project sound from the surface and sound hole of the guitar. A solid-body guitar is not going to have any significant internal resonance that shape will affect.
However, the overall neck-to-bridge resonance characteristics of the guitar, affected by many elements such as mass, wood type, bridge type, center of gravity, et al, can influence the character of the sound. Being traditional shape or any other shape is 99% style and 1% substance.
I was flipping through an online guitar magazine, and they had a review or ad for a mass you could attach to the headstock of your electric guitar, to increase its mass. It was said to increase sustain. Can anyone comment on how much of an effect it would have? I’m considering trying it out using a clamp to see if I can hear the difference (with some felt, to not do any damage).
Based on my seat-of-the-pants experience from real-world engineering vibration tests (though not of guitars in particular), attaching a mass will certainly alter the resonance profile of the guitar as a whole. Natural resonance nodes could move, appear, or disappear.
Whether this increases, reduces, or does nothing to the sustain is likely unpredictable and different guitars would show different results. I’d be skeptical of the claim that the only result is an increase. Try before you buy 
There’s an active component you can clamp to the headstock that REALLY increases sustain, even when you’re playing through headphones. It’s a transducer to impart acoustic feedback.
Folks who don’t play electric guitar (or who’ve never cranked one up) probably don’t realize how important acoustic feedback can be to the tone and response of an electric guitar. It’s one of the reasons those damn guitarists turn their amps up so bloody loud. The acoustic feedback makes a guitar handle like a revved up sports coupe. A little gas and it jumps nimbly and keeps going. Note that I’m not talking about the Jimi Hendrix style squealing feedback (also called “howlaround”, mostly by folks on the other side of the pond). It’s the same phenomenon but just not so obvious, affecting tone and definitely increasing sustain.
The clamp thingy I’m talking about is a response to a problem that folks using guitar/cabinet modelers (i.e., software) report, when playing through headphones without a loudspeaker. The guitar just doesn’t feel the same, doesn’t respond, and most obviously, doesn’t sustain the way it would with a good loud amp. The headstock clamp thingy is supposed to restore that handling while still allowing the baby and wife to sleep in the next room. Reports are, it works quite well, and I’d love to have one but never took the plunge.
Guitarists will often tell you that almost ANYTHING changes the tone of an electric guitar (the “Don’t even look at it!” scene in Spinal Tap being a good example). Steinberg and others tried to revolutionize guitars by making guitars and basses with virtually no bodies (and no headstocks); you have to use a special strap to hold them. As it turns out, bodies are very helpful for holding them! (gee!) These guitars and basses aren’t terribly popular but that doesn’t disprove the theory.
In practice, I bet most guitars could be modified quite a bit without changing the tone notably. But I also bet there’s a point of “increasing returns” where the more you go, the more it matters (especially when taking away mass).
Guitarists also debate about the differences of various woods. Ash (IIRC) was originally used by Fender because it was plentiful, cheap, and solid enough for the job. So of course, now it’s a gold standard.
Les Paul guitars use (not sure what kind but) very heavy, solid wood, and weigh a ton, and LP players will tell you that it’s crucial to the heavy metal sound they get. When LP’s came out in “SmartWood” series with lovely and much lighter woods (which I loved) most LP players hated them. Too light, not meaty enough.
It would be impossible to do a double-blind study, so we get the debate.
The ad (I’m thinking most likely it was an ad now) said you should try moving it up and down the headstock to avoid dead frequencies. That probably is a lot easier for a 6-in-line headstock than a 3+3.
Yeah, I’m going to have to try this with a clamp now.
Mahogany neck, mahogany body with a maple cap.
To be fair, the “Smartwood” guitars were LP Studios, the lower priced line, and LP players hate anything that’s not a canon, classic design LP. Like an LP Studio (no binding!). It’s kind of like Harley coming out with supercharged, aerodynamic racing bikes. They wouldn’t sell as a Harley because you’ve lost the classic design. OTOH, call a Gibson guitar some other name than “Les Paul”, like oh say this, and nobody has a problem with it. (Although maybe that’s a bad example, since the SG was actually called a Les Paul for a little big until it was renamed, now nobody calls them that.)
Just got back from doing what I do for a living … play guitar, and I’ve got about three days off. Haven’t had three in a row since this time last year.
The image you want, in order to answer your question, is visualizing a string under tension that vibrates in three dimensions. It’s not just going up and down across the top of the body of the guitar, but really in a wobbly oval pattern. This pattern is slightly different for each string, each string type, each string length, each string gauge, etc. Also, the composition of the nut and the bridge affects the shape of the string’s vibration. When you fret a string, the characteristics of the vibration change based on which fret you’re on and whether you are bending the string like a Blues guy, performing a vibrato, classical style, or damping the strings with your picking hand.
One more thing that affects the vibration characteristics of an electric guitar string is the density and resonance of the wood, and how the mass of the body, neck and headstock are distributed.
Back in the 70s, I bought a cheap Gretsch solidbody from a colleague as an “experimental” guitar. I started cutting bits off ( events I regret to this day ), and I’ll tell you, the resulting Steinberger shape I ended up with sounded NOTHING like the original guitar: same pick-ups, same pots, same wiring, same neck, same frets, same action, but I changed the body shape drastically, and with each change (cutting off a lower bout, cutting off a chunk of the end, etc.), the sound of the guitar changed significantly, at both loud and quieter volumes.
It stands to reason that the vibrational and damping characteristics of the wood would impact the vibration patterns of the strings which are attached to that body. It’s those vibrations that the magnetic pick-ups are turning into an electronic signal.
It’s possible that a such a thing could work but I’d be wary of creating an imbalance of how the guitar ‘fits’ when playing standing with a strap. It sounds like a recipe for the dreaded neck dive that some experience with guitars like Gibson SGs.
But I doubt that is a shape issue and more of a mass issue. You cut out a lot of weight off the guitar; unless it was close to an anchor point (e.g., EVH cutting wood out close to the bridge), that would be my guess.
As for mass at the end of a guitar’s headstock - that is actively discussed as a thing on acoustic message boards. Just like heavy solidbodies were a big deal in the late 70’s - brass sustain blocks, 13-lb Les Pauls - it was a “given” that nice acoustics should have their tuners swapped out to have closed-gear Grovers, Schallers, etc. But over the past 15 years, just like with electrics, there has been a big retro movement in acoustics, and an…“appreciation” for old-school, open-backed tuners. They are lighter and affect headstock mass and some folks argue that they affect tone. All I know is that my 70 and 80 year old Martins and Gibson have open-backed, low-tech tuners that function perfectly and the guitars are light as feathers and sound wonderful. I ain’t changin’ a thing.
Back to Les Pauls: LearJeff, you assert that Les Paul fans demand that they be heavy. I really would question that. Folks who love Randy Rhoads/70’s LP Customs, etc., do favor heavier models, but it is all over the map. Since the 90’s with the retro craze, the lighter the better - 1959 Historics (R9’s in online lingo) are given the lightest mahogany, R8’s a bit heavier, etc. Guys crow about getting a 9 lb Historic. I had a great Black Beauty reissue (an LP Custom based on the original 1954 with a P-90 and an Alnico pickup, not humbuckers) and it weighed 8 lbs with a one-piece mahogany body. When I brought it in for trade the dealers freaked and took it immediately - and it sold in a week. (when I brought it in, they compared it to the $35,000 1956 original Black Beauty that they had in stock - mine sounded better 
I totally loved that.
Since you can touch the headstock or the body of a solidbody guitar to a plaster wall and markedly increase the volume of the unamplified instrument, (an old tuning trick from the days of tuning with a tuning fork in hotel rooms), and since the guitar sounds different (not just louder or quieter) depending on which portion touches the wall, I surmise that the shape of the vibrating wood, and the distribution of same, greatly affects the string vibration, and therefore, the sound transmitted to the pickups.
Well, the problem with that line of reasoning is that the sound isn’t transmitted to the pickups through the wood in (most) electric guitars. The string vibrates, and induces an alternating current in the pickup, which is amplified into sound by the amplifier. If the physical change in the guitar doesn’t change how the string and the pickup interact with each other, then it’s not going to change the sound much.
The sound of the strings is determined by vibrations. The strings are connected very tightly to a wooden mass via metal fittings. The wood picks up vibrations from the strings (this accounts for any substantial acoustic volume in solidbody instruments) and resonates itself. I submit that the vibrating wood contributes sound altering counter-vibrations in the string, changing the actual shape of the string’s vibrations. When the shapes of the string’s vibratory patterns, and perhaps the nature of the standing waves present as the string vibrates, are altered by interaction with the vibrating body, the sound through the pick-ups will change based on that interaction. I believe the distribution of the mass, in relation to the individual strings, affects the sound of a solidbody guitar.
I can’t help thinking that any effect due to the shape of the guitar’s body must drop off rapidly as we get further away from the anchoring points. Like an inverse square relationship or whatever. I just don’t see how having some fancy bit of wood six or eight inches away from the bridge can make any difference.
Put any part of a solidbody against a hollow wall and you hear that the the guitar is amplified to a surprising degree. If the wood at that point is vibrating enough to transmit substantial vibrations to the wall, then I submit that the wood at that point can also affect the nature of the string’s vibrations. It’s all connected and it’s all vibrating.