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IIRC Yngwie said there are no tonal differences in unwound strings so he uses light top, heavy bottom ( 8 to 48). I don't know if that's true but I'm prepared to accept he might have more idea than I do.
I think there probably are differences if you carefully A-B different gauges but I think they would be pretty small. The examples given above of Parfitt and Young maybe it's part of their sound but they were also quite hard hitting so if they'd sported 9s they'd have been out of tune and breaking strings every five minutes.
https://www.premierguitar.com/articles/String_Myths_Part_1
And the magnetic induction difference is pure physics. Thicker strings have to make an impact on the resulting sound. Whether you hear it in a meaningful way is the debate.
Acoustics - an undeniable difference and I would definitely say heavier is better from a purely tonal point of view. You need the mass to drive the top. Of course this goes out the window if you have a very light top but it still works as a general rule for most factory guitars.
Electrics - An undeniable difference, not necessarily a better or even noticeable one - but the difference must be there, physics says so
If you want to understand any tonal difference between amplified guitars you have to consider the relationship between the string and the pickup. For example, if you want to consider how body wood or hide glue affects the tone, you have to ask whether it can influence the way the string or pickup reacts to the forces applied to it.
Changing the mass of the string must have a very direct impact on that relationship between string and pickup. The real question is "how much do I need to increase/decrease the mass of the string to hear a tonal impact?". is a 5% increase enough? Do you still think you would not hear the difference between 8's and 13's?
Playability is affected a hell of a lot more and that brings its own tonal differences. As others have pointed out with examples of ZZ top and the like, there are many other ways to achieve a big sound. It doesn't stop with that string/pickup relationship. Its just one factor at the beginning of the chain
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I'm a Telecaster player and no string swapping will ever come close to filling out the tone the way that generous adjustment of my amp's EQ does, however.
I dare say those with a better understanding of the laws of physics might explain why, but on a single coil pick-up I assume the thicker string via the magnetic field of the magnet/pick-up in turn induces a bigger current - Somehow the greater mass or vibration creates more energy for the pick up to produce the appropriate goodness
However, I'm not sure it was an entirely fair comparison as they left the pedal and amp settings untouched.
In reality, if there is a difference in tone, then the settings they use for 10.5 or 11 gauge strings won't be optimal for the thinner strings. After the string change they probably should have tweaked the settings to get the best sound they could with the lighter strings. If nothing else there should be a Fletcher-Munson type effect to compensate for.
Intuitively, you would expect bigger output from the pickups with heavier strings as there is more mass of string - a lot more more mass. If you take the extreme that they did on the Les Paul where they went from 12s to 8s (if I remember correctly), you are more than halving the mass of the top E for given length**.
If that is the case, some of the perceived loss of tone will be down to reduced volume. It won't be that simple though. For the same pick attack, you would expect that the thicker strings will have less movement. That would reduce the size of the signal which is dependent on the speed of movement. In reality they are probably digging in a lot more with the heavier strings and getting more output as a result - which will definitely result in a Fletcher-Munson type of thing going on where it sounds better just because it is louder.
I don't think there are easy one size fits all answers.
**Mass is volume times density. Volume for a cylinder is Pi r squared h. In the antiquated units we use for strings, radius of the 12s is 6/1000 of an inch, against 4/1000 of an inch for the 8s. 6 squared is 36, while 4 squared is 16. That's a 56% reduction in mass.
Edit: just saw the post by @guitars4you after I wrote this.
So far, I like them
What you are basically getting is slightly increased tension. Between 1 and 2 pounds per string. 2lb is a very noticeable difference, 1lb is a bit more subtle and most noticeable when you reach towards the top of a bend for me.
I think one of the things that makes players think 'tension = tone' is not that the tone is DRASTICALLY different but that the tone and pitch of the string is more stable.
For example, using 9s, as soon as you hit a chord hard all of the strings momentarily move out of tune and then drift back in. I suppose this can cause a sort of pitch shifting / phasing effect of sorts.
With heavier strings and more tension the pitch drift is greatly reduced so the 'purity' of the pitch is greater.
This is based on zero science and one hundred per cent opinion.
That said, I definitely hear a difference when using extreme gauge jumps, such as from a 9 to a 13, but if you increase the gauge incrementally (9, 10, 11, 12, 13) the effect becomes less noticeable.
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