Suicide_As_Alibi said:
The length of an open note always remains the same on any particular guitar. If you're tuning to the same note, then obviously the final frequency stays the same too. If you use the same gauge string, linear mass stays the same too. So in order for the equation to work, the tension has to stay the same.
I hear what you're saying, but the way you phrased it doesn't appear to be
totally accurate. For frequency to be the same for two strings of equal length and tension, their linear mass have to be the same, but
not necessarily their guage / thickness / diameter.
A .010" diameter string made of nylon will have a different 'mew', and thus require a different amount of tension to vibrate at the same frequency, than would the same diameter string made out of twine, or brass, or aluminum, or nickel with a round core, or nickel with a hex core.
Suicide_As_Alibi said:
String thickness doesn't come into it - theoretically you can tune any thickness of string to any note (even if practically tensile strength doesn't allow it).
Right, and as you increase tension to reach your note of choice, the string's thickness, though not (necessarily) its linear mass, decreases. That's an aspect of elastic deformation. As you stretch one dimension of an elastic material its other dimensions are decreased to compensate and maintain the overall volume [or mass or whatever; math and I haven't been on speaking terms in quite a while
] of its original state at rest, but when you release the tension the elastic material should return to that original state. (Provided you didn't stretch it so far that it became plastic in nature, or just plain broke it.) Picture a rubber band's width decreasing as you stretch it, then returning to normal when tension is relieved.
And again, I'm not trying to imply that tension-induced changes in string diameter have any bearing whatsoever on whether or not string length behind the nut has any bearing whatsoever on tension within the active scale length or "feel" or any of that. I just brought the thickness change up to make sure I was right in thinking that that's what happens when we tune up a guitar. I can't start struggling to wrap my brain around the behind-the-nut issue until I'm sure I have a better grasp of the fundamental mechanics than my usual "pitch go up now. fire hot." level of comprehension. :Spin:
Suicide_As_Alibi said:
Don't remember it having anything to do with Newton though - weren't he pretty much a motion guy? Momentum in closed systems and all that. I can't even remember, physics was really fucking dull.
Generally speaking, my primary area of newtonian expertise involves figgy cookies rather than complex maths, but I do know that frequency equations deal with things like a string's resistance to acceleration and 'desire' to return to a state of rest, which all sound right up Sir Isaac's alley to me.
Here's an interesting article that touches upon how increased behind-the-nut length might actually affect
apparent string tension when we play:
http://www.noyceguitars.com/Technotes/Articles/T3.html
' If you look at the length of a string on a Strat between high E machine and the nut, and that between the saddle and the back of the body, it represents 30% of the effective string length (i.e. from nut to bridge saddle). The same observation on a Les Paul represents 15% of scale length. This is a very significant factor in the string-bending feel of each guitar. To simplify this, let's look at a more exaggerated example.
Fig 3a and 3b show two strings identical in everything except overall length, but mounted such that the length between the nut and the bridge is the same for each. Since for tuning purposes this is the critical length, both strings can be seen as identical as far as what is needed to tune them to the same pitch is concerned - and will thus require equal tension. However, the string in Fig 3a will be much easier to deflect (bend) than the one in Fig 3b, as the increase in tension that happens when you deflect the string is to some extent distributed over the entire length of the string. '
That would only apply to guitars with non-locking nuts, though, I imagine.