The fundamental freqency of an ideal string is
f0 = 1 / 2L √ (F / μ )
where f0 is the fundamental (lowest) frequency, L is the length of the string that is free to vibrate, F is the force of tension on the string, and μ is the mass per unit length of the string.
In addition, the overtones or harmonics are given by the equation
fn = n f0
where n is the harmonic number or vibrational mode, and is only allowed to be a whole number.
There is a departure from the perfect harmonic modes based on the real materials and real tensions on the string. An ideal string is thought to be infitescimally thin, yet still having mass. In reality, the string has some thickness; however, the length of the string should be much longer than the thickness. Still, as the string becomes increasingly thick, it begins to behave less like an ideal string and more like a vibrating cylinder. The departure from ideal harmonics is described as "inharmonicity."
The actual, and somewhat tedious equation for the harmonic frequencies on a real string are given by
fn = n f0 ( 1 + ( n2 π2 d4 Y ) / ( 128 L2 F ) )
where d is the diameter of the string, and Y is Young's Modulus of elasticity.
The inharmonicity can be neglected for thin strings of adequate length. For example, the plain steel strings on a guitar with a length of 648 mm will have negligable inharmonicity; however, the low E string does have some noticeable inharmonicity (about 1% at the first overtone for a .042" diameter string). Tuning lower and lower yields more inharmonicity, as the tension on the string gives way to the induced tension from plucking. Using a thicker string takes us back to the problem of the string behaving increasingly like a cylinder and not an ideal string. The only ways to eliminate inharmonicity are to a) increase the length of the string, b) increase the density of the string, or c) decrease Young's Modulus of elasticity for the string.
On an eight string guitar, with a scale length of 673 mm (not the shortest available) with a .076" low F# string, the inharmonicity is about 9% at the first harmonic, so it's nine times what it was on the low E of a six string guitar. You may think that beefing up the gauge of the string is the way to solve this problem, yet a .090" sting on the same guitar at the same tuning gives 13% inharmonicity at the first harmonic, so what's going on?
Well, we are fighting the physical limitations of the materials that make the string of the guitar. I am not going to say that 13% inharmonicity is necessarily bad, just as I wouldn't say that the notes of a glockenspiel aren't bad, but - when the inharmonicity gets higher, the notes do sound less clear - what guitar players might call "muddy."
Let's come back to the guitar in the above example with a 673 mm long .076" diameter low F# string. We cannot do much to change the density or Young's Modulus without completely reengineering the basics of the guitar itself, but we could increase the scale length by, say, one fret. The extra length allows us to get the same fundamental frequency at similar tension with a thinner string, so say we reduce down to a .072" string, which would actually have a slightly higher tension. The inharmonicity is reduced from 9% to 6%, clearing the tone by about 1/3 of the muddiness.
Winding the strings of the guitar with precious metals instead of high nickel steel would increase the density of the string, and thus, decrease the inharmonicity, but at a large monetary cost. Regarless, silver wound strings are available commercially for bass guitar, which do, in fact, decrease the level of inharmonicity of the low notes, giving a clearer tone.
There are also other factors in inharmonicity, but the factor described above dominates for low notes, especially on short scale lengths. Inharmonicity also increases if a tremolo system is used, as the bridge supporting the "fixed" end of the string is yielding under impulse. Variation in density along the length of the string also increases the inharmonicity (NOT periodic variation, like the ridges in wound strings, but if a string is tapered before it reaches the bridge or the nut, which is not unprecedented for bass guitar strings), so that the strings that taper down in diameter to fit through the saddles will necessarily offer a muddier tone than those that do not (all else being equal).
Much of what I said above will likely strike up an argument with guitarists and pianists. It is all scientifically backed and quite measureable; however, because many musicians claim that they cannot "hear" inharmonicity, that it must not matter. Yet others complain about the muddy tone they get on 24 5/8" scale guitars over the standard 25 1/2" scales. The equations are not biased. As instruments cannot clean up the tone, but only further muddy it, a string that makes a muddy tone will only result in a muddy sounding guitar, which, in turn, results in a muddy tone coming out of the amplifier. Most respectable instrument builders know this full well, and that is why you see most eight string guitars with increased scale lengths.
On the other hand, a guitar made out of driftwood with a rust horseshoe for a pickup is going to sound like crap, even with a ten foot scale length. I simply think that the inharmonicity of the vibrating string, as a simple physical concept, should be a factor deciding which tuning to use. Tuning down to drop Zb is fine, as long as the performer is willing to accept the criticism that the tone sounds terrible, for the simple reason that a quality tone is not supported by the string, which is where the sound starts.
Enough picking on drop-tuners. What about tuning up? High tunings, like low tunings, are limited by the physical materials of which the string is made, and by the scale length of the instrument. Contrary to most people's first guess, a thinner string cannot be tuned any higher than a slightly thicker one. Why? As the string gets thinner, it loses its tensile strength. Less material to hold the tension means less tension can be held. It so happens that the maximum fundamental frequency that a string can theoretically reach is independent of the thickness! It only depends on the tensile strength of the string (which depends on the material used) and the scale length.
There seems to be a common conception around internet forums that no one has ever tuned a 25 1/2" instrument to A4. It just so happens that county-western guitarists have been doing just that for decades. A high tensile string, such as Ernie Ball RPS .010" can be taken up to 440 Hz with a little patience. The problem is that such a string at such a high tension feels like a cheese cutter on your finger. I have used plain D'Addario .007" strings to tune to high A with some success, and people like Garry Goodman at Octave4Plus can make strings that can tune significantly higher than that, again, with a little patience.
In my dream world, there would be a material with a super high tensile strength that could withstand enough tension to tune up to high D5, or even G5, and there would be a material with a density high enough to tune a standard instrument down to C#1, or even G#0, but I do not know of such materials. There are materials that can provide excellent tuning options, particularly when combined with multi-scale instruments, though.