Physics.
The tension formula - as given, on page 3 of
the classic D'Addario Tension Chart .pdf, but you'll find it everywhere the topic is touched upon - adjusted for pounds and inches, is as follows:
T = [UW x (2 x L x F)²] / 386.4
where T=tension in pounds; UW="unit weight"=mass of a linear inch of string, also expressed in (decimal) pounds; L=scale length in inches; F=frequency in Hertz (a reference table of notes and frequencies is provided on page 4 of the same file).
Let's take, by way of example, the very first string in the chart, i.e. the plain steel guitar string PL007 (.007"), at the start of page 5.
Its unit weight is .00001085 pounds; if we're curious about its tension on a standard (25.5" scale) guitar when tuned to A4 (tuning fork A, or 440Hz), which it can do albeit not for very long (it's a very thin string and the pitch is pretty high for it at guitar scale), we can determine it even if it's not included in the chart, by using the aforegiven formula:
[.00001085 x (2 x 25.5 x 440)²] / 386.4 = 14.13963913043478 pounds of tension.
Now, let's try and halve the frequency value, in order to know the tension at A3 (an octave lower):
[.00001085 x (2 x 25.5 x 220)²] / 386.4 = 3.534909782608696 pounds. Which is, QED, one quarter of the tension one octave above.
Note that this value is included in the chart (rounded to the nearest decimal, i.e. 3.5). Also included in the chart, the tension values at G4 -11.2- and at G3 -2.8- which are, again, one quadruple of the other.
You can repeat the procedure for whatever string in the D'Addario chart, at whatever scale length, with whatever frequency and its lower octave.
So, as we've seen, tension quadruples for every octave increase, scale and mass being equal.
By the way, tension
doubles for every
half-octave increase: everywhere in the chart, you'll see that the tension of a random string at F is double (or half) its tension at B. You can play with the formula and verify this holds true for any two notes a tritone apart. Which is useful if a tension chart is not chromatic: if you want to know the tension at Bb of a string, just look up its tension at E, and double it or halve it depending on which E we're talking about.