The usual way of writing relative notes in a scale is 1, 2, 3, 4, 5, 6, 7. In C major, this would correspond to C, D, E, F, G, A, B. Chords are usually written in Roman numerals, with capital letters for major chords and lower-case letters for minor chords: I, ii, iii, IV, V, vi, viio for C, Dm, Em, F, G, Am, Bdim.
The first problem I see with this is when we want to describe secondary chords like V/V, "five of five": we temporarily go to the major scale of the 5, and extract the V chord from that scale. In C major, the V chord is a G major; in the G major scale, the 5 is a D, so the V/V is a D major chord.
I expect that people who work with these things regularly work these out as quickly as I can do my times tables, hopping between scales with ease. But I would like to work things out in terms of modular arithmetic. In the case above, things appear to work out: there are eight notes in an octave, 5 + 5 = 10, and 10 (mod 8) = 2, and D is the 2 in the C scale.
But this breaks down if we want the V/ii chord: 5 + 2 = 7, but the ii is D, and the 5 in the D scale is an A, not a B.
The mathematically-inclined may have already spotted at least one of the mistakes in the above reasoning. There are seven different notes in the scale, so we should be working modulo 7, not modulo 8. The second mistake is with the notation: the scale should start with zero, not 1.
To do the calculation properly, we need to first subtract 1 after doing the addition. So, V/V is 5 + 5 - 1 (mod 7) = 2. And V/ii is 2 + 5 - 1 (mod 7) = 6. It works.
Having to subtract the 1 is really annoying though, and the special case of ending on a 7 (e.g., V/iii which becomes zero mod 7) needs to be handled. A better scale would be
0, 1, 2, 3, 4, 5, 6
O, i, ii, III, IV, v, vio.
Then the normal-notation V/V becomes a IV/IV, and can be calculated as 4 + 4 (mod 7) = 1, the D major chord. And a normal-notation V/ii becomes a IV/i, and can be calculated as 1 + 4 (mod 7) = 5, the A major chord.
This is much cleaner, and the only minor issue is a roman numeral for the zero , which I wrote above as a letter O.
Taking into account how little music theory I know, I figure my proposal is somewhere about as optimistic as my suggestion for question marks.