I don't want to count in twelfths, I want to count up the scale.
> call the ground floor of a building “0”
I'm a Brit, we do that here already.
For instance, “minor second”, “major second”, “minor third”, “major third”, “perfect fourth”, “augmented fourth”, “perfect fifth”, “minor sixth”, “major sixth”, “minor seventh”, “major seventh”, “octave”.
Reducing all those ordinal numbers by one really doesn’t help all that much. You still have to remember how the “minor” and “major” labels interact for every interval in the scale, and remember that sometimes the interval between the same two notes is given multiple names depending on the key, etc., which is all horribly confusing mess.
Those interactions are pretty intuitive. Where defined, major + minor = perfect (considering an octave as perfect), perfect + major/minor = major/minor. As long as you remember which notes exist, you can't get it wrong, so you'll never get confused by a piece of arithmetic in an actual piece.
If instead you used digits from –5 to 6, using arithmetic mod 12, it becomes obvious that e.g.:
-2 + -3 = -5
4 + 3 = -5
5 + 5 = -2
-5 + -1 = 6
4 + -3 = 1
etc.
Then it’s easy to see that your “major + minor = perfect” formula only works for some intervals, Etc. Overall the simple heuristics are more obfuscatory than helpful IMO.
Where does it go wrong? Do those cases come up in practice?
Counting up and down the scale is a core use case for a notation for intervals. It absolutely needs to be well-supported. A 12-semitone approach is never going to match the usability of even the existing system.
I am obliged to point out that a 12-semitone approach is in fact part of the "existing system" (see: pitch-class set theory).
(Mind you, I of course think its usefulness is overrated, because I think the "atonal" repertory is tonal.)
You need to think of it as “if the bottom note was the first note of a scale, where in the scale would the top note be?”
As with many questions of indexing, off by one errors are tricky.
It’s a system that confuses names for notes in a scale with names for intervals between notes. You’d rather they called them by cardinal numbers representing some kind of “distance”, instead of a count starting at one.
But that would be applying a later mathematical understanding on the earlier system. If that’s what you want, you should just use a log scale and count twelfth roots of two.
Ideally we’d switch all our indexing to start at zero, and use half-open intervals everywhere. Start at 0 AD, call the ground floor of a building “0”, start spreadsheets with row 0, switch Matlab to index from 0, et cetera. This is pretty unlikely to happen though.