> I think using some arithmetic rules to entice someone who knows arithmetic, but otherwise is awkward around music theory
The biggest win from that POV would actually be numbering the diatonic scale degrees starting at 0 for the tonic, and the diatonic intervals starting at 0 for the unison. Unfortunately, it heavily conflicts with all sorts of existing notations. But once you think of the third as a "2" interval, the fourth as a "3", the fifth as "4" etc. you can arbitrarily add intervals together and to scale-degrees, without having to constantly correct for the archaic 1-based counting.
The biggest win from that POV would actually be numbering the diatonic scale degrees starting at 0 for the tonic, and the diatonic intervals starting at 0 for the unison. Unfortunately, it heavily conflicts with all sorts of existing notations. But once you think of the third as a "2" interval, the fourth as a "3", the fifth as "4" etc. you can arbitrarily add intervals together and to scale-degrees, without having to constantly correct for the archaic 1-based counting.