> For example February is the 2. month, so if you have a list of the names of the months, you would expect month_names[2] to be February.
That's not 1-based indexing being good; that's conforming to (or reflecting) an externally imposed 1-based system that is itself questionable.
Should the seconds of a minute go from 1 to 60 instead of 0 to 59? Dates and times are full of poorly chosen conventions, including ones that don't match people's intuitions. For instance, many people celebrated the new millennium in January 2000. People also want decades to go from 0 to 9; the "eighties" are years matching the pattern 198?, not 1981 to 1990. Yet the 20th century goes from 1901 to 2000.
In many situations when 1 based numbering is used, it's just symbols in a sequence. It could be replaced by Gray code, greek letters, or Japanese kana in i-ro-ha order.
When the arithmetic properties of the index matter to the point that it's involved in multiplication (not merely successor/predecessor relationship), it is advantageous to make the origin zero.
If month_name[1] must be "January", I'm okay with wasting month_name[0]; that's better than supporting a one-based array (let alone making that default).
> As for music theory, a third is called so because it spans three half-notes.
No it doesn't; in that very same music theory, a "major second" interval is also known as "one step" or a "whole step"! A third is "two steps"; that's what it spans. (I don't know what you mean by "half-notes"; I sense confusion.) This nonsense was devised centuries ago by innumerates, just like various aspects of the calendar.
