But that's the point, once you think of indexes as distance things make much more sense.

The index is the distance from the first element. Thus, the third element is two elements away from the first, thus it has index 2.

I think the only reason zero-based indexing is not the standard everywhere is because some people (usually non-programmers) have a problem with the idea that the fifth element has index four.

I am suggesting that this is a linguistic problem that has messed up programming.

Consequently, when those people do music, they have to accept that something called an octave has seven notes, and inverting an interval is based on the "rule of nine", thanks to the way they defined intervals as one based.

Distances can be quantized. If you're designing something in your CAD tool, you can set a grid to, say 1cm, and never place anything on a fraction of a cm. The coordinates of points in the design are then de facto addresses.

Indices can be regarded as measures. We speak about an array having a "size" or "length": that is measurement language. Something is "3 words wide": ditto.

A given record in a file can be 25 words from the beginning, or 100 bytes, or 800 bits. All of these tell us how much storage immediately precedes that record and we can easily convert among them.

If indices support calculation, they should be displacements, and displacements should originate at zero.

Indices not intended for calculation (beyond simple successor/predecessor, perhaps) can place items into correspondence with any ordered set: natural numbers, letters of the alphabet, and so on. This is where we can get away with 1 based.

Indices not intented for any calculation whatsoever can use a set: like associating character strings with objects via an "associative" array or whatever.