Or how about
architectures not using a word length that's an
integral number of bytes?
You mean like any 36-bit machine?
No, they usually either ran as 6 6-bit bytes or 4 9-bit bytes, from
what I understand. (The era of 36-bit machines was before "byte" had
drifted to its current synonymity with "octet".)
Byte sex is really an issue of conversion between numbers and their
serializations. For example, it does not make sense to speak of the
number 2271560481 (0x87654321) having an endianness in its own right;
endianness becomes a meaningful concept only when a number is made up
of smaller units with some kind of ordering among themselves. Most
commonly these are memory-addressing units, with the order being
increasing address order, but the terminology can also be used when,
for exmaple, discussing serializing bits onto a bit-serial transmission
medium, with the order being transmission order (eg, when sending
characters over a serial line).
To address the comment, I don't know enough about any of the 36-bit
machines to know whether they had sub-36-bit-word addressing or any
such, which would be necessary for concepts such as endianness to make
sense. I think at least one early (by modern standards) machine
supported addressing memory as if it were an array of bits, extracting
an arbitrary block of those bits not longer than a machine word; if
true, that addressing order would induce an endianness....
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