Unable to set a new Apple ID password

I'm trying to change the password associated with my Apple ID. Once at the appropriate page you're presented with three fields in which to key the current password, the new password and the new password again for confirmation. My first attempt failed because my new password apparently contained 'illegal characters'. After devising one devoid of exotic characters I tried again, but that failed - as have all subsequent attempts - with the message "An unknown error has occurred" appearing in red before I actually finish typing the password. I've tried on my iPhone and my MacBook but the result is the same on both. Thinking the page may have got itself in a tizz I also tried logging out of the account site, then back in for another attempt. Same outcome.

Is it just a case of letting it be for a day, then trying again? Is there some unwritten procedural oddity whereby I have to first sign out of my Apple ID on all devices in order to then change the password?

Help or advice would be greatly appreciated.