How to monitor server for error establishing a database connection

I know there are tools like pingdom.com and uptimerobot.com that will monitor your site and let you know if they’re down, but I find that my WordPress site is experiencing the dreaded “error establishing a database connection” more often than desired, and I’d like to get some sort of notification when that happens (but not in the form of a user complaint!). Pingdom and Uptimerobot do not consider that an error, and thus do not push any notifications.