There are a few methods suggested for doing this online, but none of them seem to work.
For example:
http://blog.makezine.com/archive/2008/11/youtube-in-720p-hd-viewin.html
That article is about making it start in 720p, but it doesn't even work on their own video.
Does anyone here know how to do it?
(This answer was updated, as the previous method using vq isn't recognized anymore.)
Specifying the height of the video will change the quality accordingly. example for html 5;
<iframe style='width:100%; height:800px;' src='https://www.youtube.com/embed/xxxxxxxx'></iframe>
If you don't want to hardcode the width and height you can add a class to the iframe for css media queries.
Tested on a working server + passes the w3.org nuhtml validator.