My original question for this was how to both have an element of a fixed aspect, but to fit that within a specified container exactly, which makes it a little fiddly. If you simply want an individual element to maintain its aspect ratio it is a lot easier.
The best method I've come across is by giving an element zero height and then using percentage padding-bottom to give it height. Percentage padding is always proportional to the width of an element, and not its height, even if its top or bottom padding.
So utilising that you can give an element a percentage width to sit within a container, and then padding to specify the aspect ratio, or in other terms, the relationship between its width and height.
.object {
width: 80%; /* whatever width here, can be fixed no of pixels etc. */
height: 0px;
padding-bottom: 56.25%;
}
.object .content {
position: absolute;
top: 0px;
left: 0px;
height: 100%;
width: 100%;
box-sizing: border-box;
-moz-box-sizing: border-box;
padding: 40px;
}
So in the above example the object takes 80% of the container width, and then its height is 56.25% of that value. If it's width was 160px then the bottom padding, and thus the height would be 90px - a 16:9 aspect.
The slight problem here, which may not be an issue for you, is that there is no natural space inside your new object. If you need to put some text in for example and that text needs to take it's own padding values you need to add a container inside and specify the properties in there.
Also vw and vh units aren't supported on some older browsers, so the accepted answer to my question might not be possible for you and you might have to use something more lo-fi.