Hi Stephan, and welcome to the forum!
Unfortunately, you can't multiply just two parts of a ratio like you did: you have to multiply ALL THREE of them. So based on your example, you would get 2 : 2,28 : 2,78. Which is, of course, identical to the original ratio of 1 : 1,14 : 1,39. You change nothing at all like that.
Also, there is no such thing as a "golden ratio". That's a myth that you often see repeated in some audiophile web sites, and home theater magazines. Reality is a lot more complex.
In fact, there are many good ratios, and even more bad ones. A good ratio is simply one where the modal response of the room is spread out evenly, with no modes bunched up together, and no mode too far from its neighbors. Many scientists have done work on figuring out modes over the years, so you will often see people here refer to "Sepmeyer's best ratio" or "Louden's third" or "Bolt's second best" or things like that. And in fact, even those are not 100% accurate: it turns out there are good regions and bad regions, so as long as you are close to one of the good ratios then your room will sound fine. It will still need treatment, of course, but at least the modal response will be somewhat even.
The best way to approach this is to use one of the many room mode calculators on the forum, or elsewhere on the internet, and just use the dimensions of the space you have available. Plug those numbers into the calculator, look at the results, and see if you can then get closer to a good ratio by changing one or more of the dimensions a bit. Sometimes a change of just a few cm is enough to improve the modal response.
- Stuart -