Two excellent semi-open cans are the Beyer DT 880 (+/- Pro) and AKG Q 701. Since I can't recommend the Beyer 80 Ohm version, both of these cans need to be amped. Therefore, they probably will not work for you with your current equipment. The AKG K 240 Studio is the next step down. It is very good for only $61 or so that we can now snap it up in the U.S. at Amazon. It was good in former $100 +/- range.
The Shure SRH 840 and SRH 940 have different sound signatures. The 840 has a mid-bass hump and somewhat recessed highs, while still being not too far from neutral. The 940 is very close to neutral and has great clarity. It is my favorite closed headphone in its price class and would be a very strong choice for listening to classical music with a closed can.
The 8400 is more neutral than the 840. It has less bass and is brighter. The 8400 is the better choice for classical music. The 8400 is a very good can in its price class, but clearly a tier below the 940 in terms of quality (and price).
Hmm I'm not sure what you meant by you couldn't recommend the Beyer 80 Ohm and that they need to be amped. Sorry! I just looked up the Shure SRH 940 and unfortunately it is too expensive for me. Also, I had a quick look on my favourite store's website and they do stock the DT 880 Pro, but at CHF275, it is also too high above my price range. However, they do have the DT 990 Pro for CHF190 for some reason. I presume with the higher model number, it is the better one but then it is much cheaper. What do you think? But would I need extra hardware to make these work? Otherwise, I might just go with the 8400.