I'm not sure if any of these bands/albums have been mentioned before, but they certainly deserve some attention.
This Town Needs Guns - Animals
I'd definitely classify them as Math Pop, but they're still a fantastic math-y band. Beautiful instrumentation, and although the vocals certainly aren't going to be enjoyed by everyone, I think Stu has an incredible voice. As a caveat, they recently went through some MAJOR lineup changes and subsequently changed their name to TTNG. They're really not the same band any more, and while I do like what they've become, I miss what they were a lot more.
[VIDEO]https://www.youtube.com/watch?v=DOHGTJoh498[/VIDEO]
Charts and Maps - Dead Horse
A fantastic math rock / jazz fusion band. Not a whole lot more to say. I love em.
[VIDEO]https://www.youtube.com/watch?v=9WjHr4d9YgY[/VIDEO]
Mouse on the Keys - An Anxious Object
You know what, I'm not going to try and describe this one too much. Pianos. Drums. Mathy as hell.
[VIDEO]https://www.youtube.com/watch?v=goD4y00Aeug[/VIDEO]
Giraffes? Giraffes! - More Skin With Milk Mouth
Another really weird band that kinda bends the rules of everything. Worth checking out.
[VIDEO]https://www.youtube.com/watch?v=JZqZCS0rCmE[/VIDEO]
The Fall of Troy - Ghostship Demos, Self Titled, Doppelganger, and Phatom on the Horizon
A little bit of a caveat - these bands definitely draw a lot from the Hardcore/Post-hardcore genres. If you don't like your music fast, loud, and with a bunch of screaming, just skip this one.
The first two of these albums (Ghostship and their eponymous album) have relatively low production value, but more than make up for it in sheer energy. Not to mention that Ghostship was recorded during the band's spring break when they were all about 17, these albums are absolutely astonishing. Doppelganger is more or less a re-recording of their self-titled album with a few songs added and a few songs nixed, with MUCH higher production value, but (in my opinion) more lacking in energy and emotion. Phantom on the Horizon is Ghostship's Doppelganger (see what I did there?)
All of them are FANTASTIC albums that have their own strengths and weaknesses, and all four of them are easily within my top 20 favorite albums of all time. I'm not very pleased with TFOT's other albums (Manipulator and In the Unlikely Event), but I'm hoping that their upcoming reunion album will bring them back to their roots. Below I've selected key tracks from each album, but I highly recommend buying all of them.
One final note before you begin: As it stands right now, I CANNOT find a legal way to obtain the Ghostship Demos. It was a limited run that the released while still in high school, and I'm sure the masters have long since been lost or destroyed. However, if you're willing to bend the rules a bit to get something otherwise unobtainable, there are some relatively high quality rips floating around. Use your own judgement.
Ghostship - Part 1
[VIDEO]https://www.youtube.com/watch?v=K0sLkvbzH9I[/VIDEO]
Phantom on the Horizon - Chapter 1: Introverting Dimensions
[VIDEO]https://www.youtube.com/watch?v=Nly5Bhtp4Ew[/VIDEO]
Self-Titled - F.C.P.S.I.T.S.G.E.P.G.E.P.G.E.P. / The Adventures of Allan Gordon (Couldn't choose just one for this one)
[VIDEO]https://www.youtube.com/watch?v=sdvacK6Cmxo[/VIDEO]
[VIDEO]https://www.youtube.com/watch?v=4AazI5Epz90[/VIDEO]
Doppelganger - F.C.P.R.E.M.I.X. / The Hol[ ]y Tape (Same deal as the self-titled album)
[VIDEO]https://www.youtube.com/watch?v=_spVxhWEQWc[/VIDEO]
[VIDEO]https://www.youtube.com/watch?v=AE0jSfYli9s[/VIDEO]