How to tame brightness

Get an equalizer.

Based on the measurements and the comments in this review of the Epos M12.2, which I am assuming is not particularly different than the M12 in relevant respects, the comments by Duke (Audiokinesis) about listening height appear to be right on the mark. As well as the comments by Sideways about tubes.

The M12.2 has a considerable frequency response peak centered just below 1kHz, and another one around 4kHz, which the review indicates can be mitigated by listening from a point a few inches below tweeter level. That listening height apparently helps to smooth the upper treble response as well.

Furthermore, the impedance vs. frequency characteristic of the speaker is, in the upper mid-range and lower treble, close to being the exact inverse of the frequency response plot. Meaning that with a typical solid state amplifier having near zero output impedance, more power will be drawn at exactly the parts of the spectrum where the response peaks occur, presumably either causing them or further reinforcing them.

So first try various listening heights. If that does not satisfactorily resolve the problem, consider changing to a tube amp having relatively high output impedance (= low damping factor), although not too high or you will degrade bass performance.

Regards,
-- Al

Is it that far-fetched of an idea to take a disc shape of foam, similar in density to what used to be used on some manufacturer's speaker grills in the 70's and 80's, and just cover the tweeter to reduce the brightness, of course being careful not to physically touch the tweater and afix to the baffle with tape??
You don't mention whether your shrillness problems are at all volumes or just loud volumes. The following may be a bit of an aside. Here's what I have done on my B&W 805S, standmounts, experimenting when I have played select CD rock recordings that are just too shrill when playing them rocking loud, examples are My Bloody Valentine "Loveless" or Sonic Youth's "Daydream Nation", both great records and really shine when played loud, IMO, except for the shrillness. I use the foam cylinders (approx 2" diameter, 3" in height) that came from B&W with the speakers which are provided to optionally stuff in the reflex port in situtations where the speaker is to boomy, BUT instead use them on the tweeter, holding them in place over the front of tweeter (co-axial with the tweeter) with a small rubber band. With the B&W tweeter you can use the rubber band since the tweeter is in a small separate housing on top of the cabinet as most well know. It works from me and can be put in place in seconds. I notice a slight degradation in imaging but would rather live with that then having my ears be victim to the shrill guitar notes within these recordings yet enjoy the benefits of loud volume in the bass and mids which apparently my ears are better adapted.

Nisna, folded tissue tapped to the baffle sometimes works.

Do what Kal said. I had my room treatments done after I purchased speakers and it is a world of difference. Like upgrading speakers.

If that doesn't work buy some flour. :)