Strange thing is my mirrorless seems sharp enough at 1/60sec for same subject with the 16-70/F4.
We're you referring to your Sony a6000? It's an APS-C camera.
A friend of mine moved from a 1/1.5" 10MP PASM P&S camera (advanced in its class) to an APS-C (DX) 16MP D5100. He kept complaining during the first few weeks of using the D5100 that images used to be "so sharp" with that less expensive camera!
I still say my Canon 70d is much sharper when you zoom in and pixel peep. I think it has to do with pixel density on the smaller sensor. I thought part of my problem was too slow shutter speed too. The other day I did headshots indoors at shutter speed of 160 on my 100mm lens and kept getting very soft images. I was a good 10 feet away shooting at f2.8 and still just ugh. 1 out of 5 were sharp. I was also shooting around 400ISO so its not noise.
Bottom line, I'm having a hard time with the term "tack sharp" and my D750 and I have Auto Tuned every one of my lenses.
My humble suggestion.
I believe that Canon and Nikon JPEG engines output different levels of perceived sharpness.
My own opinion, is that this comes out markedly in sensor processing tests. Where the JPEG and then the RAW's are compared. Canon's JPEG engine seems to give more punchy, whereas when the RAW's are examined, very often, the Nikon RAW is close or even better, in resolution and contrast.
I use a D300, I don't have a D750 and am now awaiting my newly acquired D610.
But, my own RAW workflow, yields substantially better output than the in camera JPEGs. Really, I think the out of camera JPEG rendering is crap. Although, C1 has had six years to get to C1 7 & 8 algorithms improved since Nikon issued the D300 with its then software.
This is without haloing or other severe artifacts. Sure, sometimes the contrats etc may be pumped or whatever, but it is not with artifacts and only a matter of interpretation.
But what I am saying is that a good RAW workflow will make all the difference, other things being equal (AF fine tune etc).
As an example, someone who shoots the same event regularly asked me, and surprised me, why my images were sharper than hers.
I was surprised a bit, but all I could say was that I use C1 Pro for my images. But truly, whatever software you are comfortable with, whether C1, Lightroom, ACDsee etc, should yield better results than the JPEG's.
By the way, I use a D300 and older Tokina 28-70 2.8. The lady uses a D600 + 24-70 2.8 AFS.
So, her equipment is at least comparable.
So, try a RAW workflow and give it a few months and see what you think.
If you already do use a RAW workflow, then I really do not know what to say.
Regards
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Wishing You Good Light.