Does APS-C sensor provide better bokeh than FF and why?

Bokeh is the way a lens renders or draws out of focus points of light. Being a product of the lens, sensor format does not matter.

Now how much bokeh matters is a product of depth of field, in which case FF has the advantage as it can render a shallower depth of field than APS-C (as the subject magnification is larger for FF than it is for APS-C).

As for long lenses, they look like they provide a shallower depth of field or "better bokeh", but this is a product of the narrower field of view. This means that there is effectively less background captured meaning it is easier to isolate your subject.

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What you're really asking is, "Does a larger format sensor allow me to more easily shoot with a shallow depth of field?" The answer is, in practical terms, is, "Yes."

But as Craig points out, the way that out-of-focus blur is actually rendered, the aesethic qualities of the blur (NOT how blurry it is), depends on the lens you're using. You could use three different 50mm primes at f/2.8, and three different f/2.8 zoom lenses that happen to cover the 50mm focal length, and all six of those lenses used on exactly the same subject matter, at the same distances (between you, your subject, and the background) at the same focal length and aperture, would produce:

1) Essentially the same amount of depth of field, and thus the same amount of blur in the foreground and background. This is selective focus. This is a natural byproduct of any and every lens used in the same way.

2) Possibly very different looking blur. Some of those lenses could produce "buttery" or "creamy" blur, and others a more jaggy, nervous, or edgy-looking blur with visible artifacts. This is what we're talking about when we use the word "bokeh." Not how out of focus is it, but what does the out of focus area look like.

So, the sensor format doesn't have much to do with it at all, other than that for a given shot (say, a head-and-shoulders portrait from two meters away), you'd use a different lens (a difference focal length) for each of those two camera formats. And because you might choose a different actual lens (perhaps from different manufacturers, or from different price points in Canon's product line) to suit the different sensor format's field of view issues for a given composition, you may find that the lens you choose for one or the other has different bokeh characteristics. Different because of how each lens is designed, what sort of glass is used, the number of iris blades, etc.

Longer focal lengths on a given camera format do narrow the field of view. Maintaining the same framing (say, of a portrait subject) also means backing up farther and farther away as you go longer. Which also means you're changing your perspective on the subject, as you also reduce how much background is visible in the frame. Depending on the image, the elements seen in it, the light, and your creative purpose, issues like perspective and field of view may be far more important than the way the background blur is rendered.

Depth of field and background blur are entirely different things goverened by entirely different lens properties. Check out http://www.bobatkins.com/photography/technical/bokeh.html for more info.

The answer to your question is that the quality of the background blur does not depend on format. It depends only on the lens.

If you want to determine the quantity of background blur in a print of a given size taken with a given lens used on a camera with a given format size, this calculator will do it - http://www.bobatkins.com/photography/technical/bokeh_background_blur.html

In general, for any given angle of view using lenses appropriate for the format (i.e. with different focal lengths), a larger format will tend to throw a distant background out of focus more.

To the OP: No.

To the OP: Yes.

What the others say is technically true, but in practical terms, mostly irrelevant. There was a "relevant truth" repeated by several people, that the definition of bokeh as the "quality" of the out of focus parts of the image, the freedom from false "bright edges" and "double lines." Although it is possible to measure the quality of the bokeh, this is seldom done, and it's usually expressed in subjective terms, like "good", "creamy", or "smooth" for the bokeh that the majority like, or "poor", "harsh", or "busy" for the sort that the majority don't like as much. There's always a minority who prefer the sort of bokeh that the majority dislike, too...
Bokeh is a function of the lens design.

• Fast normals, like the typical 50mm f1.8 or 1.4 are designed to get added sharpness at the expense of bokeh, so to gain a bit more detail in the part of the image that's supposed to be in focus, they give up smoothness in the out of focus background. Even a legendary one, like the 50mm f1.4 Zeiss, is especially ugly.
• Macros tend towards ugly, too, but this is a trend that lens makers are addressing, as more people are making macros "do double duty" as "portrait lenses".
• Short teles are usually deliberately designed for "good" bokeh, with the Nikon 85mm f1.4, 105mm f2.0 DC, 135mm f2.0 DC, and older 105mm f2.5 Ai, or Canon 85mm f1.2 being classic examples.

This is independent of format. Put that 135mm f2.0 DC on a FF camera or a 1.5x crop APS, and it will still deliver good bokeh, because that's how it's designed. It will also deliver lots of background blur, because it's long and fast. So, it's got quality and quantity, you don't have to pick between them.

The problem is "suitability for purpose". A "conventionally trained" photographer, such as myself, will tend to work around 8-10 feet from a portrait subject. At that distance, on FF, the 85mm is the right focal length for "waist up" portraits or couples, the 105 a classic "head and shoulder portraits", and the 135 a favorite for headshots. That's three lenses that Nikon, Canon, and Sony make in versions with good bokeh.

On APS, you need 50-60mm for a waist up, about 75mm for a head and shoulder, and 90-100mm for the headshot. Nikon, Canon, and Sony don't make "good bokeh" lenses in those focal, although they're getting closer. For example, the current 60mm f2.8 Nikon macro has much nicer bokeh than its predecessor. But it's not in the same league as the 85mm f1.4 I'd use on FF. Pentax, on the other hand, makes some great bokeh lenses for APS, a 55mm f1.4 and 77mm f1.8 that you have to use to appreciate how good they are. Especially the 77mm.

So, the "meaningful" answer is that, if you shoot Nikon, Canon, or Sony, you can't get as good bokeh from APS as you can from FF, because you can get "good bokeh" lenses in the lengths you are most likely to want. If you shoot Pentax, you can get a variety of "good bokeh" lenses in APS.

And, for the four thirds, NEX, and Samsung systems, the "good bokeh" lens landscape is famine-scorched and dead...

So, yes, overall, with the exception of Pentax, FF provides much better bokeh than APS.

Well to follow up on this - the image circle is produced by the lens, and the crop sensor just uses the center part of that, but the image is the same, and the depth of field should be the same with either sensor, all else being equal. But I wonder if the pixel size has any effect on that - not the megapixels per se, but the size of the individual pixel sites. Does anyone know if that makes any difference in the depth of field?

and the depth of field should be the same with either sensor

Bzzzzzz. Sorry, wrong answer - http://www.bobatkins.com/photography/technical/digitaldof.html

Pixel size doesn't affect DOF

Sherman would be right if the photographer didn't move or care about composition differences when comparing that same lens on the two different format bodies. But that's not a real-world situation. In the real world, photographers have favorite working distances (because of perspective, logistics, etc), and that means choosing the right focal length for each job. And as I and J.W. have each pointed out, you may not have the same choices of lenses for use with each format ... and so there is an indirect impact, possibly, on real bokeh-ishness, because different lenses are involve, and all lenses are different in that regard.