It's an old habit. I remember Linotype imagesetters that had a resolution of 1,200 dots per inch. When printing images, we could set the halftone screen to whatever we wanted. We still called it 1200 dpi (the resolution of the printer) even when printing an image with a 100 dpi halftone screen.
Your offset press doesn't know a native dpi or lpi, in theory it can print say a 250 LPI raster next to an image that had a 150 LPI raster. The (photo) imagesetter that creates the halftone film or plate can have a 1200 LPI resolution (usually higher though) which means its laser can put 1200 lines with 1200 equal sized spaces in between. With that laser it can build all kinds of shapes (fonts) and patterns on lith film or offset plate, among them halftone screens. Up to 120 LPI halftone raster in this case as there is a rough rule that the imagesetter has to have about 10x the resolution of the halftone screen it has to create. It is unusual to qualify the offset printwork by the resolution of the imagesetter, a printshop will quote its maximum halftone screening if it specifies quality. LPI is somewhat confusing too though, halftone screens usually have elliptical dots or round dots, the last can get a square shape at 50%. There are line screens, wave screens, whatever way you can split paper white and ink black at 50% and let it gradate either way.
With inkjet printers, I get confused. The printer itself may be able to place ink dots with 1/1440 of an inch accuracy, but it can also vary the amount of ink (hence size) of that dot. Not only that, but few inkjets use a traditional halftone screen with a regular halftone dot pattern. Many printers use proprietary (and undocumented) dithering patterns where there might not be anything corresponding to a traditional halftone dot.
Inkjet printers can use all kinds of tone representations by varying:
droplet size and by that dot size
ink density and by that dot density
and related to the two:
squirting more/less small droplets at roughly the same spot
frequency modulation of identical dots (stochastic)
simulate haftone screens
and use any combination of the methods mentioned.
In offset printing you will see next to halftone screening, also frequency modulated screening, for example with Agfa CrystalRaster.
Offset however can not vary the ink layer density per dot but it can split a tone range in two screens and two print runs, with two different ink densities to give smoother gradations.
Related to litho/offset printing is Collotype (Lichtdruck) and the "dot" varies in size and density (to a degree) and the screening is non-linear and can have a high frequency.
In conventional rotogravure the linear screen dot (cell) size didn't vary but the ink amount per dot did, later replaced by a mix of halftone and continuous tone screening. Photogravure that led to rotogravure was originally done with an aquatint base that gives it an irregular screening.
So totally new are the tone rendering methods used in inkjet printing not but the technology allowed the use of all methods and hybrids at the same time. One color pixel can be translated to several colored dots that can vary in hue, density, size, shape, distance to one another, the total then is often called a cell. The cell resolution is way lower than the 1440/2880 maximum droplet frequency (dpi) of an inkjet printer. it will be more related to the native (input) resolution of a printer expressed in PPI.
met vriendelijke groeten, Ernst
Try:
http://groups.yahoo.com/group/Wide_Inkjet_Printers/