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The 300 dpi standard carried over into the photographic world in the 90's with the development of continuous tone digital photo printers using dye-sublimation technology. This continued with the development of LED and laser based printers designed for chemically processed silver halide photo papers. All based on the idea that 300 dpi is above the threshold for the human eye to discern the individual dots, giving the appearance of continuous toned images. Output to these printers is 1 to 1 pixel relationship. Each 300 pixels yields 1 inch on the print. If you need to vary the size you must resize the image - the 300dpi must remain constant. Here the computer is controlling the exact color of each discreet dot on the paper, since each dot can be any individual color within the gamut of the printer. While some drivers can resize the image before printing it, you are better off doing it yourself, where you have more control.
What gets a little confusing is comparing the DPI output of a continuous tone printer such as these (where this 300dpi standard originated in photography) to the DPI settings of inkjet printers. Inkjet printers depend on sophisticated dithering and very high dot densities to achieve their affect. A "240dpi" print from an Epson printer still has far more dots (such as 2880x1440) used to render its image. Even at "180 dpi" many inkjet prints are high quality. Here the printer driver is usually more than capable of rendering the image to a final output size without much regard for the "dpi". There really isn't much need to force your print into a specific DPI for these printers, and some argue it is detrimental, since the print driver has to RIP the image into hundreds of millions of dots anyway. As long as your DPI doesn't fall too low (180 - 200 seems to be the bottom limit) or too high (400 - 480 seems to be the high limit suggested by most), the printer will do a better job than a resize in photoshop to some "artificial" DPI number. The suggested workflow is set your print size without resampling the image, letting the DPI fall where it may, perform appropriate output sharpening, and send the file to the printer for it to size.
If your print falls below or above those limits, what DPI should you use? Here again, DPI is such a misnomer with inkjet printers. Many now advocate a different approach. Rather than trying to force the file into a specific "DPI", use a technique that will guarantee the best quality when resizing .. in this case bicubic at an _even_ increment. So upsize 200% with bicubic smoother or downsize 50% with bicubic sharper. After resizing set your print size without resampling, perform appropriate output sharpening, and send the new file at its new native resolution to the printer. The listed DPI will most likely be some very bizarre number, but again the only real use for that number is to make sure you image falls within a certain parameter.
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