Halftones and grey levels - part 1 of 2

Some principles of halftones and the myth of grey level capability.
This information is basically true for all vendor's offerings.

A halftone dot is formed inside a halftone "cell" The cell is a grid of pixels which are turned on to form the dot. The cell begins with no pixels turned on (0% tone) and as pixels are turned on the dot grows until all the pixels within the cell are turned on and the cell is filled (i.e. 100% tone).For example.
If the cell size is 2 pixels wide by 2 pixels deep the halftone cell will contain a total of 4 pixels. As a result the following halftone dot tone values can be created:
0% = all pixels off
25% = 1 pixel turned on
50% = 2 pixels turned on
75% = 3 pixels turned on
100% = 4 pixels turned on.

So, with a 2x2 pixel halftone cell it is only possible to have 5 tone levels (grey levels). I.e. the total number of tones possible equals the total number pixels available plus one. In this case 2x2=4 4+1 = 5.
If the number of pixels is increased within the cell by making them smaller - i.e. cell size remains the same but the pixels are smaller - then the number of possible grey levels goes up.
For example:
For a 3x3 cell the number of possible grey levels is 10 (3x3=9, 9+1=10
For a 10x10 cell the number of possible grey levels is 101 (10x10=100, 100+1=101
For a 16x16 cell the number of possible grey levels is 257 (16x16=256, 257+1=257)
In a basic AM screen the dot is formed by turning on pixels starting from the center of the cell. For a basic FM screen the pixels within the cell are turned on pseudo-randomly.

So, as resolution (the "dpi" of the recording device) increases - grey levels increases. As resolution decreases grey levels decrease.

If the resolution (dpi) is fixed but the number of adjacent cells is increased (lpi, i.e. going from 100 lpi to 175 lpi) then the number of pixels available for each dot decreases and therefore the number of grey levels decreases.
This principle is captured by the classic formula:
(dpi/lpi) squared + 1 = number of grey levels

So for a 2400 dpi output device:
At 100 lpi:
2400 dpi/100 lpi = 24 squared = 576 plus one = 577 tones possible. No problem - more than enough grey levels.
But at 175 lpi:
2400 dpi/175 lpi = 13.7 squared = 188 plus one = only 189 tones possible. A big problem because when the ratio of dpi to lpi drops below 16, the number of available grey levels drops to below 256. This can result in tonal reproduction that is inaccurate and uneven, causing visible shadestepping (a.k.a. banding or contouring) in gradients. Color steps abruptly from one tone to the next without a smooth transition.