Designing a half-step is kind of a systems engineering problem. There are two systems to design. First is an ideal half-step system which ignores the facts that you can only count gear teeth in whole numbers, and that you can only buy certain rear and front sprockets in today's market, and other practical goals and limitations. Second is to design a real-world half step system, or more likely a bunch of real-world systems. These are based on cassettes/freewheels and chainrings that are either on hand at your workbench, or can be bought on today's market. The real world half-step systems are evaluated against the ideal one to determine which real half step system is the closest to the ideal one, and which still meets practical constraints.

It's the requirements of the ideal half step system that define what's unique about them. The goal of a half-step system is to use a zig-zag shifting pattern, in which each the gear inch ( or gain ratio) increment between adjacent gears is the same percentage over the whole gear range. For example, suppose you want a 14-speed system (2 chainwheels, 7-speed rear end) with a top gear of 100 inches and a bottom gear of 35 inches. The total ratio is 100/35 = 2.857.

The ideal incremental ratio is (2.857)^(1/(1-14)) = 1.084, or 8.4%. This step ratio is the half-step. A whole step for this system will be twice the percentage, 16.8% or 1.168. The chainwheels are selected to be separated by the half-step ratio, and the freewheel cogs are to be separated by the whole step.

If we want the smallest rear sprocket to be 13 teeth, the ideal line-up of rear sprockets is as follows:

13, 15.18, 17.73, 20.71, 24.19, 28.26, 33.01

If we round these off, we should look for a freewheel or cassette with 13, 15, 18, 21, 24, 28, 33. I have not seen anything quite like this.

The chainwheel ideal ratio is 1.084, or 8.4%. For 100 gear inches, a 27 inch wheel diameter, and the 13 tooth small rear sprocket, we need a 48.15 idea chainring, or 48 teeth in real-world parts. This we can buy. The inner chainring must be 8.4% smaller, or ideally 44.42 teeth. We can round this down to 44 teeth. So we have designed one real-world system using a 48/44 chainset with the 13/15/18/21/24/33 freewheel.

How good is this? Well first notice that the real chainset ratio is 48/44, or 1.091. This is not the same as 1.084. It's as close as you can get, but not the same. We should also look at how good the freewheel is. Here we need to calculate (I use Excel for all of this) the ratios of adjacent pairs:

13
15 1.154
18 1.200
21 1.167
24 1.143
28 1.167
33 1.179

Again using Excel I calculate the standard deviation of this right hand column, and in this case get a value of 0.0198. This indicates that the freewheel cogs are pretty good approximations to the ideal case.

Problem is, I haven't seen any odd numbered cogs in the 30's for decades. I know I can get 32s, if I shop Ebay long enough - both SunTour and Sachs made them. So, what should a 13/32 look like, noting we are trading away some gearing range to target cogs we might be able to get: ideal ratio 13/32 is 2.462, and the ideal increment is 1.162. If we just substitute a 32 for the 33 above we get a standard deviation of 0.0239 - still really good!

The ideal chainring ratio is now 1.081, which requires a 48/44.4. Again we round to 48/44.

Khatfull, if you would like to put these just for example on the Sheldon Calculator, you could see how they work.

As far as your two designs, their problems are related to chainwheel ratios. For the 14/34, you need an ideal chainwheel ratio of 1.097 (50/46 closest), and you have 1.111. For the 14/28, you need 1.074 (50/47 closest) and you have 1.111 again. In both cases the middle rings need to be larger than the 45 you have chosen.