Originally Posted by
Arktiger
I recently purchased a vintage bike with what I have come to learn is a half-step gearing with granny set-up. The triple up front is 50-45-24 with a seven speed freewheel of
13-14-15-17-19-21-23. My question: Is this an appropriate half-step arrangement? Thanks for for your input!
short answer: the freewheel has a pretty consistent step size from cog to cog, which is important for half step gearing. The downside is that you'd only have two teeth separating the chainrings, such as 50-52. Since you are starting with a narrow range freewheel like a 13-23, you already know that you are getting a narrow range of gearing.
longer answer:
To evaluate a freewheel for suitability for half step gearing, you'll want to see how even the gear steps are and how well the ideal step size maps onto physically available cogs.
In this case, with a 7 speed freewheel, if you had even step sizes from one cog to the next, the size of this step is X.
The first cog is 13 teeth. The second would then be 13 x X, the third cog would be 13 x X x X, or 13 x X^2, the fourth cog is 13 x X^3, etc.
The final cog is 13 x X^6, which we know is equal to 23.
This lets us solve for X: X = sixth root of (23/13), or (23/13) to the (1/6) power.
Running this through the calculator shows that X = 1.0998.
If you now calculate what the ideal cogs would be, you get this: 13 - 14.3 - 15.72 - 17.3 - 19.02 - 20.92 - 23
Practically, to build this, you'd pick 13 - 14 - 15 - 17 - 19 - 21 - 23.
For a half step gearing arrangement, the step size between the chainrings should be the square root of the step between the cogs.
In this case, it should be the square root of 1.0998, or 1.049.
If the big ring is 52, then the small ring would be 49.59.
That's not a great answer, because you have to use either 49 or 50 teeth, and that's going make your gear spacing more irregular.
If the big ring was 42 teeth, then the small ring would be 40.04 teeth, which is close enough to 40. This would produce more regular gear spacing.
These calculations are a bit more useful than just plugging random arrangements into a gearing calculator, in the sense that you can look at how well a physical freewheel can match the ideal freewheel. If the ideal cogs are mostly fractional tooth counts, then you know it's not going to deliver evenly spaced gears.
Of course... when designing your gearing arrangement, the first step is to determine what gear range you want. Then you can decide whether half step or cross-over or compact gearing (or even 1x) works better.
Steve in Peoria