Talk to me about fixed and single speed

For devotees of the "Every chainring tooth must be visited by any given link" school of thought, the Fibonacci numbers 21x55 look good. If my calculations are correct, (no guarantee of that) it will take 28 revolutions of the cranks for a link to visit all 55 teeth, which (I think) it does in it's own systematic way, like some crazy postman: "Let's see, I've just been to number one, I'd better go to number twenty-two next, then forty-three, then I'll go back to number nine.."


That's a 69 inch gear on road wheels, just like doctor-bond's 13x34... or about 65 inches on 26" wheels.

Addendum:

Well... I don't mind admitting that the behaviour of this system is baffling me now. I've just been doing some empirical tests on a 12-speed with 42 & 52 rings, a 13-20 block, and 106 (edit: 108) links in the chain. I tied a bit of thread around one chain link, and counted how many revolutions of the chain it took for that link to return to the same
tooth on the chainring.

No matter which sprocket was engaged, in the 42 ring it took only six chain revolutions for the link to return to the same chainring tooth it started from, evidence that any given link only ever visits six teeth on the chainring. Whether the link visited the same six teeth in every gear I don't know, but I think it must have.

On the 52 ring, no matter which sprocket was engaged, it took twelve chain revolutions for the link to return to the same chainring tooth, evidence that any given link only ever visits twelve teeth on the chainring. 52 is not even divisible by twelve. What's going on?

Over forty odd years of cycling, I have accumulated a small collection of chainrings, but none of them have an odd-number tooth count, let alone a prime number. So that's about as far as my empirical tests can go... and all that stuff about prime number tooth count remains, at least for me, theoretical. Phew!!

what the heck have I started??

Single Speed is good though...

A manic episode?
A chain reaction?

I'll get me coat... but before I do:

This was bugging me, so I numbered all the teeth on the 52 ring 1-52 counter-clockwise, and shifted through the gears again. This is I guess dependent on the number of links in the chain, but on my 106 (edit:108) link chain, the marked link visited the teeth in this order no matter which sprocket was engaged:
Tooth 1
Tooth 5
Tooth 9
Tooth 13
Tooth 17
Tooth 21
Tooth 25
Tooth 29
Tooth 33
Tooth 37
Tooth 41
Tooth 45
Tooth 49
So any given link visits not twelve, but thirteen evenly spaced teeth around the 52 ring, in any sprocket.

I'm probably the last schmuck in cycledom to whom this is news..

the sprockets are irrelevant to the relationship between chain length and number of chainring teeth. taking that further, you could drop it off the sprockets completely and it would still visit the same chainring teeth in the same order, which I think makes that a bit easier to understand.

And similarly, the chainring tooth-count will make no difference to a given link's relationship with sprocket teeth.

What I'm struggling with more is how these two relationships interact with each other, and whether or not it actually makes any difference to anything at all...

Gear ratio in inches is simply a throwback to the old ordinary, when gear quoted was the diameter of the front wheel.
So 48 / 16 x27 = 81 inches.
When in veteran/junior road races in the 70s the maximum gear ratio allowed was 52 x 16 theoretically. I always used 53 because of the small tubulars being less than 27 inches.
I would have thought a suitable fixed gear for most would be in the low 70s.

Keith

Well...

Just tried what I have lying around which was a 42t chainring and 19t sprocket attached to the £2 carboot sale wheel that set me on this path of madness.

<note to self: must not 'stop' pedaling...>

Looks good but I would wear shin pads if you intend using those pedals. with clip in pedals you just get a reminder nudge if you stop pedaling, at first your reflexes may not twig what is happening so beware. Perhaps with flat pedals you may actually learn quicker than the shin scars take to heal.

Basically combinations that don't divide by anything. 41 and 43, both primes.......

Also, as mentioned, prime numbers (or odds at a push) mean that you keep swapping inner link to outer link as you pedal. Evens out wear. One reason (amongst many) that road doubles went from 42/52 to 39/53.

FWIW, the stuff at the back is irrelevant to the stuff at the front, it's simply a matter of chainring size and chain length.

We've had some right ding dongs trying to syncronoise cam systems (chain or belt) to ensure that the vibration orders don't cause the entire thing to vibrate itself to pieces......