What is the maximum step size that maintains virtually 100% reliability (presuming there's no combination that won't hiccup once in a great while)? I ask wondering if there's a combination of rings and cassette that would give 27-30 gears without overlap. IOW, run up from 1-9 or 1-10 on the small ring and then go to 1 on middle ring and that's the next gear up from 10 on small ring. Run up 1-9 or 1-10 on middle ring then go to 1 on large ring for next step. I hope that makes sense, what I'm asking. I'm not sure the idea makes sense, hence the question.

No, there will always be a little overlap. It may not be exact but gears will overlap.
Do you have a project in mind or just wondering?

Search the forum. Play with some gear calculators.

https://www.sheldonbrown.com/gears/

https://home.earthlink.net/~mike.sherman/shift.html

I was just curious what the largest jump that's reliable is so that when I'm looking at bike manufacturer websites I could imagine various combinations. I've played with the sheldonbrown site some but haven't tried this particular exercise with it.

53/39 which is 14t is pretty standard and you could probably go to 15 or 16t steps without a problem. One thing to consider is that when shifting large steps like that, you want to be using outers (or also middle on triples) with pins or ramps to aid the shift.

Another consideration is the phasing (hyperglide by any name) which makes for smooth shifting. Most high end original chainrings are made in pairs (or sets of 3) which are correctly phased, as in a 53t made for a 39t. when you mix and match, you'll lose that perfect phasing, and some of the smoothness. It isn't earth shaking, but on large jumps every little bit helps.

It appears that with a 10 speed cassette and wide chainring spacing one can get at most about 24 ratios that don't overlap using 1-10 on the small ring, 3-10 on the center and 6-10 on the large chainring. If one were to go to an outrageous spread on the chainring maybe 25 or 26 without overlap but I don't think it would ever be possible to get all 30 without overlap. It was one of those things that pops into your mind and you have to ask about it and look into it to satisfy it.

There is some advantage to having the ranges of adjacent chainrings overlap a bit as it gives you more options without having to double shift and sweep the entire cassette to get the next gear.

even with 10sp and 11sp cassettes, you don't want to cross chain your gears.

small ring and smallest 2 to 3 cogs on your cassette as well as large ring and largest 1 to 2 cogs makes for a loud drive train and quickly wearing chain.

Sure you can. Use half step gearing. While it makes less sense with a 9 speed setup than traditional 5,6, or 7.... it works just the same. For instance, run rings with a 4 or 5 tooth differnece on the crank, with a 9sp. widespread rear .

For example, a 44/48 crank and a 9sp. 11-32 cassette. Run it through a gear chart. https://home.earthlink.net/~mike.sherman/shift.html ... and you'll see, no overlapping gears. (The high 44/11 and 48/12 do overlap, but that's not a bad thing, you get 16 distinct gears) Add a 26t granny, and you add more. You can use any progression you want. Stay in one ring, and use you FD to fine tune your gears, or vise versa. The only limits are your imagination. People can get all stupid when comes to half step gears. You don't have to follow a steady left/right steup up/down progression, especially with anything above 7 speeds. Use the FD to fine tune your selection.

With a 48/44 and wide cassette it's about all overlap and you really have 11 distinct gears/speeds out of 18 (using a 9sp cassette). What I was talking about was going 1-9 or 1-10 on the small ring then 1-9 or 1-10 on the middle ring and finish with 1-9 or 1-10 on the big ring to get fully 27 or 30 gears. That's impossible though without insane gear sizes and not sure if it would work even then.

No offense meant , but how long have you been riding? The only way to get a steady left-right progression is with one chainring. You'd need an internally geared hub, like this one https://sheldonbrown.com/harris/nuvinci.html Infinite speeds, but you have limits on how high, and how low with one chainring.

Overlap is good. I wish I could have more, and with closer ratios. Shifting in the back is quicker and smoother than in the front.

I question the usefulness of such an exercise - there's nothing that says you need 27 or 30 distinct gears. But for argument's sake let's just look at the result. Assume a 9 speed 11-21 "corncob" 9 speed cogset and a 24 tooth small chainring. The highest gear would be the 24-11 (a very noisy combo as noted above). The next gear up from that using the 21 tooth cog would require a 50 tooth chainring. Even avoiding cross chaining (eliminating 2 gears) would require going up to a 42 tooth chainring. Both combos are obviously impractical.

Beyond the mathematics you need to think about what you do while shifting from the outer all the way to the inner cog (or reverse) in order to find that next gear. Whistle a tune, hope that your bike does not slow down so much that you need to shift again right away, or worse shift back to the gear you were just in? Or perhaps you will need to pull over your bike after you shift all the way from the largest to the smallest cog because your chain kept on going and jammed?

+1, adequate overlap is very valuable to you for riding in the real world. It's no accident that almost all (if not all) new bikes are spec.'d with overlapping gearing. Without overlap, you have two places where adjacent gears require shifting the front, and and the RD across it's entire range.

In the real world, you'll use the front to establish the range, then shift the RD ot pick the right gear within that range. You want overlap so for example on a small flat zone or dip within a long climb, you'll be able to shift only the rear, then shift back when the grade resumes. Likewise, you want some climbing ability off the large ring for small rises on otherwise flat areas.

As usual FB adds the voice of reason and rationality to my tongue in cheek (though accurate) post. The other thing I would add is that I know from personal experience that it is an exercise mostly of futility to find the "perfect" shifting pattern and offset on a triple. The goals of small increments between gears, no near-duplicates, good chainwheel shifting and having to shift only one lever most of the time are at odds with each other.

As with everything mechanical it is a matter of compromise. One might keep in mind that the choice of even non-duplicate gears that one has on a modern 8-10 block + triple is way, way beyond what cyclists had available in the 70's heydays. Imagine trying to find the right gear for a given condition when you have a choice of only 5, maybe 6 gears in the rear over a 13-32 range and usually a double chainwheel in the front, or with a triple half-step plus granny having to double-shift every other time with downtube levers!

As far as difference between chainrings. I run triples with 10 or 12 tooth differences between the big/middle rings

14-16 between the mid and small ones. redundant ratios are in edges of chain line displacement one doesn't use.

bigger mid-low gives new ratios on the low side 3 or 4 good, less hardly makes the granny gear worthwhile.

larger differences like using a TA cyclotourist cranks . big differences can be made, small mm bolt circle down to a fitting a 26t..

traveled with a fellow , his double was a 50 -28, most riding on the big ring, the 28 for climbing steeper slopes.
up shift on the crest of hills, makes the big upshift when the load on the chain is least.

not an uncommon practice on Classic French bikes in the 40's and 50's..


Leaving derailleurs aside, the 2 speed planetary Crank offers a 250% difference,
as if a 50 and 20 tooth gear combination.

My Brompton has a AW3 Sturmey Hub 15t cog, 54 tooth chainring , second low range as if a 21.6t gear
chain never moves off the cogs, 6 ratios are largely in 30~40% differences, with no overlaps.
lowest in high range to highest in low range slightly bigger, but no big notice

Rohloff engineered a similar set with a range shift, [7 speed x2] but engineered the gaps between ratios
to be within fractions of a percent of being 13.x % of each other from top to lowest.


One thing to bear in mind is OEM build ups are as much about price as specifications..

for buying a pallet of cases of a component from shimano will be a standard gear set.
Shimano engineers a fixed set to work together , so OEMs can expect after a quick installation they will work.

Customer can pick and choose at point of sale with dealer, for different than came out of the box ,
cost to be negotiated with dealer at that time...