Originally Posted by Andrew R Stewart

A few comments-

An aprox 15" gear (24x42 on a "27" wheel") is a very low gear and one that being able to balance the bike (especially a tandem), at the 3ish MPH that this low ratio is often run at, would be hard to do for many to actually do. But I am a proponent of low gearing and do spin (as well as have a lot of tandem miles in my past) so if you can get it all to work that's great IMO.

Do know you're mixing different "generational" versions of what is "the right way" to design a gear system. The triple crank is "old school" (all my bikes that I ride much have them) but the cassette range is "new school". The ders that the triple crank was intended to be used with generally didn't have the large rear cog capacity to handle more than a 36 at most (and pretty much all of that era had a 34T max as the listed limit). trying to use a cassette with so huge a large cog was never their intention. As the industry has moved on from front shifting in general and have had to instead go to larger rear cogs for the low gear ratio the chain wrap capacities have reflected this.

An attempt at a work around that can work for some is the der hanger extended adaptor (aka "Wolf Tooth"). This mounts the rear der farther away from the cogs, thus allowing a larger diameter/greater tooth count. One issue with these adaptors is that the smaller cogs are also further from the der's guide pulley and the additional chain links, in between the pulley and cog underside, all have to go through their amount of lateral flex/slop before the "top" link will even try to climb off the current cog it's on before the shift is tried. When all is new and not too worn this can be OK... Add wear or other issues and...

For the rear der to "like" shifting into a cog it wants the guide pulley to be lower down then the cog's underside. Too far under and things get sluggish, too close and the chain might not be able to climb onto the next larger cog well or at all. This balancing act is why slanted parallelograms and sprung B pivots have been a standard (since the 1980s aprox) when a wide range of chain wrap is needed. (A notable exception was the Huret DuoPar). The book "The Dancing Chain" is a great resource for those looking to better understand all this.

One other issue might be that the guide pulley bushing is worn enough to have more lateral slop then when new. If that pulley is rocking about it can't control where it's feeding in the chain WRT where the intended cog's teeth tops are. Pulleys can be replaced if that were the only issue, however with the miss match between old and new this is not the only issue.

On the der's backside of the inner parallelogram link is where Shimano usually places the actual version/model number. RD-M761 is an example of one Deore XT version. These model numbers can be hard to see and often covered with grime. For all the flack Shimano gets for various reasons they have done a far better job at stamping their parts with model numbers then other companies.

Now besides all the mechanical crap affecting shifting there's that elephant on the back of the tandem, so to speak. A tandem with its far longer cables and that added power unit (stoker) which usually lags behind any "commands" from the captain just increases the challenge of shifting.

I suspect this tandem is your first as a team. I also suspect that who ever was servicing this tandem didn't pay attention to der capacity limits or such either. My first tandem was long before my stoker was always the same person. My rides were far less pleasant and more challenging to "work" the bike than after I married. (That first tandem came from a couple who never could work in harmony. My second and third tandems saw over 30K miles with my late wife, you do the math). Besides my stoker learning how to be steadier in their movements I learned how to better communicate and we learned what it took for the bike to work better. This didn't happen on the first ride and we "forgot" our techniques far more often when either distracted (like my wanting to keep up with others) or tired.

When I worked in a tandem specialist shop I got to watch many teams go through this as they tried to see if their hope of a more intimate riding experience was going to turn out well. When we did service work on our customer's bikes the mechanic would test ride after the work alone first then have ne of the other shop people stoke for them. That added power and lessening of control over that power really changed the dynamics and the bike would sometimes need a revisit to the repair stand before the work was done as well as we wanted. Andy