Please forgive a semi-novice mechanic if this is a dumb question....In a recent thread the discussion centered around a guy wanting to use a short cage RD with a triple crank. The thing that struck me was that the rear cassette was 7 speed and the crank was from a 9 speed set. I thought these would require different width chains but nobody mentioned it. What am I not understanding?

Not a dumb question. Chain width is determined by the number of rear cogs, because the cogs get more narrowly spaced the more gears they have. Chainrings have lots of room in between them no matter what, so they can use pretty much any chain.

This, along with linked documents on the page, will probably help with a general understanding...
https://www.sheldonbrown.com/speeds.html

In a nutshell, speeds are more critical when you use index shifting. From a simplistic view, you need an "index shifter" that matches number of speeds, and the drivetrain "should' be from the same mfg; except the crank.

For many years and over a number of speeds Shimano had a standard rear derailleur actuation ratio of 1:7. This meant that the rear derailleur would move over 1.7mm for every 1mm of cable the shifter pulled, or released. If there are more speeds the shifter pulls/releases less cable so the rear derailleur moves over less. It doesn't matter which rear derailleur you use, as long as it has a compatible actuation rate, it will only move the amount the shifter dictates.

Chains have a slightly narrower outside width as you go from 6 to 7/8 to 9 to 10 speeds. While there are some crank chainring width and spacing concerns, it is "generally" the cassette spacing that drives the size chain to use. Using a wider 7/8 speed chain on a 10 speed cassette is an issue because of the distance between cogs will make it tough to run the chain smoothly without clinking against an adjacent cog.

But you can run a narrower chain, "within reason," on a wider, lower speed cassette, without problems.

It is not uncommon to see someone running a 10 speed crank, with a 9 speed FD, an 8 speed cassette, 8 speed shifters, and a 7 speed rear derailleur. Typically a 9 speed chain works well in this instance.

John

I know you meant to type this, John, but just for clarity's sake -- Shimano's actuation ratio here is 1:1.7, not 1:7. Your second sentence confirms the thought...that 1mm of cable pulled creates 1.7mm of movement of the derailleur.

Jason

Yep! Thanks for the catch.

John