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Assume it's not possible to bodge anything less than 10speed since it won't be able to move far enough?
The rear dérailleur travel is the same for 8, 9 and 10 speed (they just squeeze more sprockets into the same width by narrowing the chain), so you can use an 8 or 9 speed RD on a 10-speed cassette. In fact, you can probably get away with a 6 or 7 speed RD too, as they had only about 3mm less travel and there was always some wiggle room to allow for dropout geometry variation, hence the limit screws. Potentially, using a lower speed RD makes shifting slower/less precise as the cage plates get progressively further apart as you drop down the speed range, but it only really gets bad if there is a large mismatch.
Building a 1x10 frankenbike since I've got a random 10speed rear wheel - gonna use a downtube shifter but will I need a 10 speed rear mech? Assume it's not possible to bodge anything less than 10speed since it won't be able to move far enough?