A "third minute" (of 1/60 second) could be useful in talking about human reaction times. We could say that people commonly have a reaction time of 7 to 15 thirds, and that alcohol consumption may slow that reaction time by a further 8 thirds or more.
Or, speedrunners and professional musicians can practice motions (relative to their own motions or to an ongoing rhythm) that are accurate to about 1 third.
I assume the terminology would be too easy to confuse with ⅓, so maybe they would have to be called "terces" or "tertians" or "tertiaries" or something. (Specifically, we want to say (1/60)³ hour rather than (1/3) minute!) And maybe it's not a great thing to introduce a non-decimal unit when we're also trying to get rid of them in many other contexts.
I should follow up and say that I learned that this unit has already been referred to under the name "tierce", although it's not at all common.
It could also be useful for measuring latency of Internet connections, although these have commonly gotten faster over time and many latencies can be under a tierce. On the other hand, the round-trip latency to a satellite in geostationary orbit is about 14 tierces.
Or, speedrunners and professional musicians can practice motions (relative to their own motions or to an ongoing rhythm) that are accurate to about 1 third.
I assume the terminology would be too easy to confuse with ⅓, so maybe they would have to be called "terces" or "tertians" or "tertiaries" or something. (Specifically, we want to say (1/60)³ hour rather than (1/3) minute!) And maybe it's not a great thing to introduce a non-decimal unit when we're also trying to get rid of them in many other contexts.