While "perfect third" may not exist as a name for an interval, it does make sense to apply the adjective "perfect" to the interval "third" in order to distinguish a just intonation third (4/5) from a well-tempered intonation third (1/2)^(4/12).

You can construct a system under which the phrase makes sense but musicians and musicologists will look at you weird if you say it. The LLM just forges bravely on, however.

Example usage here,

https://archive.org/details/onsensationston01helmgoog/page/3...

(by Helmholtz of Helmholtz Equation)

I'm very impressed that you were able to find this.

> (1/2)^(4/12)

Not a single one of those numbers is 3 like "third" sounds like. Or is the 4/12 == 1/3 the actual numeric connection here?

Still doesn't explain why a doubling of frequency (2) is called an octave (8, which isn't even the correct value for neither the 7 major notes nor the 12 notes) though.

I don't think of it as a failure. The LLM figured out a reasonable interpretation for a term it did not know. It might be better to come back with "do you mean pure ptolemaic major third?" instead of assuming, but the way it does it now might be better for programmatic requests.