Precise etymology of "harmonic" (including h. mean and h. series)?

It seems to be very much a common knowledge now that harmonics of an (almost) periodic audio signal do evidently stem from Ancient Greek investigations into musical harmony, which was intertwined and likely gave us harmonic means and harmonic sequence/series, means likely being derived from the sequence like it can be done with cases of arithmetic and geometric sequences and means. The details are hovewer not clean at all. Presumably, not even in the Hellenistic period had it been known about harmonics in sound. One question is then, if a noun parallel to harmonic (which should stem from Greek, obviously) was in use, what meanings did it have in mathematics and music theory of the time. I see at least two options as very probable: (a) it hadn't been used in a way parallel to how today's harmonic is used; (b) it denoted sounds or musical pitches that you can derive from the "original" sound of an open string by subdividing it into k parts. Note how right now, a very much related technique of playing stringed instruments is indeed called "harmonics" in English (but " flageolets " in some other languages). The second question (that's less related to this sub) is precise etymology of this name: I would expect flageolets should be a technique older than both the discovery of Fourier series and connecting string and air channel lengths to their fundamental frequencies. Also a question 1.5 is relating the answer to the first question to harmonic means and harmonic series. For example if (a) held, the answer is very much warranted and shouldn't be deduced without historical sources. There are probably a bunch of additional minor questions that I can't refine out of the mess in my head right now, that would be obvious when one tries to set things straight on this entire topic and encounters important details that answer them. For example as far I know, frequencies of sound weren't known in antiquity (but that's really a question to answer definitively as well!), so we can't attribute "reciprocalness" of harmonic mean and so on to the inverse relation between length and frequency . Instead it should probably be entirely due to there being a sort of a natural order to harmonics (in the sense of (b)) coming one after another at 1/1, 1/2, 1/3, ... of the string's length —but is that the factual reason that had been used? You see even a mathematicians' brains would otherwise be glad to fill in the "obvious" blanks and I feel that a great lot of people think they sorta know the story about this. I'd of course expect it of non-historical music theorists even more because in their case it's actually quite important to talk about time to time —whereas in mathematics, harmonic mean and harmonic series (and wasn't there something else?) is just a neat and somewhat old-fashioned (as is, again, common) term flavoring that evokes associations that no one has to unpack to work with the actual math. But that may still give birth to mythology when one gets interested just a little. So it's a good point to look around and try remembering while the ancient sources may be still laying around! And if there's already a piece of work on this very topic already written, the easier it is to find it again (I couldn't, but my work with references is abysmal).