A single harmonic series offers quite an assortment of intervals, but they won't necessarily be conveniently positioned with respect to each other. For example maybe you want to follow a 5:3 with another 5:3 – there's no way to do that, at least not without interspersing a 6:5 interval.
On the other hand, if you add another harmonic series, aligned so that its 3rd harmonic is at the same pitch as the 5th harmonic from the first series, then you can follow the 5:3 from the first series with another 5:3 using the second series. Moreover, every member of the second series will be 5:3 of the corresponding member of the first series, including the fundamental.
Add a third series in the same relationship to the second as the second is to the first, and not only do you get the opportunity to chain three 5:3 intervals together, and a new set of 5:3 intervals between the members of the second and third series, but you also get a set of 25:9 intervals between the corresponding members of the first and third series, again including between the fundamentals.
I use 'related harmonic series' to mean series that have members with the same pitch at different harmonic numbers, and I use 'harmonic structure' to collectively refer to two or more harmonic series that are related in this sense, either directly or through a chain of such relationships. So far as I am aware, these are my own terms, and I'd welcome a pointer to any standard terminology having the same meaning. I'm not out to stake a claim to the namespace.
It might not be obvious why there's any advantage to treating sets of related harmonic series collectively, as a structure. One reason is that doing so simplifies certain operations, both conceptually and in terms of the code required to perform those operations programmatically. Another reason is that it more closely approximates the notion of a musical scale than does a list of harmonic series and their relationships.
That said, I wouldn't advocate for casting any such structure in stone. What's the point of having composable structures if you can't edit them? In my humble opinion, any software written to enable working with harmonics using such structures should place emphasis on making them easily editable.
It's worth noting that there are other ways of arranging pitches related to each other by integer ratios of frequencies, besides placing harmonic series adjacent to each other, for example the tonality diamond. My original goal was to arrange nodes representing related pitches into an editable tree structure. Unfortunately I never came up with a way of keeping such a tree structure from becoming disproportionate and messy on-screen, and I've generally found harmonic series easier to work with, both in abstract terms and for the purpose of building software around them.
Next I'll address the question of how to deal with a harmonic structure as a single unit.
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