Just as it's easier to carry a bucket that has a handle than one that doesn't, it's easier to manipulate and make use of a collection of related harmonic series if they are all tied to a single frame of reference that both specifies the relationships between them and relates them to an external context.
That external context is the pitch scale, frequencies measured in Hz. This is a bit like Schrödinger's cat in that it doesn't matter for the purpose of the relationships between harmonic series and their members, not until those relationships begin to participate in the generation of sound, whether tinkering, composition, practice, or performance. The relationships between series in a structure are pitch-independent (beat frequencies excepted).
However, in the interest of being able to apply that external pitch scale when the time comes, it will be convenient to have a sort of handle to connect to, a node which is stable with respect to the rest of the structure (at least between edits), so that by assigning a frequency to it you also assign frequencies to every member of every series in the structure. That node need not actually be a member of any of those series. The only requirement is that it be in integer-ratio relationship with them; exactly what's used is somewhat arbitrary. There are probably better choices and less good choices, based on the complexity of the ratios needed to specify the fundamentals of the component series, and it may be desirable to recompute this central point of reference after an edit, for the sake of keeping those ratios as simple as possible, but these issues are out of scope for the moment.
I'll be referring to this recomputable central point of reference as the 'anchor'. I say this with some trepidation as I have also used 'anchor' to mean something different, but this seems both the best word for what is intended and the best use of the word, so I'll let it be. Again, I welcome pointers to any standard terminology having the same meaning.
The anchor might start out as the fundamental of the first series in a structure, before anything else is added, but later you might find yourself moving that series either higher or lower with respect to the rest of the structure, meaning that either the anchor follows it, and the ratios relating it to the other fundamentals will need to be recalculated, or it remains stable with respect to the rest of the structure and the ratio relating it to that initial series will need to be recalculated. However one deals with this, after it's over the anchor must remain in integer-ratio relationship with the fundamentals of all series in the structure.
There's also the matter of how to specify the anchor within the external pitch-scale context. You can simply assign it a frequency, and then recompute that frequency as needed, however I recommend using the combination of an infinitely variable scalar (a floating-point value) and an integer ratio, multiplied together to produce the anchor's frequency. The ratio makes consonant transposition simple, and the scalar allows fine tuning.
Next, the nature of the 'highest common fundamental' and what it's good for.
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