From this point on I'll (mostly) refer to the Highest Common Fundamental as HCF.
While you might actually choose an anchor that initially matches the pitch of the HCF, they have different purposes, and different behavior resulting from changes to the structure. The anchor is like a handle, a means of hanging onto the structure as a whole, and exactly where it's positioned is somewhat arbitrary; it is free to remain stable, or not, in the wake of editing of the structure. The HCF, on the other hand, even though it will end up being defined in terms of the anchor, is not arbitrary at all. It's position is dictated by the fundamentals of the series composing the structure. Change one of those fundamentals or add another series to the structure and it's likely that the HCF will also need to change, and its relationship to every member of the structure along with it.
The anchor is the only component of the structure which is directly tied to the pitch-scale; everything else is connected to the pitch-scale through the anchor. For the purpose of sound generation that connection also passes through the HCF. Starting with the ratios relating the fundamentals of the series composing the structure to the anchor, expressed in lowest terms, the denominators of those ratios will need to be reduced to their prime factors, for example 1/6 becomes 1/(2 x 3) and 1/9 becomes 1/(3 x 3). For each prime used, the greatest number of times it is used in a single denominator (in this example 2 and 3 x 3) is included in a multiplication to produce the denominator of a ratio relating the anchor to the HCF (2 x 3 x 3 = 18, yielding a ratio of 1/18). This is the application of prime factors for which I can make an argument in favor of their relevance.
To determine which members of the harmonic series defined by the HCF represent the fundamentals of the series composing the structure, we'll divide the ratios relating those fundamentals to the anchor by the ratio relating the HCF to the anchor, or rather invert and multiply (1/6 x 18 = 3, and 1/9 x 18 = 2).
At this point it will be helpful to introduce some shorthand notation, to differentiate between native harmonic numbers (multiples of the fundamentals of the harmonic series composing the structure) and harmonic numbers in the series defined by the HCF, I'll prefix the latter group with "HCF-" (HCF-3 and HCF-2 in the above example).
To go on to determine which members of the harmonic series defined by the HCF represent the members of the other series, we'll multiply their native harmonic numbers by the HCF harmonic number of the fundamental of the series to which they belong (using the 5th harmonic in each case, 5 x HCF-3 => HCF-15 and 5 x HCF-2 => HCF-10).
We can carry this one step further, crafting complex tones using harmonics of the harmonics, which are also part of the structure, by simply adding another multiplication factor. Continuing with the above example, the 2nd harmonic of the 5th harmonic of each series will be two times the basic harmonic (2 x HCF-15 => HCF-30 and 2 x HCF-10 => HCF-20) and the 3rd harmonic of the 5th harmonic of each series will be three times the basic harmonic (3 x HCF-15 => HCF-45 and 3 x HCF-10 => HCF-30).
Note that it's quite possible to arrive at multiple instances of the same HCF harmonic, and there may be algorithmic efficiencies to be built around that.
What we now have is a chain of connection, beginning with the pitch-scale, passing through the anchor, through the HCF, to the fundamentals of the harmonic series composing the structure, and from them to the other members of those series, and from them to their own harmonics, expressed in terms of multiples of the HCF.
So how do we get sound out of this? That'll be next.
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