Pursuing clarity through openness, part 5: the highest common fundamental

Given a harmonic structure, two or more harmonic series all connected together through identically pitched members at different harmonic numbers, an anchor having nominally stable integer-ratio relationships to the fundamentals of those series, and a means of specifying the pitch of that anchor, what more do you need?

Perhaps nothing else, depending upon what you have in mind to do, but if you intend to use this assemblage to synthesize sound there's another important piece to the puzzle.

Say you have two tuning forks that are very nearly the same pitch, within a fraction of 1 Hz of being exactly the same, and you strike them both and hold them near each other and close to one ear, the sound you hear will rise and fall as the sound waves from the tuning forks alternate between constructive and destructive interference. The rate at which this happens is the beat frequency, and the closer the tuning forks are to being exactly the same pitch the lower the beat frequency.

With digital sound, it is quite possible to have two sound sources at exactly the same constant pitch, meaning that how they interact at first is how they will continue to interact for as long as both sources persist, whether constructively, destructively, or something in between. What sort of interference you get depends entirely on the relative alignment of their phases at the outset, something that isn't easy to control if, for example, you're passing user events to code that spawns a new render thread for each new note. That scenario will reward you with random results.

To backtrack a bit, a latent property of harmonic structures is that they always imply a harmonic series that includes all of the members of all of the harmonic series composing the structure. The fundamental of that implied series will generally be lower than any of the fundamentals of the series composing the structure, although it might be the same as the lowest of them. In reality, there are many such fundamentals, since, if x defines a harmonic series which includes all the members of a structure, so too will x/2, x/3, x/4, and so forth. To keep things simple, what we really want is the highest pitch satisfying the requirement that it define a series including every member of the structure. This is what I'll call the 'Highest Common Fundamental' (HCF).

You can think of that highest common fundamental as a continuous sine wave — a constant tone, albeit one which may be well below the threshold of human hearing, and, in any case, doesn't participate directly in the sound produced. It serves a function vaguely analogous to that of a metronome, establishing a phase alignment that advances at a steady pace and can be used to drive the phase alignments of all of the members of the structure, enabling control over how they interact, whether constructively or destructively. I'll save the details of how this works for later.

For now just soak in the idea that every harmonic structure implies a fundamental defining a series that includes every member of the structure, and that, by providing a common basis for phase alignment, this implied fundamental can be used to drive sound generation while controlling for constructive vs. destructive interference.

Part 6: the anchor, HCF, and tracing the chain of tonality