The “classical” counterpoint curriculum usually goes like this: you start with first species in two voices (note against note), then second, third, fourth, and fifth species—still in two voices; then you move to three, four, five… all the way to eight voices. Only then (finally!) do you get to canon, double counterpoint, triple and quadruple counterpoint, and at last the fugue.
It sounds obvious because it feels “progressive”: from little to much, from simple to complex. Except that—mathematically speaking—it’s often the reverse of what actually happens when you build a polyphonic texture. In other words: we train the student to drive uphill on bald tires, and then act surprised when the car spins out.
We’re not “downgrading” Cherubini (he’s still a giant). We’re saying something more useful: the traditional method, as it stands, is prone to mistakes because it confuses the number of voices with combinatorial difficulty. And those are not the same thing.
Want to try it yourself? The paper presents the method in a practical, hands-on way: matrix, number→interval conversion, and control criteria.
Go to the paper (Zenodo) →1) The paradox: with eight voices you have less freedom… and that’s exactly why you make fewer mistakes
School counterpoint plants a simple idea in your head: “more voices = more difficulty.” But if you treat it as a problem of allowed choices (i.e., legal combinations), the opposite happens: as you add voices, the choice space shrinks. And when the choice space shrinks, many moves become almost mandatory. That’s not romanticism: it’s engineering.
Put bluntly: in two voices you have too much freedom. And too much freedom—especially in the hands of someone learning—produces the “taste” mistake that later becomes a “grammar” mistake. In eight voices, instead, if you impose a coherent load-bearing structure, the rails are tighter and the locomotive derails less.
That’s why an eight-part counterpoint, built with clear constraints, can be paradoxically more manageable than a two-part one. Difficulty is not “how many voices”: it’s how many real degrees of freedom you still have after you’ve excluded what is forbidden.
2) Counterpoint’s skeleton is a triad: three numbers, not a thousand rules
Counterpoint, before it becomes style, is a skeleton. And that skeleton, reduced to essentials, rests on just three positions: 1, 3, 5. Two stacked thirds (1–3 and 3–5) plus the fifth (1–5). That’s it.
If you reason in terms of distances, you’re working with a tiny set of possibilities. And when the set is tiny, you can build a system that prevents mistakes by construction, instead of chasing them with corrective rules.
This is the point traditional teaching often misses: it teaches “how not to make mistakes” through a thousand local precautions, but it rarely teaches how to design a global structure that makes mistakes unlikely.
3) Canon is the simplest thing: why do we teach it after eight-voice counterpoint?
Here’s the question that irritates people (which is precisely why it’s useful): if canon is an imitation machine that practically builds itself, why does it arrive so late in the manuals?
A well-set canon is an exercise in structural coherence more than in “invention.” If all voices imitate the same skeleton, “difficulty” doesn’t grow: it gets distributed. And, crucially, the student immediately learns a basic truth: polyphony is not “many melodies,” but one generative rule.
The didactic paradox is this: students are forced to do the most exposed homework first (two voices with maximum freedom and maximum risk of parallels), and only later are they shown the exercise that actually clarifies what structure is.
If you want to teach “counterpoint as skeleton,” the reasonable order is often the reverse: start with extremely rigid exercises (like an eight-voice canon), where choices are few and controllable, and then widen the space of freedom step by step.
4) How to build an eight-voice canon (no theatrics: just method)
In a fully imitative eight-voice canon, the options are few—almost “countable on your fingers”—because total imitation imposes very strong constraints. And pedagogically, that’s an advantage.
The basic logic, in a dry version:
- Avoid repeated pairs within the same structural progression: if you repeat the same pair in two voices, you easily generate parallel octaves or fifths (the same distance repeated at the same moment).
- Control saturation: you can’t have too many occurrences of the same value in the same “column” (vertical moment), otherwise you block the legal exits into the next column.
- Don’t place 5 in the bass as a prudential skeletal rule: this avoids having to manage, too early, the fragility of the 6–4 sonority and keeps the focus on the stability of the frame.
The point is not “being clever”: it’s understanding that, with a correct structure, polyphony stops being a labyrinth and becomes a path with guardrails.
5) Why the school system is prone to mistakes: the domino effect
Traditional teaching often drills local prudence: “keep common tones,” “bind the harmony,” “play it safe.” It sounds sensible, but in dense textures it can be a false friend: prudence accumulates repetitions, repetition creates saturation, and saturation forces technical “patches” that set up the final mistake.
It’s the classic domino effect: a seemingly conservative choice at bar 2 sends you the bill at bar 9. Not because the composer is “bad,” but because the strategy lacks a global theory of saturation: it navigates by sight.
When the structure is designed from the start with distribution constraints (how many 1s, how many 3s, how many 5s per column), freedom doesn’t vanish: it gets channeled. And parallel motion (fifths and octaves) is not “avoided” afterwards: it is made mathematically unlikely beforehand.
Conclusion: invert the ladder to restore meaning to “difficulty”
If you want to train an actual composer (not just a recipe-executor), you must teach them to see counterpoint as structure: a few numbers, a few clear constraints, and a global design that reduces mistakes by construction.
And yes: at that point you can return to Cherubini with even greater respect, because you read him not as a catechism but as a great historical workshop. The difference is that today we have the tools to see where didactic prudence turns into a trap. And music—when it’s done well—doesn’t forgive traps: it amplifies them.
