A rule, exactly stated

The useful thing about a convention

Conventions have a reputation for vagueness, which is unearned. The rules of part-writing are not advice about taste. They are closer to a specification, and the most famous of them is exact enough to implement in about fifteen lines:

No two voices may move from one perfect fifth to another, or from one octave to another. Both voices have to move for it to count; if one holds still, no rule is broken.

Nothing in that needs interpreting. Given two chords and which note each voice is singing, it is decidable. That is unusual enough to be worth building the whole course on, because a rule you can run is a rule you can measure rather than argue about.

Two questions follow, and this lesson answers both. Why would anyone adopt it? And how much does it actually forbid?

Why the rule exists

Foundations lesson 2 established that a pitched sound is a stack of partials at whole multiples of its frequency. So ask what happens when two such stacks sound together.

If the two tones stand in the ratio p:q, then one in every q of the upper tone's partials lands exactly on one of the lower tone's. An octave is 2:1, so every partial of the upper tone is already present in the lower one. A fifth is 3:2, so half of them are. A major third is 5:4, so a quarter.

The more two tones share, the less evidence your hearing has that there are two sources rather than one. Move a pair of voices in parallel at such an interval and they stop being a pair.

Which is the whole list

Sort the intervals by how much the two tones have in common and the forbidden ones are the top of the list, with a gap under them. Unison and octave share everything. The fifth shares half. Nothing else reaches beyond a third.

So the rule is not arbitrary and it is not about taste. It is a device for keeping four voices audible as four voices, aimed precisely at the intervals where two of them would otherwise collapse into one. Somebody worked this out by ear, centuries before anyone could have written down the reason.

Two honest wrinkles. The perfect fourth shares a third of its partials, the same as a major sixth, and yet counterpoint treats the fourth as a dissonance requiring preparation and the sixth as a free consonance. The fusion account does not explain that, and nothing else here does either. Second, the figures use just ratios while the music uses equal temperament, where a fifth is two cents narrow and a major third nearly fourteen cents wide. The tempered versions of the forbidden intervals are almost exactly just, so they fuse; the tempered third is far enough out that its partials beat instead. Temperament happens to preserve the fusion of exactly the intervals the rule bans.

What it costs

Now the second question, which is the one a convention is rarely asked. Take two chords, generate every arrangement of them across four voices that stays in range and does not cross, and count how many of the transitions between them the rule removes.

About a fifth of everything

Between 7,830 and 11,067 transitions per progression, and the strict prohibition removes between 16.8% and 21.6% of them depending on the pair. Add the weaker rule about arriving at a perfect interval by contrary motion and the range runs from 22.6% to 28.6%, so on the worst progression better than one arrangement in four is ruled out.

That is one rule out of dozens. Part-writing has prohibitions on voice crossing, overlap, spacing, doubled leading tones, unresolved sevenths and augmented seconds, and lesson 3 applies them together to see what survives. The answer is not many, which is the point of teaching them: the rules do not describe a style so much as fence it in until only the style is left.

Worth noticing which progression sits highest in that figure. IV to V is the most constrained, and Harmony lesson 5 found it also has the roughest voice leading available in the key - six semitones of movement, nothing held. It is also one of the most used progressions in the music. The rules and the practice point different ways there, and the practice is not the one that gives.

And it is still not a law

Everything above is a case for the rule. It has a mechanism, the mechanism is real, and the mechanism explains the exact list of intervals. It is about as well-founded as a convention gets.

Organum, the earliest written European polyphony, consists of voices moving in parallel fifths and octaves for minutes at a time. It is not breaking the rule; the rule would not exist for another four hundred years. What it wants is exactly the fusion the fusion argument predicts: one enormous voice rather than several small ones.

Rock guitar does the same thing for the same reason. A power chord is a bare fifth, and a riff played in power chords is parallel fifths from beginning to end. The fusion is the product. Nobody wants to hear two independent voices in there.

Debussy and Ravel used parallel fifths deliberately against a tradition that forbade them, and it reads as colour rather than as error, which only works because the prohibition was in force.

So the rule has a genuine perceptual basis and remains a choice. It says: in this repertoire we want four independent voices, and here is what that costs. A repertoire that wants something else declines it, and is not wrong. That distinction is what this course is for, and lesson 1 spends it on the strongest case the other side has.