Why the exercise is in four parts

Open any harmony textbook printed since about 1750 and the exercise is the same shape: a bass, a figure under it, and four staves to fill. Not three, which would need no doubling and would state every chord exactly once. Not six, which is what a real choir often has. Four.

This ladder has spent eight rungs inside that number without once asking about it. The cheapest assignment of voices is computed over four. The geometry of chord space has four coordinates. The price of each prohibition is priced in four parts, and the essay that computed those prices said out loud that the compass of the voices was the one free parameter in the apparatus and moved it to see what happened. Nobody moved the count.

So this rung moves it, which is the last thing about this model left to move.

One choir, cut into n parts #

Varying the number of voices needs a family of ensembles, and the family has to contain the choir or the comparison measures the family instead of the count.

This one holds the total compass fixed. The lowest voice starts at E2 and the highest starts at C4, exactly as the choral ranges do; each voice covers twenty semitones, which is the choral average; and the n voices are laid evenly between the two ends. Every ensemble in the family therefore occupies the same span of pitch and differs only in how many parts that span is cut into — which is the right control, because a six-part texture that was also allowed to be two octaves wider would be answering a different question.

At four voices the family gives E2–A♭3, B2–E4, F3–B♭4 and C4–A♭5. The choral ranges this site has used since the rules were first priced are E2–C4, C3–G4, F3–D5 and C4–A5: within a semitone at every voice. And they produce exactly the same number of complete voicings — 145 of the dominant triad and 200 of the tonic, against the site’s own SATB machinery, to the unit. The family is the choir at four, and it is something else everywhere else.

Two voices cannot state a triad #

The family starts at three rather than two, and the reason is the first result rather than a convenience.

A triad has three pitch classes. Completeness — every one of them sounding — is enforced here for the reason the four-part machinery has always enforced it: a realisation that drops the third is not a cheaper realisation of the chord, it is a realisation of a different chord, and a minimisation allowed to drop notes will buy its cheapness by leaving them out. Under that rule a duet has no complete voicings of any triad at all. Not few. None.

That is worth stating plainly because it is the whole of why two-part writing is a different subject with different rules. Two parts cannot assert a harmony; they can only imply one, which is why the treatises on two-voice counterpoint spend their time on intervals between the parts and the four-part ones spend their time on chords. The ban on parallel fifths is stated for two voices and enforced in four, and the reason it can be stated for two is that it is not a claim about a chord.

Three voices, and the rules are free #

Now the census. Take every complete voicing of the dominant triad and every complete voicing of the tonic, pair them all, and count how many of those transitions contain a parallel fifth or a parallel octave between any pair of voices.

Ninety-one per cent legal is a prohibition with almost nothing to prohibit. And the cheapest three-part realisation of I–vi–ii–V–I obeys both rules without being asked to: the minimum with the rules off and the minimum with them on are the same number, twelve semitones, and the same set of voicings.

What the doubling does, and it is not symmetric #

Splitting the census by which prohibition catches the writing turns the result from a curve into a mechanism.

Zero is not a small number here; it is a structural impossibility. A parallel octave needs two voices a twelfth or an octave or a unison apart, moving together — which needs two voices sounding the same pitch class. Three voices carrying three pitch classes never do. The moment a fourth voice arrives it has nothing new to sing, so it doubles, and the doubling is what makes the second prohibition possible.

n parts against a triad double n − 3 notes. One doubling at four voices, two at five, three at six, four at seven. The octave rule’s reach follows that count almost exactly, and by seven voices — a census of 25.7 million pairs, run once away from the page because it is not a thing to recompute — 91.4 per cent of all transitions contain a parallel octave somewhere.

Behind the whole curve is one number that can be backed out of it. If the voice pairs were independent, the clean fraction would be (1 − p) raised to the number of pairs, where p is the chance one pair goes parallel. Solving for p gives 0.030 at three voices, 0.047 at four, 0.078 at five and 0.108 at six. So it is not merely that six voices have fifteen pairs where four have six: each individual pair is three and a half times likelier to offend, because the doubling has put two voices on the same note and given them somewhere disastrous to go together.

The price, and where it turns #

The census counts what is forbidden. What it costs is a different quantity, and it is the one this ladder has always used: the extra semitones of motion the cheapest legal realisation needs over the cheapest realisation of any kind.

The gap between the two lines is the price, and it is the sharpest number this rung has: 0.000 semitones at three voices, 0.063 at four, 0.150 at five, 0.542 at six, and 0.821 at seven. From four to six it multiplies by nearly nine.

Those are one progression’s numbers and the limitations below run two more. What holds on all three is the zero at three voices and the direction of everything after it; what does not hold is the size at four, which is four times larger on a progression whose roots move by step. The six-voice figure barely moves. So the shape of the curve is the finding and its steepness is partly a property of I–vi–ii–V–I.

The two lines going opposite ways is the part worth dwelling on. More voices make the unconstrained problem easier, because motion can be spread over more parts and each one has less to do. More voices make the constrained problem harder, faster, because the number of ways to offend grows quadratically while the number of ways to move does not. Four parts sit at the crossing: enough voices for the prohibitions to mean something, few enough that obeying them is nearly free.

What four is, stated as a claim #

Put the three measurements together and the answer has a shape.

Two parts cannot state the chord. Three parts state it and the prohibitions are vacuous — nine transitions in ten are legal, and the cheapest writing obeys them by accident. Five and six parts state it several times over and the prohibitions dominate — most transitions are illegal, the price of obedience is most of a semitone a voice, and what a student would be learning is how to dodge rather than how to move. Four parts are the only place where both halves are true at once: the chord is complete, there is exactly one doubling, three quarters of the writing is still legal, and obeying costs between a sixteenth and a quarter of a semitone per voice per change depending on how the roots move.

Priced on its own, over every melody the progression admits, each prohibition costs something different: the two parallel rules sit near the cheap end and the dearest rule is not the famous one. That is the four-part reading, and the census above says the ranking is not stable in the count — the octave rule is free at three voices and dominant at six, and no measurement made at four could have said so.

The saturation at six is the part of that worth carrying furthest, because it says something about the pedagogy the essay is explaining. A student writing in four parts finds the difficulty of an exercise depends heavily on which progression they were given; a student writing in six finds it does not, because at six voices the prohibitions are the whole of the difficulty and the chords have stopped contributing. That is a fair description of the difference between a four-part exercise, which can be easy or hard, and a six-part one, which is uniformly a puzzle.

It is not a claim that four is optimal for music. It is a claim about what four is optimal for, which is teaching a constraint: the exercise needs a rule that bites and a rule that can be satisfied, and four parts is the narrow band where one rule is both.

That reading also predicts something the treatises do. In five- and six-part writing the textbooks stop insisting and start allowing — doubled thirds, omitted fifths, “unequal fifths” tolerated between inner parts. They are not being lax. They are working in a region where the constraint set has become nearly unsatisfiable, and the exceptions are what is left of a rule that can no longer be kept.

Which computation produced the numbers #

The realisations come from a layered-graph shortest path — the same Viterbi recursion the four-part machinery has always used, generalised over the voice count. Every complete uncrossed voicing of every chord is a state, every legal transition is an edge weighted by total semitone motion, and the answer is the exact minimum. That matters more here than usual, because every headline number is a subtraction between two of these, and a merely good solution would price the search instead of the rule.

Uncrossed is enforced during the enumeration rather than filtered afterwards, and that is what makes six parts computable: the ordered count of a triad’s voicings is 21, 145, 556 and 1,541 at three to six voices where the unordered count is 28, 334, 2,350 and 11,785. The transition census is every ordered pair — 3.5 million at six voices, 25.7 million at seven, and 117 million at eight, which is where this stops.

The two prohibitions are tested exactly as the four-part code tests them, including the requirement that both voices move: a fifth held between two stationary parts is not a consecutive fifth, and treating it as one would ban a repeated chord.

Whose music, and when #

The rules being priced are the eighteenth-century chorale style’s, as every harmony textbook since has transmitted them, and the four-part exercise is that tradition’s pedagogy rather than a fact about music. Renaissance polyphony is routinely in five and six parts and does not obey these prohibitions in this form; Baroque trio sonatas are in three; a great deal of the world’s music is in one.

What travels beyond the jurisdiction is the arithmetic underneath, and it is a claim about counting rather than about style: any prohibition stated over pairs of parts gets quadratically harder as parts are added, while the freedom to satisfy it does not. That is why every polyphonic tradition that went past four parts also loosened something, and it is the same pressure that shows up in what a choir does that a soloist cannot — sixteen singers on one line are not sixteen independent voices, and a tradition that wants more parts than four buys them by making some of them not really parts.

What the picture cannot show #

The compass is a choice and the family is a construction. Every ensemble here spans the same forty-one semitones, and a six-part texture with genuinely wider outer voices would score better — the earlier finding that widening every compass by a fifth drops the price of the parallel-fifth rule from 0.54 semitones to 0.10 applies here too. What the family fixes is the total range, deliberately, because the question is about the count.

One progression, and it is a friendly one — which is worth running rather than conceding, because the price is the essay’s whole result. I–vi–ii–V–I is all root-position triads and all descending-fifth or descending-third motion. Re-running the price curve on progressions that move by step, where parallels are hardest to avoid:

The prediction is right that a stepwise progression is dearer and wrong about where. The four-voice price quadruples across these three — 0.063 to 0.250 — while the six-voice price moves by a third, from 0.50 to 0.67. So the six-part figure is nearly the same whatever is being written and the four-part figure is not: at six voices the constraint has saturated and the progression barely matters, and at four there is still enough slack for the choice of chords to be most of the answer.

That inverts the factor of nine as well. From four voices to six the price multiplies by 8.7 on the friendly progression and by 2.7 on both step ones, so the nine is a property of a numerator that happened to be small rather than of a denominator that is large. The honest range is between three and nine, and the reason it is a range is that only one end of it is stable.

What survives untouched is the three-voice result, and it survives on every progression tried: zero, exactly, three times. The cheapest three-part realisation obeys both prohibitions without being asked, whether the roots move by fifth, by step, or back and forth between two chords. That is the structural claim of this essay and it does not depend on the example.

And the crossing survives too. On all three progressions the free minimum falls as voices are added and the ruled minimum rises, so four remains the place where the two lines are nearest — it is the size of the gap there that is a property of the music rather than of the count. A quarter of a semitone per voice per change is still small beside six parts’ two thirds, and it is four times what this essay reported.

Nothing here doubles a note deliberately. Which note gets doubled is left free for the minimiser to choose, which is exactly what the treatises have opinions about and this model does not. A real six-part texture is written by somebody choosing the doublings first; the number computed here is what an ensemble can achieve at best, not what it typically does.

And no rung of this ladder has ever heard a voice. The streaming rung is the one that measured the difference, and its answer applies with more force here rather than less: at six parts the adjacent voices are on average four semitones apart, well inside the boundary below which two lines cannot be heard as two at any speed. Some of those fifteen pairs are not pairs of anything a listener has.

The ladder ends here #

voice-leading closes at nine rungs. The model is a chord as a set of pitches in a register and a change as a motion between two of them, priced by total semitone displacement, and the ladder now bounds it in every variable it has: what the metric is (one), what shape the space it induces has (two), what motion costs beyond distance (three and five), what the metric can be run over (four), where two ways of stating it disagree (six), where the metric picks the wrong answer (seven), whether the assignment it computes is heard at all (eight), and now how many voices there are.

Name a variable the model has, and it is on that list. What is not on it belongs to other models: which chord follows which is progression, whether a chord is a chord at all is the-triad, and how a notation collapses the whole space is somewhere else again — that rung counted 2,042,672 realisations of a figured bass and 59,418,496 of a Roman numeral, in four parts, because four is what it inherited.

The count that started this was a count over four voices. A chord symbol, a Roman numeral and a figured bass admit different numbers of four-part realisations, and the numbers say what each notation thinks a chord is — but every one of them is a count over four, and the four was never chosen, on this site or in the tradition the numbers come from.