What the rules cost

A harmony textbook is a list of things not to do. No consecutive fifths, no consecutive octaves, no voices crossing, no gap of more than an octave between the upper parts, no doubled leading note, resolve the leading note, and no augmented melodic intervals in any voice.

They are presented as one kind of thing, in one list, with one tone of voice. They are not one kind of thing, and the difference is measurable — because every one of them is a constraint on a problem this site has been solving since its first rung on this ladder: find the assignment of voices that moves the least.

Constrain a minimisation and the minimum gets worse. The amount it gets worse by is the price of the constraint, in the units the minimisation is in, and here those units are semitones of voice motion.

The progression is I–ii–iii–IV, roots rising by step, which is the case every treatise warns about. Even there, three of the seven rules are free.

What is being minimised, and over what #

The pitch-class metric of the earlier rungs cannot state any of these rules, and it is worth being precise about why.

A parallel fifth is a relationship between two voices in a register: two parts a seventh apart in pitch class terms may be a fifth apart in the actual music or a twelfth, and the rule cares. A voice crossing needs voices that have an order, which a set of pitch classes does not have. A doubled leading note needs four parts and three notes to put in them. None of that survives the reduction to pitch classes, so from this rung on the arithmetic works in actual pitches with a compass for each voice — the ordinary choral ones, E2 to C4 for the bass up to C4 to A5 for the soprano.

A realisation of a progression is then one pitch for each of four voices in each of its chords, with every note of the chord sounding somewhere. Its cost is the total distance every voice travels, added up over the whole progression. The cheapest realisation of I–ii–iii–IV, with no rule in force at all, costs 18 semitones.

That figure is the whole argument in one picture. The cheapest answer breaks the most famous rule in Western music, and it breaks it for a reason: parallel motion is what minimising total distance wants, because every voice taking the shortest available step is exactly what makes two voices move by the same interval at the same time.

Which computation produced the price #

Each rule here is either a property of one chord — is it crossed, is it spaced, is the leading note doubled — or a property of one move between two adjacent chords — parallels, resolution, augmented steps. Nothing depends on a chord three bars back.

That makes the search a shortest path through a layered graph, and the Viterbi recursion returns the exact minimum without any heuristic, ordering or preference anywhere in it. The point of saying so is not the algorithm, which belongs to another subject entirely; the point is that a price computed from an approximate solver would be pricing the solver. Subtracting one exact minimum from another gives a number about the music.

The melodies are enumerated rather than chosen. A progression admits a finite number of soprano lines inside a one-octave compass — 144 for the four chords here — and every rule is priced against every one of them. That matters because a rule’s price is not a constant: the same prohibition costs four semitones under one tune and nothing under the next, and reporting the price for one hand-picked melody would be reporting the melody.

Three rules that are free #

No voice crossing costs 0.01 semitones a melody, and is free for 143 of the 144. No gap over an octave above the tenor is free for all of them. The leading note resolves costs nothing in this progression at all.

A rule that costs nothing is not a weak rule, and it is not quite the same as a rule the minimisation already obeys. Two different things produce a price of zero, and the difference matters. The crossing and spacing rules are obeyed anyway: the cheapest way to connect two chords almost never crosses two voices, because crossing means one voice travelling past another and the travelling is what the objective is trying to avoid. The parallel rules on a fifth-related progression are not obeyed anyway — they are simply avoidable for nothing.

So a large part of the counterpoint syllabus is not a constraint on the answer at all, and another part is a constraint that happens to cost nothing where the roots move by a fourth or a fifth. Both were arrived at by ear two centuries before anybody could compute them, and both are taught as restrictions because a student who is not minimising anything will break them.

The dear one is not the famous one #

The two parallel rules do cost, and they cost differently. Parallel octaves run at 1.27 semitones a melody, are free for only 29 per cent of tunes, and reach four semitones at the worst. Parallel fifths run at 0.54, are free for 83 per cent, and reach five.

That ordering is the reverse of the emphasis. Consecutive fifths are the prohibition with the reputation, the one students are hunted for and the one this ladder has already given a mechanism; consecutive octaves get half a sentence. On the arithmetic the octave rule is the one that bites, twice as often and at more than twice the average cost.

The reason is a doubling. Four voices and three notes mean one note is sung twice, and two voices on the same pitch class are the cheapest possible pair — they move together, at no extra cost, for as long as the doubling holds. The minimisation likes doublings and it likes keeping them, and a doubling kept across a chord change is a parallel octave or unison. The fifth rule is dearer to break and easier to avoid.

Four semitones over four chords, spread across four voices, is a small price for the most heavily policed body of rules in the subject. It is worth holding on to that number when the rules are described as a straitjacket.

The compass is a choice, and only one of the two rules notices #

Every price above is an average over 144 melodies, and 144 is what a one-octave soprano compass admits. That octave was picked to keep the enumeration finite rather than because hymn tunes stay inside one — they stay inside about a ninth — so it is a parameter, and the two parallel rules respond to it in opposite ways.

The octave rule does not notice at all. Its mean sits between 1.23 and 1.36 across a fivefold increase in the number of tunes, its worst case is four semitones in every compass, and the share of melodies it is free for stays near three in ten.

The fifth rule gets steadily cheaper, halving from 0.54 to 0.29 and going free for nine tunes in ten by the widest compass. So the essay’s headline strengthens rather than weakening: the octave rule is twice as dear as the fifth rule in a one-octave compass and four and a half times as dear in a twelfth.

The reason is the doubling again. Extra room lets a voice go somewhere else, which is exactly what breaking a would-be parallel fifth requires — so more room is cheaper escape. A parallel octave comes from a doubling the minimiser wants to keep, and no amount of extra room makes keeping a free pair of voices less attractive. One rule is escaped by having more space and the other is not, which is why the price of the famous rule is the one that turns out to depend on a parameter.

Where the prices actually live #

The prices above are for one progression, and the progression turns out to be most of the story.

That is a sharp result and it deserves stating plainly. Over every pair of diatonic triads whose roots move by more than a step, the ban on parallel fifths and the ban on parallel octaves are free. Not cheap: free, in all 252 cases, exactly.

Chords a third apart share two notes; chords a fourth or fifth apart share one. Shared notes are held, held voices do not move, and a parallel requires both voices to move. Two triads a third apart share two of their three notes, which is the same fact the neo-Riemannian transformations are built on. The prohibition cannot bite on a change that has a common tone to sit on, and the changes with common tones are the cheap ones — so the whole of the parallel problem is confined to the small corner of the map where two chords share nothing and every voice has to walk.

That corner is where root motion by step lives: I to ii, IV to V, V to vi. It is exactly the list of progressions a counterpoint teacher marks most carefully, and the reason is now a number rather than a tradition. It also says something about the map of a key: the moves that are cheap in motion are the moves on which the rules have nothing to forbid, so the two accounts of what makes a progression easy agree without having been made to.

One chord change with its roots a tone apart, priced over all twelve melodies it admits, costs on two rules only: the octave rule at a third of a semitone and the ban on doubling the leading note at half of one. That is the deceptive cadence, and it is a two-chord progression with exactly one move for either of them to happen in. Over every pair of diatonic triads whose roots move by more than a step the two parallel bans are free — not cheap, free, in all 252 cases — because chords a third apart share two notes and chords a fourth or fifth apart share one, and a parallel needs both voices to move. The whole of the parallel problem is confined to the corner of the map where two chords share nothing.

Two rules that are paid in a different currency #

Two of the seven never cost a single semitone and are not therefore free.

The leading note resolves — a voice singing the seventh degree in a dominant-function chord moves up one semitone to the tonic — has a price of zero on I–IV–V–I and is impossible for a quarter of the melodies. No augmented melodic interval — no tritone leap, and in the minor no step from the flat sixth to the raised seventh — has a price of zero and is impossible for eight per cent of them, rising to a quarter in a minor key.

Impossible means what it says: there is no realisation at all. If the tune itself sings the leading note and then goes somewhere other than up a semitone, no amount of rewriting the inner voices can rescue the rule, because the rule is being broken in the soprano.

So these two are not constraints on the harmonisation. They are constraints on the melody, and they are enforced before the harmony exists. That is why they cost nothing when they can be obeyed and everything when they cannot, and it is why a composition teacher applies them at a different moment from the parallel rules — one is a comment on the tune, the other on the setting.

The strict form of the resolution rule — every voice, not merely the outer ones — prices identically, which is worth reporting because it was expected to cost something. An inner voice holding the leading note can always be given the tonic by choosing the next chord’s voicing differently, so the stricter version constrains nothing the weaker one left free. The textbooks’ concession that an inner part may frustrate its leading note buys nothing at all.

Whose rules, and when #

The prohibitions priced here are the eighteenth-century codification of sixteenth-century practice, which is a specific claim about a specific corner of music.

Johann Joseph Fux’s Gradus ad Parnassum of 1725 abstracted them from the style of Palestrina, a century and a half after the fact, and taught them to Haydn, Mozart and Beethoven; the chorale-harmonisation exercise that most students still meet is a nineteenth-century descendant of that book crossed with Bach’s four-part settings. Fux’s own justification is not the one in this essay. He argues from variety and from what good composers were observed to do, which is an appeal to a repertoire rather than to a mechanism — an honest one, and the only one available at the time.

Outside that jurisdiction the rules are not weakened; they are absent. Ninth-century organum moves in parallel fifths as its entire technique. Guitar-based popular music uses parallel fifths as a texture and doubles at the octave as a matter of course. Debussy and the composers after him wrote parallel triads deliberately, having set aside the goal the rules served rather than the rules themselves. The intervals in question are perfectly pleasant, which is why the prohibition needs a mechanism rather than an appeal to taste. None of that is a counterexample, because a price is only a price for somebody who is buying — and what the eighteenth century was buying is four separately audible lines.

Why the octave is the interval the arithmetic keeps stumbling over is a fact about partials: two notes an octave apart share every partial the upper one has, so a parallel octave is two voices whose spectra are nested rather than two voices at all. A parallel fifth shares four of twelve. That ordering — octave worst, fifth next — is the ordering of the prohibitions, and it was arrived at by ear four centuries before anybody counted partials.

What the prices do not settle #

A price is not a reason. The parallel-octave rule is dear and the crossing rule is free, and the tradition treats both as absolute. If cost explained absoluteness the cheap rules would be the strict ones, and the ordering here is the other way up. What the prices explain is where a rule does work — which progressions it constrains and which it merely describes — and that is a smaller claim than a justification.

The compass is the one free parameter, and it moves the numbers. Give every voice a fifth more room and the parallel-fifth price on I–ii–iii–IV falls from 0.54 semitones to 0.10, because the extra register is an escape route. Take a fifth away and it rises to 1.58, with only 24 of the 144 melodies harmonisable at all. The free minimum, though, is identical at the wider compass: the cheapest answer never wanted the extra room. Every price in this essay is a price for a choir.

There is no metrical position and no duration. A parallel fifth on a downbeat between the outer voices and one on a passing quaver between the alto and the tenor are the same event here and are not the same event in any treatise. That is the same weakness the canon scan had, arriving from the other direction.

The rules are stated in pitch and the treatises state them in spelling. An augmented second and a minor third are three semitones; the difference is in the notation, which this site does not carry. The two cases that occur in the repertoires concerned — the tritone leap and the harmonic minor’s step — are caught, and the diminished intervals Fux also forbids are not.

Nothing here says a rule-obeying realisation is good. It says what obedience costs. The cheapest legal answer is frequently a dull one, and every composer in the tradition spent motion on things this objective does not know about: a melodic shape in the tenor, a bass that states the harmony, a suspension held over a bar line, and whatever makes an arrival sound like an arrival, which is a five-component object with no total and none of its components in view here.

And the measurement this ladder began with is the same one underneath: two chord changes with their voices assigned to minimise total motion, which is what every price on this page is a departure from. The rules do not replace that metric; they constrain the search it runs over, and the price is the difference between the constrained minimum and the free one.

What the pricing establishes #

Three things, none of which is available from a rule book.

The rules are not one kind of object. Two describe the minimum, two constrain the setting at a measurable cost, one constrains the doubling, and two constrain the tune before the setting begins. Presenting them as one list is the pedagogy’s own compression.

Their cost is a property of the progression, not of the rule. The same prohibition is free across two-thirds of the diatonic map and dear in one corner of it, and the corner is nameable: root motion by a step.

The whole syllabus is cheap. Four semitones over four chords, on the progression chosen because it is the awkward one. A tradition that took these rules as absolute was not accepting a large constraint on what could be written; it was accepting a small one, which is a very different historical fact and one that a list of prohibitions does not communicate.

Where the ladder goes #

The next rung takes one line out of that list — move the least, and keep the common tones — and asks whether it is one instruction or two. Enumerating every pair of chords answers it, and the answer depends on whether the phrase is being used to connect two chords or to say which chord is nearer.

After that the ladder turns to the dominant seventh, where the rules above are at their strictest and the metric that started it all has nothing useful to say; and then to the listener, for whom a voice is not an assignment at all but a stream, and who may not be following the parts the arithmetic worked so hard to keep apart.