Two criteria that are taught as one

Every account of how to connect two chords contains the same sentence, in some arrangement of the same words: move each voice as little as possible, and keep the notes the two chords have in common in the same voice.

It is delivered as one instruction. It is two functions of a pair of chords, and they can be evaluated separately. One returns the smallest total distance the voices can travel; the other returns the largest number of notes that can be held still. Nothing guarantees that the assignment achieving the first also achieves the second, and nothing guarantees that a chord scoring well on one scores well on the other.

Both are cheap to compute over the whole space, and the space is small enough to leave nothing out.

Connecting two chords: one instruction, not two #

The first question is the one a part-writing exercise actually asks. Two chords are given; the voices have to be assigned; does move the least ever recommend an assignment that drops a common tone it could have held?

Over all 552 ordered pairs of the twenty-four major and minor triads: never. Over all 3,540 ordered pairs of the sixty dominant, major, minor, half-diminished and diminished sevenths: never. Over all 1,440 pairs of a seventh chord with a triad, where the sizes differ and one note of the triad has to be doubled: never.

That is 5,532 pairs of chords with no exception. And it holds in the other direction too — in none of those pairs does holding every available common tone cost so much as one extra semitone of motion elsewhere.

So as an instruction for connecting two given chords the phrase is redundant, and provably so rather than usually. Keep the common tones is not a second criterion; it is a description of what minimising motion already does.

The reason is short, and it is worth following because the same argument shows what it would take to break it. A common tone held costs nothing, and every other destination for that voice costs something. Suppose a minimal assignment moves a voice off a note it could have kept: some other voice has to arrive on that note instead, and swapping the two voices back leaves one of them at zero and moves the other no further than the first one was already going. The swap therefore cannot lengthen the total, so an assignment holding the common tone is minimal too.

What that argument does not do is rule out a tie in which only one of the minimal assignments holds the note, and it says nothing about chords where two voices are equidistant from one destination and both have somewhere else to be. Those are the cases the enumeration is for, and among triads and sevenths they never arise.

The uneven case is the one where an exception looked most likely, because the doubling is a free choice on top of the assignment. A dominant seventh going to a triad has four voices and three destinations, so one destination takes two voices, and a wrong doubling could in principle be cheaper than the right one. Over all 1,440 seventh-to-triad pairs it never is. Taking G7 to C major holds the G, resolves the leading note up a semitone and the seventh down one, and doubles the tonic — which is both the cheapest answer and the one every textbook prints.

Which computation produced the numbers #

For a pair of chords of the same size, every way of pairing the voices is tried — six orderings for a triad, twenty-four for a seventh — and two things are recorded for each: the total distance, taking the shorter way round the circle of twelve for every voice, and how many voices did not move at all. The minimum of the first and the maximum of the second are then compared.

For chords of different sizes the map has to be onto rather than one-to-one: four voices arriving on a triad means one note of the triad is doubled, and which note is part of what gets minimised. Requiring every note of the target to be sounded is what makes the model four-part writing rather than an abstraction, and it is the same construction the next rung uses on the dominant seventh.

Nothing is sampled. The sets enumerated are complete: all twenty-four consonant triads, all sixty sevenths of the five standard types, and all their ordered pairs.

Saying which chord is nearer: two instructions after all #

The second question is a different one, and it is the question a map of chord relationships answers. Given a chord, which other chords are near it? Here the phrase does come apart.

From C major, three triads share no note at all and are still closer in motion than a triad that shares one. C sharp major, A flat minor and B major each sit three semitones away — every voice moving a single semitone, nothing held — while G minor, which shares the note G, costs four.

Drawn as voice motion the two cases are unmistakable and unmistakably unalike. From C major to C sharp major every voice moves one semitone and nothing is held: three semitones in total, no common tone. From C major to G minor one voice holds and the other two travel further: four semitones, one common tone. A criterion counting common tones puts the second nearer; a criterion counting motion puts the first nearer; and nobody who has heard both would call them comparable.

Counted across the whole set, the two criteria order a pair of destinations oppositely in 72 of the 6,072 comparisons available among triads — about one in eighty-four. That is rare, and it is not negligible, and every instance of it is a chord related to the reference by sliding every voice a semitone.

The semitone slide is the whole of the disagreement, and it is exactly the relation the common-tone account is blind to by construction. Two triads a semitone apart in every voice have nothing in common by definition — a pitch class either coincides or it does not — while in motion they are as near as two chords that share a note and closer than most that do.

That is not a curiosity of the arithmetic. It is the chromatic third relation and the parallel slide that nineteenth-century harmony runs on, and it is why a map built on shared notes draws those moves as remote while the music treats them as smooth.

The sevenths, and the set the pedagogy uses #

Extending the count to four-note chords makes the disagreement about twice as common: 2,352 opposite orderings in 102,660 comparisons, or 2.3 per cent against the triads’ 1.2.

The interesting part is that the increase is not spread evenly, and where it is concentrated is the opposite of where the teaching is.

Among the twelve dominant sevenths alone, the two criteria never disagree at all — 660 comparisons, zero inversions. Among the major and minor sevenths taken together the rate is 6.3 per cent, five times the triads’ rate and the highest of any set examined here.

Running the same count on each type separately says that the first of those is not a fact about dominant sevenths:

Every single type gives zero, and the disagreement appears only when two types are mixed. The reason is structural rather than musical: inside one transposition class both quantities are functions of the transposition interval alone — how far the chord has been moved decides both the motion and the common-tone count — so the two are perfectly rank-correlated by construction and cannot cross. Mixing types breaks that, because two chords the same distance apart in transposition are now different objects.

So the sentence above should not be read as a property of the dominant seventh. It is a property of any set closed under transposition of one shape, the dominant seventh included, and what the 6.3 per cent measures is the cost of putting two shapes in one space rather than anything about the shapes chosen.

The triad disagreements are worth characterising the same way, because the claim made about them can be checked exactly rather than described. All seventy-two of them have the same shape: motion of three semitones and no common tone. Not most — all, with nothing else appearing in the census at all. So the semitone slide is not merely the commonest source of the disagreement, it is the only one, and the two criteria order every other pair of destinations the same way.

That makes the disagreement a good deal easier to state than a rate suggests. There is one relation on which a common-tone map and a voice-motion map disagree, it is the relation nineteenth-century harmony is built on, and everywhere else the two maps are the same map.

Every dominant seventh is a transposition of every other, so both quantities depend only on how far the roots are apart, and for this chord the two functions happen to run together: the transpositions that share two notes are also the ones that cost two semitones, and the rest tie at four. A worked example built from dominant sevenths therefore cannot show the criteria disagreeing, however many of them a book works through.

That is the sense in which the pedagogy is safe exactly where its examples are. Nothing is being hidden and nothing is wrong: the chords a harmony course spends its time on are the chords for which the two halves of the sentence really are one instruction, and the sets where they part company — chromatic slides between triads, chains of major and minor sevenths — are the material of repertoires the course reaches last or not at all.

The seventh-chord version of the disagreement is the same shape one note larger. From G7 to F sharp 7 every voice slides a semitone — four semitones in total and not a note in common — and from G7 to D major seventh one voice holds while the others travel, at five semitones and one note shared. The count of shared notes and the count of semitones put those two changes in opposite orders, exactly as they did for the triads.

What the two criteria are actually measuring #

Both are called distance, and they measure different things about the relationship between two chords.

Motion measures work. It answers a question about performance and about the score: how much do the singers have to move. It is continuous, in the sense that a change of one semitone in one voice changes it by one, and it is the quantity the map of a key is drawn from.

Common tones measure overlap. It answers a question about content: how much of this chord is already sounding. It is a count, it moves in whole numbers, and it saturates — two triads either share two notes or they do not, and there is nothing between.

A count that saturates cannot resolve the cases motion separates. Six of the twenty-three triads other than C major share exactly one note with it, and their motion costs run from two semitones to four; the common-tone criterion calls all six equally near.

The Tonnetz is the clearest case of a map built on one of the two criteria and blind to the other. Two triads sharing two notes are triangles sharing an edge on it, which is its strength: adjacency there is a fact about content, and it is why a lattice built for the arithmetic of ratios turned out to be a usable map of voice leading at all. What it cannot place is a chord sharing nothing, however near that chord is in motion — and the map the second rung of this ladder built has the same gap for the same reason, since every edge on it joins two triads sharing two notes. The three chords that break the ranking belong nowhere on either picture, while in motion they sit closer than several chords both draw as neighbours.

Whose instruction, and when #

The paired instruction is thoroughbass pedagogy in origin. A player realising a figured bass at a keyboard has to place the upper voices in real time, and hold what can be held, move the rest as little as possible is a rule that can be followed at speed without deliberation. It arrives in the harmony textbook by descent from that practice, and by way of the chorale-harmonisation exercise, where it is a rule about which finger stays where.

Stated that way it contains something this essay’s arithmetic cannot see. Keep the common tone in the same voice is a claim about register and about parts, not about pitch classes: it says the alto that had the G should be the voice that still has the G. Every measurement here works in pitch classes, so it can say that the note is held and not that the same singer holds it — which is what the rules of part-writing are about and is where the two accounts have to be joined.

Where the criteria disagree is where the music changed. The semitone slide between triads is a nineteenth-century device, standard in Schubert and Wagner and inherited by film scoring; the chains of major and minor sevenths are the harmonic language of jazz standards after about 1940. Both are repertoires in which composers demonstrably treat chords sharing nothing as close neighbours, and a theory that measures nearness by shared notes says they are doing something remote.

There is a third case, and it is the one that makes the disagreement audible rather than merely countable. Two voices sliding by a semitone in the same direction keep whatever interval they had, so a semitone slide between triads moves every interval in the chord in parallel — including, when the chord is spaced that way, the fifth and the octave that stop two voices being two. The cheapest change on one criterion is the one the part-writing rules of the previous rung most want to forbid, which is a disagreement between two of this site’s own measurements and not merely between two textbook phrases.

Where the model stops #

Pitch classes have no register and no voices. The whole of this essay is about assignments of pitch classes, so it cannot see whether the held note stays in the same part, cannot see a bass, and cannot distinguish an alto holding a G from a tenor picking it up.

Doubling is left free. Where the sizes differ the doubling is chosen by the minimisation, which is not what a chorale does — practice doubles the root, avoids doubling the leading note, and pays for it. The price of that preference is a third of a semitone a melody.

Neither criterion has a direction. Both are symmetric: the distance from V to I equals the distance from I to V, and no count of common tones distinguishes an approach from a departure. Function is asymmetric and neither of these is.

The sets are chord types, not music. Enumerating all sixty sevenths counts the diminished seventh twelve times when it has only three distinct forms, and counts chords no repertoire uses alongside chords every repertoire uses. A census of what composers actually wrote would be a corpus study, which this site does not have.

Twelve is doing work that is invisible here. Both criteria are computed on the circle of twelve equal steps, and the semitone slide is only cheap because a semitone is the smallest step there is. In other divisions of the octave the smallest step is smaller and the triads sit differently, so the rate of disagreement is a fact about twelve rather than about chords.

One in eighty-four is a rate, not an importance. The disagreements are rare and they are concentrated on exactly the relation that carries a great deal of chromatic music, so their share of the comparisons and their share of the interest are not the same number.

What the enumeration settles #

Three statements, each of which was available only by counting.

The instruction is one instruction where it is used as one. For connecting two given chords, over every pair of triads and sevenths and every pair of unequal size, minimal motion holds every common tone available. A student following either half is following both.

It is two criteria where it is used as a map. Ranking destinations by shared notes and ranking them by motion give different orders, rarely but systematically, and the systematic part is the semitone slide.

The examples decide what a rule appears to say. A harmony course working in triads and dominant sevenths will never meet the divergence; a course working in major and minor sevenths meets it five times as often as the triad rate. The rule did not change.

Where the ladder goes #

The next rung takes a single chord — the dominant seventh, the one whose resolution the whole tradition treats as the clearest case there is — and ranks every triad by its distance from it. The expectation is that the tonic will be at or near the minimum. It is not, and neither is the chord a deceptive cadence goes to instead: what selects those two is not a distance at all.

After that, the last rung of this ladder asks whether a listener follows the voices these assignments so carefully construct, which turns out to depend on how far apart the parts are sitting and how fast the music is going.