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Lesson 5

Voice leading

Four lessons have been about chords one at a time. Music is chords one after another, and what happens in between turns out to be measurable: how far each note has to travel, and how many do not have to travel at all.

A chord change is an assignment problem

Going from one chord to another, each note of the first has to become some note of the second. Which note becomes which is not given: with three notes there are six ways to pair them up, and they cost different amounts.

The cost is how far the voices move in total. Minimise it and you have the voice-leading distance between two chords, which is a property of the pair rather than of how anyone happened to arrange them.

The figure below solves all six pairings and reports the cheapest. Notes that appear in both chords are the interesting ones, because their cheapest move is not to move: they are held, and a held note is what makes a change sound like a change rather than a cut.

What a chord change costs

Each voice takes the shortest route it can, and the pairing is solved rather than assumed.

From

To

  • CB1 semitone
  • ED2 semitones
  • GG0

Every pair in the key, ranked by total motion

  1. I - iii1
  2. IV - vi1
  3. I - vi2
  4. ii - IV2
  5. ii - vii°2
  6. iii - V2
  7. V - vii°2
  8. I - IV3
  9. I - V3
  10. ii - vi3
  11. iii - vi3
  12. ii - V4
  13. iii - IV4
  14. iii - vii°4
  15. IV - vii°4
  16. I - ii5
  17. I - vii°5
  18. V - vi5
  19. vi - vii°5
  20. ii - iii6
  21. IV - V6
Total motion
3 semitones
Common tones
1
Which
G
Largest single move
2

Read the ranking rather than the chord you picked. The smoothest pair in the whole key is I to iii, one semitone of movement, and hardly anybody writes it. The roughest is IV to V, six semitones with no common tone, and it is one of the most-used progressions in existence. Smoothness is real, measurable, and not what decides which chords follow which.

What the ranking says, which is awkward

That figure ranks all twenty-one pairs of chords in the key, and the result does not say what a tidy lesson would want it to say.

The smoothest change available is I to iii: one semitone of movement, two notes held. Almost nobody writes it. The roughest is IV to V: six semitones, nothing held, the worst score on the board. It is one of the most-used progressions in the history of the music.

The famous ones sit in the middle. ii to V costs four, V to I costs three, I to IV costs three. None of them is anywhere near optimal.

So voice-leading smoothness is real, it is measurable, and it does not predict which chords follow which. Any account of harmony that leans on smoothness alone is quietly picking the examples where the two agree.

Two forces, not one

What the common progressions do have is root motion by a fourth or fifth, and in the case of V to I, the tritone resolution lesson 4 derived. Those are functional considerations: they are about where the music is going, and they come from the structure of the key rather than from the distance the notes travel.

So there are at least two things being optimised, and they pull against each other. Functional logic wants roots a fifth apart. Voice-leading economy wants notes to stay still. IV to V satisfies the first and violates the second as hard as it is possible to do inside a key.

Which is exactly why part-writing exists as a craft. The rules students learn - keep common tones, move the other voices as little as possible, avoid parallel fifths and octaves - are not rules about which chords to use. They are techniques for executing a functionally chosen progression with as little motion as the choice allows. The two forces are reconciled by voicing, which is why the same chord sequence can be clumsy or graceful.

The parallel-fifths prohibition is worth flagging as genuinely conventional. There is a decent perceptual story for it, that two voices moving in parallel fifths fuse into one and the texture loses a part, but it is a stylistic rule of a particular repertoire and not a fact about hearing. Organum is built from parallel fifths on purpose, and so is a great deal of rock guitar.

The three smallest moves

Set the question differently. Instead of asking which chords belong to a key, ask: from a given consonant triad, what is the smallest change that produces a different consonant triad?

There are exactly three answers, and each holds two notes still while moving the third by a semitone or a whole tone. They have names from nineteenth-century theory:

  • P, parallel: move the third by a semitone. C major becomes C minor. Cost 1.
  • L, leading-tone exchange: move the root down a semitone. C major becomes E minor. Cost 1.
  • R, relative: move the fifth up a whole tone. C major becomes A minor. Cost 2.

Three moves, twenty-four chords

Each keeps two notes and shifts the third. Walk anywhere from here.

C majorC E G
  • P parallel: same root, other quality
  • L leading-tone exchange
  • R relative major or minor
Path
start
Moves taken
0
Back to C major in
0 moves
Common tones kept
2

The number beside each move is its total voice-leading motion, and it is always 1 or 2. Every one of the twenty-four major and minor triads is reachable from here by these three moves, so the entire consonant vocabulary is a single connected space, and the distance readout is the shortest route home. Nineteenth-century composers found their way around this graph long before anybody drew it.

One connected space

Walk that figure and you will not get stuck. Every one of the twenty-four major and minor triads is reachable from every other using only those three moves, and the furthest any of them sits from C major is five moves.

That is worth stating plainly, because it reframes what a chord change is. The consonant triads are not a list to be memorised per key; they are a connected graph, and a progression is a walk on it. Modulation, in that picture, is not a special operation at all. It is walking far enough that the notes underfoot no longer belong to the key you started in.

The composers who exploited this hardest, Wagner and Liszt and the late Romantics generally, were doing it by ear about eighty years before anyone drew the graph. The theory did not tell them what to write. It arrived afterwards and explained why the things they had already found sounded connected rather than arbitrary.

The distance between two chords is a real quantity. The smallest available steps are three specific moves, and the space they define is connected. What none of it does is tell you which step to take. That decision comes from function, from the key, and from what the music has already done.

One lesson remains, and it is the one where this course has the least to offer by its own standards. Melody is the least derivable part of the subject, so lesson 6 is careful about the difference between what is perceptual, what is statistical, and what is simply style. If you want the reason two notes sound smooth together in the first place, Foundations lesson 3 still has the only physical answer in either course.

Previous4. Chords in a keyNext6. Melody

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