Three notes at once, and why these three

Take a note. Add the note four semitones above it. Add the note three semitones above that. The result is a major triad, and it is the single most used object in four hundred years of music.

Swap the two intervals — three semitones then four — and the result is a minor triad, which is used almost as much and sounds entirely different. Use three and three, or four and four, and the results are unstable sonorities that nobody rests on.

Four arrangements of two thirds. Two of them are places to stop and two of them are not — and the reason is not legible in a single quantity, which the section on roots below establishes by trying two.

4:5:6 #

The major triad’s frequencies, in just intonation, stand in the ratio 4:5:6.

That is the simplest three-term ratio available above the octave, and it is not a coincidence that it is also the most consonant chord. Its intervals are 5:4 between the lower pair, 6:5 between the upper pair, and 3:2 between the outer pair — all of them among the simplest ratios there are.

The consequence, in terms of what actually reaches the ear: the three tones’ partials coincide extensively. A note at 200 hertz has partials at 200, 400, 600, 800, 1000, 1200. A note at 250 has partials at 250, 500, 750, 1000, 1250. A note at 300 has 300, 600, 900, 1200. The three stacks share 1200, 600, 1000 — and the whole assembly could be read as partials 4, 5 and 6 of a fundamental at 50 hertz that nobody is playing.

Partials four, five and six of any harmonic tone stand in the ratio 4:5:6 and are a major triad in root position — so a single note already contains one, faintly, and the triad is the first chord a vibrating object supplies rather than the first one anybody chose.

That last point is the strongest thing that can be said for the major triad’s status. It is a subset of the harmonic series. Partials 4, 5 and 6 of any note are a major triad on that note’s double octave, so every string, every pipe and every voice contains a faint major triad inside its own tone.

The minor triad, which has no such story #

The minor triad is 10:12:15. That is a good deal less simple, and it does not appear as a contiguous subset of the harmonic series anywhere.

This is a genuine asymmetry and it has generated a great deal of bad theory. The most persistent bad theory is the undertone series — a mirror of the harmonic series descending from a note — which would make the minor triad as natural as the major one. There is no physical mechanism that produces undertones, no instrument generates them, and the idea survives only because the asymmetry is uncomfortable.

The honest account is that the minor triad is nearly as consonant as the major one by any roughness measure, because its intervals are still simple enough for the partials to behave, and that it lacks the harmonic-series pedigree the major triad has. Both facts are true and they do not resolve into a neat story.

Whether listeners hear minor as “sad” is a separate question with a disappointing answer: the association is strong within the European tradition, weak or absent outside it, and appears to be learned. What is not learned is that both triads are stable and the other two are not.

The two that do not settle #

The diminished triad stacks two minor thirds and the augmented stacks two major thirds. Both are symmetrical, and symmetry is the problem.

A diminished triad’s outer interval is a tritone, which is the roughest interval inside a diatonic scale and which resolves by contrary motion in two voices. An augmented triad divides the octave into three exactly equal parts, which means it sounds the same from any of its three notes — there is no root, no orientation, and no way to tell which note it is built on.

That last property is the same one the whole-tone scale has, and it has the same consequence: an equal division has no landmarks, so it cannot establish a position. Both chords are used as transitions, and both are used deliberately when a composer wants to suspend the sense of where the music is.

Inversion, and what it costs #

Move the bottom note of a triad up an octave and the chord is inverted. Every theory of harmony treats the result as the same chord, and it very nearly is.

The pitch classes are unchanged, the harmonic function is unchanged, and a listener will identify it as the same chord. What changes is the spacing, and spacing has audible consequences that the theory quietly discards.

In root position, the chord’s intervals from the bass are a third and a fifth, matching the low partials of the bass note’s own harmonic series. In second inversion, the bass has a fourth above it — and the fourth above a bass note is, notoriously, treated as a dissonance in classical practice. A second-inversion triad cannot end a phrase, which is a strange rule if inversions really are the same chord.

The resolution is that inversions are the same chord in pitch-class terms and different sonorities in acoustic terms, and functional theory chose to track the first. That is a defensible simplification and it is a simplification, and the six-four chord’s awkward status is where the seam shows.

The lattice, which draws all of this at once #

There is a picture that makes triads and their relationships simultaneously visible, and it was devised by Euler in 1739.

On the Tonnetz — pitch classes arranged so that a step east is a fifth and a step south is a major third — every major and minor triad is a triangle and neighbouring triangles share two of their three notes. That is the same fact as the roughness ordering read off a different axis: the chords that share partials are the chords that share vertices.

Place the pitch classes so that one axis is the fifth and another is the major third. Every major triad becomes a triangle pointing one way; every minor triad becomes a triangle pointing the other. Two triads that share two notes share an edge.

The picture makes several things immediate that a list of chords does not. Chords sharing two notes are adjacent, which is exactly the shortest voice-leading relationship. The three transformations that swap one note — relative, parallel, and the one that turns C major into E minor — are reflections across the three edges of a triangle. And the whole space tiles infinitely without repeating, which is a geometric statement of the fact that the chain of fifths does not close.

The seventh changes the arithmetic #

Three notes is where the theory starts and four is where the practice mostly lives, and adding the fourth changes the character completely.

Add a minor seventh to a major triad and the result is the dominant seventh: 4:5:6:7 if the seventh partial is admitted, or 36:45:54:64 in the ratios a keyboard actually supplies. The first is smooth and the second is not, and the difference of 31 cents between them is why a barbershop quartet’s seventh chord rings and a piano’s does not.

What the added note contributes is a tritone, between the third and the seventh. That interval is the only one in the scale that has a unique resolution, and its presence converts a stable chord into one that points somewhere. The whole of functional harmony rests on that conversion.

Extend further — ninths, elevenths, thirteenths — and the added notes are further up the harmonic series, more dissonant by any roughness measure, and increasingly treated as colour rather than as tension requiring resolution. Jazz harmony is largely the history of that reclassification.

Whose music, and when #

The triad’s dominance is specific to a tradition and a period, and it is younger than most people assume.

Medieval European practice treated the third as a dissonance. Cadences resolve onto bare fifths and octaves, and a triad in the modern sense is not the unit of harmony — the unit is the interval between two voices. The shift happened over the fifteenth century, arriving from English practice, and by 1500 the triad is the basic sonority.

It stayed basic for about four hundred years and then stopped. Extended chords in jazz make the seventh and ninth structural rather than decorative; quartal harmony stacks fourths instead of thirds; and a great deal of twentieth-century music abandons the triad entirely.

Outside Europe the triad is not usually the unit at all. Indian classical music is melodic and modal over a drone; a great deal of African music is built on interlocking melodic parts; gamelan uses stratified textures with fixed melodic layers. Harmony in the triadic sense is one tradition’s solution, and a fairly recent one.

Root, and where it comes from #

A triad has a root, and the root is not simply the lowest note — it survives inversion, which is what makes it a useful idea and a slightly mysterious one.

The best available account is that the root is the fundamental the chord’s notes would be partials of. For a major triad at 4:5:6, that hypothetical fundamental is two octaves below the lowest note, and the auditory system’s pitch-inference machinery — the same machinery that supplies a missing fundamental — appears to supply it.

That explains why the root is stable under inversion: the set of partials is unchanged when one of them moves an octave, so the inferred fundamental is unchanged too.

It also explains why the minor triad’s root is less secure, and this collection has the machinery to say by how much rather than to assert it. Handing the four triads to the site’s own residue matcher — the one that supplies a missing fundamental from a set of partials, scored by how many harmonic slots it has to leave empty and by how far off the best fit is:

The major triad’s fit is exact and free. It is partials four, five and six of a note at 50 hertz, no slot is left empty and no tone is out of place, which is the strongest statement the model can make. The minor triad’s fundamental is 20 hertz — below the bottom of hearing, an octave and a third further down than the major’s — and it costs three empty slots to get there. The two-century argument about whether a minor triad’s root is its lowest note or its fifth is an argument about a hypothesis the machinery only reaches by paying for it.

The augmented triad is where the account breaks properly, and it breaks in the right place. Its best fundamental costs three empty slots and puts one of its tones thirty cents out, which is a worse fit than any of the others by either measure — so a listener supplying a root for it is supplying one the evidence does not support, which is exactly the rootlessness this essay attributes to its symmetry, arriving from the other direction.

And the diminished triad is the surprise: its fit is as cheap as the major’s in slots and only eleven cents off, because 5:6:7 is a genuine stretch of the harmonic series. What makes it unstable is not that it has no root. Roughness says the same thing and says it more sharply. Scoring the four at middle C, the diminished is much the roughest at 0.389 and the other three are within seven per cent of each other — major 0.288, minor 0.298, and the augmented smoothest of all four at 0.279. So the division into two stable chords and two unstable ones is not a division either instrument here makes: roughness isolates the diminished and the root inference isolates the augmented, and neither isolates the pair.

And drawn a second time with the minor triad marked, the lattice makes the asymmetry between the two triads visible as an orientation: a major triad’s triangle points one way and a minor triad’s the other, and no rotation of the plane turns one into the other. That is the same fact the ratios report — 4:5:6 against 10:12:15 — with no arithmetic in it.

Spacing, which the theory discards #

Pitch-class analysis says a chord is a set of three classes. A player has to decide where to put them, and the decision is not free.

Low intervals are rough. Critical bandwidth is proportionally much wider at the bottom of the range, so a major third in the bass falls inside one critical band and grates, while the same third two octaves up does not. Every orchestration text therefore says to space low chords widely and allows close spacing higher up, and the rule is a direct consequence of cochlear mechanics.

The traditional voicing that results — wide at the bottom, close at the top — is also the spacing of the harmonic series itself, whose partials get closer together as they rise. A chord voiced that way is imitating the internal structure of a single note, which is a reasonable guess at why it fuses well, and it is checkable rather than a guess.

Scoring all 480 four-part voicings of a C major triad within the standard ranges: the smoothest is C3, C3, G4, E5 — nineteen semitones from the bass to the tenor and nine from the alto to the soprano — and the roughest is C3, E3, G3, C4, the close bass triad every orchestration text forbids, at five and a half times the roughness. Among the smoothest five per cent, ten voicings have gaps that never widen going up against one whose gaps never narrow; across all 480 the same counts are 96 and 50. So the shrinking-gap shape is about twice as common as its opposite in general and ten times as common at the smooth end, which is the textbook rule recovered from a roughness model that was told nothing about it.

Where the model stops #

Just ratios, in a tempered world. The 4:5:6 argument uses just intonation. On any equal-tempered instrument the major third is 14 cents sharp of 5:4, which is enough that the partial coincidence is imperfect and the chord beats. Every major triad on a modern piano is audibly rough compared to a sung one.

Three notes only. Real harmony uses four, five and more, and sevenths and ninths change the analysis substantially.

Spacing is ignored. The figures draw pitch classes. A triad spread over three octaves and a triad crammed into a fifth are the same chord in these pictures and completely different sounds.

No time. A chord in isolation is not what harmony is about. Every question of function, tension and resolution requires knowing what came before, and none of these figures has a horizontal axis.

Why three notes and not two #

Two notes make an interval and three make a chord, and the difference is not merely one of quantity.

An interval is ambiguous about its context. A perfect fifth C–G belongs to C major, C minor, A-flat major, F major and several others, and on its own it commits to none of them. That ambiguity is why medieval cadences on bare fifths sound open: they genuinely are.

Three notes resolve it. C–E–G is C major and nothing else; C–E-flat–G is C minor. The third is the note that decides quality, which is why it is the note left out when a composer wants to withhold the decision — the “power chord” of rock guitar is a bare fifth, and its usefulness is precisely that it is neither.

Four notes, adding the seventh, resolve a further question: not what the chord is but where it goes. So each added note answers one more question, and the sequence interval, triad, seventh chord is a sequence of increasing commitment.

That is a reasonable reason for the triad’s dominance in a system organised around keys. It is the smallest object that states a quality without also stating a direction, which makes it the right unit for a music that wants both to be composable independently.

Four chords, one tonic #

A last observation about the four triads: they exhaust the ways of stacking two thirds, and the exhaustion is complete rather than approximate.

A third is either major (four semitones) or minor (three). Two of them stacked gives four combinations: major-minor, minor-major, minor-minor, major-major. That is the major, minor, diminished and augmented triads, and there are no others.

So the four chords in the opening figure are not a selection of useful ones. They are all of them, and the fact that exactly two are stable is a fact about which of four arithmetic possibilities produce simple ratios — not a decision anybody made.

The ladder from here #

Later rungs: the dominant seventh and the tritone that wants to move. Extended chords. Inversion and the six-four problem. The undertone series and why it is wrong. Chord function versus chord quality. Neo-Riemannian transformations on the lattice. Quartal harmony. Voice leading as the real subject. And the question of whether harmony is heard at all as a succession of objects, or whether it is heard as several melodies at once — which is how it was written for most of its history.

Euler’s 1739 Tentamen contains the Tonnetz, drawn to display the ratios of just intonation, in a book that also proposed a numerical measure of the pleasantness of an interval. The measure has not survived. The diagram is still the best picture of chord space anyone has.