Chord origin and affinity

A C major chord sounds stable, almost inevitable. An old question of theory is why: whether that chord is one culture's invention or whether it is somehow already inside sound itself. The harmonic series gives a tempting answer, and also shows how far that answer goes.

Chords inside the series

In theory, chords can be found in the harmonic series (Med 1996, ch. LVI). Over a low C, harmonics 4, 5 and 6 are C, E and G: the major triad, in its closest position. Add harmonic 7, a B flat, and the dominant seventh chord appears; add harmonic 9 as well, a D, and you have the dominant major ninth (Med 1996, ch. LVI):

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Take away harmonic 4, the root, and two more chords are left. Harmonics 5, 6 and 7, E, G and B flat, make the diminished triad; harmonics 5, 6, 7 and 9 make the leading-tone seventh chord (Med 1996, ch. LVI):

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In practice there is a caveat. The notes of the series do not match those of the tempered system exactly, and the seventh harmonic strays the furthest (Med 1996, ch. LVI): it sounds well below the piano's B flat. The seventh chord the series produces is a relative of the one played on the keyboard, not the same chord.

Chords without a root

The series hints at a relationship that harmony confirms. Med compares three pairs of dissonant chords (Med 1996, ch. LVI):

Dominant chord	Without its root
dominant seventh	diminished triad
dominant major ninth	leading-tone half-diminished seventh
dominant minor ninth	diminished seventh

In C major the dominant seventh chord is G, B, D, F. Without the G, what is left is B, D, F, the diminished triad on the seventh degree:

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That is why chords built on the leading tone sound and behave like dominants. They hold the tritone, B and F, which drives the dominant, and all they lack is the note that names the chord. Much analysis treats them as incomplete dominants.

And the minor chord?

Here the series is less generous. The minor triad does not appear among the first harmonics. Med finds it higher up, in harmonics 6, 7 and 9, or in harmonics 10, 12 and 15 (Med 1996, ch. LVI):

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The first form uses the seventh harmonic, precisely the most out of tune, and the minor third between 6 and 7 is far too narrow. The second sounds much better, but it sits much higher in the series, where the harmonics are weaker.

The German theorist Hugo Riemann tried to resolve the asymmetry with a "descending" series, the harmonic series mirrored downwards. In it, harmonics 4, 5 and 6 make a minor chord, and the minor chord becomes the exact counterpart of the major one (Med 1996, ch. LVI). The idea is elegant, but that series has no physical basis: no string produces sounds below its fundamental.

Blatter urges caution with explanations of this kind. There is probably some truth in the idea that the series influenced harmony, but the link has never been proven beyond dispute, and the influence, if any, is limited (Blatter 2007, part 6, appendix II). The series explains the major chord well; the minor chord, which music uses as much as the major, it explains poorly.

Affinity: shared notes

The second half of the story needs no physics. Two chords are related when they have notes in common, and the more notes they share, the closer the affinity (Med 1996, ch. LVI). Among the triads of a key, the distance between the roots tells how many notes they share (Med 1996, ch. LVI):

Roots a … apart	Shared notes	Example in C major
fourth or fifth	one	C major and F major: C
third or sixth	two	C major and A minor: C and E
second or seventh	none	C major and D minor

One shared note, by a fourth or fifth:

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Two shared notes, by a third or sixth:

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None, by a second:

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What affinity explains

Affinity helps make sense of three things harmony does all the time.

Moving smoothly. When two chords share notes, the voices can hold them and move only the rest. That is what you hear from C major to A minor: C and E stay, and only G rises a tone to A. Between chords with no shared note, such as C major and D minor, every voice has to move, and the connection needs more care.

Substituting one chord for another. Chords with two shared notes sound related, and one can take the other's place without changing the sense of the phrase much. A minor and E minor each share two notes with C major, which is why both can stand in for the tonic.

Progressing with strength. The fifth relationship, with a single shared note, is the one that most seems to move: it changes enough to sound like motion and keeps one note that stitches one chord to the next. It is no accident that the strongest cadence, dominant to tonic, is a fifth relationship.

CITonicDiatonicAmviTonicDiatonicFIVSubdominantDiatonicGVDominantDiatonicCITonicDiatonic

Hear the progression

In this progression, from C to A minor two notes stay; from A minor to F major, two; from F to G, none; from G to C, one.

Enharmonic chords

Med closes the chapter with a third kind of kinship, that of chords that sound the same in the tempered system but are written with different notes, the enharmonic chords. Enharmony can be partial, swapping some notes, or total, swapping all of them, and it turns up often in modulations, although not every enharmonic respelling is usable in harmony (Med 1996, ch. LVI).

The dominant seventh chord of C is G, B, D, F. Writing the F as E sharp, the sound does not change, but the chord now holds an augmented sixth, and its tendency changes:

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The same sound, read two ways, points to two different places. That is how enharmony opens doors to distant keys.

What comes next

Affinity between chords is the ground of chord connection: which chords follow one another well, and how harmony moves from one to the next to form progressions.