Everything so far has been one note at a time. A chord is three or more sounding together, and the whole of tonal harmony is built from one method of choosing which.
Why thirds
The method is to stack thirds: take a note, add the note two letters above it, then two letters above that. From C, the stack is C, E, G. Skip one letter each time and stop at three notes and you have a triad.
Thirds rather than any other interval, for a reason from the first lesson. The harmonic series of a low C runs 65.41, 130.81, 196.22, 261.63, 327.03, 392.44 Hz, and partials 4, 5 and 6 are 261.63, 327.03 and 392.44, in the ratio 4:5:6. That is a note, a major third above it, and a minor third above that, which is exactly the C major triad, sitting inside the spectrum of a single sounding C. The chord is not a human invention layered onto the physics; it is three of the loudest components of one note, made explicit.
Stacking seconds would give clusters of adjacent notes, which the previous lesson called dissonant. Stacking fourths gives a usable but unstable sound, used deliberately in twentieth-century music precisely because it is not this. Thirds give the two consonant intervals, 5:4 and 6:5, and their sum, the perfect fifth at 3:2, which is why a triad is stable.
The three notes have names: the root the stack starts on, the third, and the fifth, named by their interval above the root. The root is not necessarily the lowest sounding note, which the section on inversions makes precise.
Four qualities
Two thirds can each be major or minor, giving four combinations, and all four are used.
A major triad is a major third then a minor third, so 0, 4 and 7 semitones above the root. C E G. A minor triad is a minor third then a major third: 0, 3, 7. C Eb G. Both contain a perfect fifth, and both are consonant; the difference is only which third, and it is the difference between the two moods most listeners can name.
The other two stack identical thirds and lose the perfect fifth. A diminished triad is two minor thirds, 0, 3, 6, so its outer interval is the tritone: B D F. An augmented triad is two major thirds, 0, 4, 8, an augmented fifth: C E G#. Both are unstable and both are used for exactly that, the diminished constantly and the augmented sparingly.
Example. What quality is the triad F# A# C#?
F sharp to A sharp is 4 semitones, a major third. A sharp to C sharp is 3 semitones, a minor third. Major then minor is a major triad, so this is F sharp major.
Now you. What quality is the triad G Bb Db?
Answer
G to B flat is 3 semitones and B flat to D flat is 3 semitones, two minor thirds, so it is a diminished triad: G diminished. Its outer interval, G to D flat, is the tritone.
The seven chords of a key
Now restrict the stack to notes of one scale. Build a triad on each degree of C major, using only white keys, and every quality is decided for you by where the semitones fall.
On C: C E G, major. On D: D F A, minor, because D to F is only three semitones thanks to the E to F semitone. On E: E G B, minor. On F: F A C, major. On G: G B D, major. On A: A C E, minor. On B: B D F, diminished, since it contains both of the scale's semitone pairs.
The result is the same in every major key, because it depends only on the step pattern T T S T T S T, not on the letters. Major, minor, minor, major, major, minor, diminished, in that order, always.
These are written as roman numerals, one per degree, with case carrying the quality: uppercase for major, lowercase for minor, a small circle after a lowercase numeral for diminished and a plus sign after an uppercase one for augmented. So any major key has
The numerals are the reason this course spends so long on scale degrees. They describe a progression without naming a key, so the same three symbols describe the same music in every key at once, and a musician who learns that ii goes well to V has learned it for all twelve.
Example. Spell the ii chord in A major and give its roman numeral and chord symbol.
A major is A B C# D E F# G#. Degree 2 is B, and the triad on it is B, D, F#. B to D is 3 semitones and D to F sharp is 4, so it is minor: the numeral is ii and the symbol is Bm.
Now you. Spell the chord on degree 7 of E flat major, and give its numeral.
Answer
E flat major is Eb F G Ab Bb C D. Degree 7 is D, and the triad is D, F, Ab. Both thirds are 3 semitones, so it is diminished: vii° in E flat, spelled D diminished.
Minor keys, and where the dominant comes from
Minor is less tidy, because the previous lesson gave it three forms. Using natural minor alone, A minor gives i, ii°, III, iv, v, VI, VII: the chord on degree 5 is E G B, which is minor.
That is a problem for the same reason the natural minor scale was a problem. A minor v has no leading note in it, so it does not pull back to the tonic. Raising degree 7 from G to G sharp, which is what harmonic minor exists to do, changes every chord that contains that degree: v becomes V, since E G# B is major; VII becomes vii°, since G# B D is diminished; and III becomes III+, since C E G# is augmented.
Practice takes what it wants from both forms. Real minor-key music uses i, ii°, iv, VI and the raised V and vii°, and the augmented III+ is a rarity. The working summary is that minor-key harmony is natural minor with a major dominant borrowed from the harmonic form, and the reason is entirely the leading note.
Inversions
A triad has three notes and any of them can be lowest. Root position puts the root in the bass. First inversion puts the third in the bass, so C E G becomes E G C. Second inversion puts the fifth in the bass: G C E.
The chord is the same chord in all three, since it holds the same three pitch classes and the same root, and it does not sound the same, because the bass line is what a listener tracks most strongly. First inversion is lighter and more mobile than root position and is used to keep a bass line stepping smoothly. Second inversion is genuinely unstable, since the interval from bass to root is a fourth, which the previous lesson noted is treated as a dissonance above a bass, and it is restricted to a few standard uses, the commonest being the cadential six-four in the next lesson.
Two notations exist. Figured bass, from the seventeenth century, writes intervals above the bass note: root position is 5/3 and normally left unmarked, first inversion is 6/3 abbreviated to 6, second inversion is 6/4. Combined with the numeral, that gives V6 for a first-inversion dominant and I6/4 for a second-inversion tonic. Chord symbols, from popular music, write a slash: C/E is a C chord with E in the bass. The two systems say the same thing, one relative to the key and one absolute.
One more third
Add a fourth note, another third above the fifth, and the triad becomes a seventh chord, named for the interval from root to top. There are five worth knowing, and they are distinguished by the triad underneath and the size of the seventh.
A major seventh is a major triad plus a major seventh, 0, 4, 7, 11 semitones, written Cmaj7. A dominant seventh is a major triad plus a minor seventh, 0, 4, 7, 10, written C7. A minor seventh is a minor triad plus a minor seventh, 0, 3, 7, 10, written Cm7. A half-diminished seventh is a diminished triad plus a minor seventh, 0, 3, 6, 10, written Cm7b5 or with a slashed circle. A fully diminished seventh is a diminished triad plus a diminished seventh, 0, 3, 6, 9, written Cdim7, and being three stacked minor thirds it is perfectly symmetrical, which is why it can resolve almost anywhere.
Doing the same restriction to one key gives the diatonic sevenths of C major: Cmaj7, Dm7, Em7, Fmaj7, G7, Am7 and Bm7b5, written Imaj7, ii7, iii7, IVmaj7, V7, vi7 and viiø7. Look at what the arithmetic has just delivered. Exactly one dominant seventh occurs naturally in a major key, and it is on degree 5. That is not a naming coincidence; it is why the chord is called what it is, and it is the single most important fact in the next lesson.
Sevenths invert too, with figures 7, 6/5, 4/3 and 4/2 for root position and the three inversions, so V4/2 is a dominant seventh with its seventh in the bass.
Example. Spell V7 in D major.
D major is D E F# G A B C#. Degree 5 is A, and the seventh chord on it stacks A, C#, E, G. Check the intervals: A to C sharp is 4, C sharp to E is 3, E to G is 3, so it is a major triad with a minor seventh, a dominant seventh. The symbol is A7.
Now you. Spell ii7 in B flat major and give its chord symbol.
Answer
B flat major is Bb C D Eb F G A. Degree 2 is C, and the stack is C, Eb, G, Bb: a minor triad with a minor seventh, so Cm7.
What a chord label leaves out
Three limits, all of which matter when analysing real music.
A chord symbol says which pitch classes, not how they are arranged. C E G, E G C an octave up, and a spread with three Cs in different octaves are all "C major", and they sound substantially different. Voicing, spacing and doubling are choices a label does not record, and the next lesson's voice-leading rules are largely about them.
A chord label also assumes there is a chord to label. Real music contains notes that belong to no chord at all, and the last lesson of this course deals with them; until then, be aware that picking the chord from a bar sometimes means deciding which notes are structural, and that two competent analysts occasionally disagree.
And the stacking-thirds scheme is a description of one repertoire, not of music in general. Jazz continues the stack to ninths, elevenths and thirteenths, and adds sixths and suspensions in which the third is replaced by a second or a fourth. Debussy and much film music use chords whose function this scheme cannot name. Rock and folk often use plain triads in progressions that break the rules the next lesson sets out, and they sound fine. The scheme is a very good account of European music from about 1600 to about 1900, and a partial account of everything since.
Chords now exist, but nothing so far says why one should follow another. A single triad, however well spelled, does nothing at all. What turns a list of chords into a progression is the subject of the next lesson.