Lesson 4
Chords in a key
Everyone learns that a major key contains three major chords, three minor ones and one diminished, in a fixed order, and almost everyone learns it as a list. It is not a list. It is what you get when you apply one instruction seven times to the step pattern from lesson 2.
One instruction, seven times
To build a chord on a scale degree, take that note, skip the next one, take the one after, skip again, take the one after that. Three notes, alternating. Do it on every degree and you have the chords of the key.
The instruction never changes. What changes is where it lands, because "skip one" is a generic move: it means two steps up the scale, and the scale's steps are not all the same size. Sometimes two steps is four semitones and sometimes it is three, depending on whether you crossed one of the two semitones on the way.
That is the entire mechanism. Nobody decided the chord on the second degree is minor. It is minor because starting there and skipping alternate notes happens to produce a three-semitone gap and then a four-semitone one.
The chords a scale contains
Built by skipping alternate notes. The qualities are read off, not looked up.
Three major, three minor, one diminished. Nobody chose that distribution; it is what stacking alternate notes of this step pattern produces.
- Qualities
- 3 major, 3 minor, 1 diminished
- Tritones in the scale
- 1
- Which notes
- F-B
- Chords holding one
- 1
The outlined chords are the ones containing both notes of a tritone, and they are exactly the diminished ones. In the major scale there is a single tritone, F to B, which is the 1 at the end of the interval vector lesson 2 computed, and exactly one triad manages to contain both of its notes. Change the scale and both numbers move together.
What the figure is not doing
There is no table of Roman numerals in that component. Each chord is assembled by the skipping rule and then named by measuring the two intervals it turned out to have: four then three is major, three then four is minor, three then three is diminished, four then four is augmented.
Which is why switching the scale works at all. Choose harmonic minor and the row rearranges itself into i, ii°, III+, iv, V, VI, vii° with nothing edited, including an augmented triad on the third degree that no theory book presents as a goal. It is a side effect of raising the seventh, and the raised seventh was wanted for an entirely different reason: it turns the chord on degree 5 major, which natural minor cannot manage.
The single tritone
Lesson 2 computed the major scale's interval vector as 2, 5, 4, 3, 6, 1 and moved on. That last entry is worth coming back for: it says the entire scale contains exactly one tritone. In C major it is F and B.
Now look at which chords contain both of those notes. Exactly one does: the triad on the seventh degree, B-D-F. And that is the only diminished chord in the key.
Those are one fact, not two. A diminished triad is two stacked minor thirds; two minor thirds span six semitones; six semitones is a tritone. So the number of diminished triads you can build is bounded by how many tritones there are to build them from. One tritone in the scale, one diminished chord. Switch the figure to harmonic minor and you get two of each.
Worth stating the limit of that argument, because it is tempting to promote it to a law. Across all 66 seven-note scales, tritone count and diminished-chord count agree in only 13 of them. Plenty of sets contain a tritone whose two notes never land in the same skip-alternate triad, so they have a tritone and no diminished chord. The relationship holds for the diatonic set and its close relatives, which is what this lesson is about, and it is not a general theorem.
Why V wants to go to I, partly
The dominant seventh chord, G-B-D-F, contains that same tritone. That is the structural fact behind the most reliable gesture in tonal music, and it is worth separating the part that is derivable from the part that is not.
Derivable. The tritone is the scale's one maximally unstable interval, both by roughness and by being the only interval that divides the octave exactly in half. It occurs once. It therefore points at exactly one place: the two notes either side of it, B and F, are a semitone below C and a semitone above E, and both of those belong to the tonic chord. Move each by a semitone in the direction it is nearest and you land on C major without having tried.
Not derivable. That this should feel like arriving home, that a piece should establish a tonic at all, that ending anywhere else is unresolved: none of that follows from the arithmetic. Those are conventions, learned by exposure, and traditions exist that do not share them.
So the honest summary is that the scale supplies a unique unstable interval with a unique nearest resolution, and a culture spent four centuries building a grammar on top of that affordance. The affordance is structural. The grammar is not.
What Roman numerals actually are
Two pieces of information in one symbol. The numeral is the scale degree, so it says where. The case says what: upper for major, lower for minor, a ring for diminished, a plus for augmented.
The reason this notation is worth having is that it is transposition-invariant. ii V I means the same progression in every key, which is exactly the property the chain of fifths bought in lesson 3: every key is the same shape in a different place, so a description in terms of degrees rather than note names survives moving it.
It is also where the spelling rule from lesson 3 finally makes sense. Each scale degree gets its own letter, so a key never uses the same letter twice, which is why B flat major is spelled with a B♭ rather than an A♯: it already has an A, and using A twice would make the degrees ambiguous.