Lesson 6: The chords of a key

Lesson 3 built chords on any root; lesson 5 built keys. This lesson stacks chords inside a key, one on each degree, which gives every key the same pattern of chord qualities, named with Roman numerals. It then asks what each chord does in the key, its function, and compares all of it with Guitar Alchemist (GA): its chord catalog and its compatible keys agree with the theory, its function names stop short for minor keys, and its MCP tool spells some keys wrong.

All GA links point to commit a826864. The outputs come from:

dotnet run --project code/music-theory-ga/GaTheory -c Release -- l6

Triads on every degree

The idea

Take the notes of a key and, on each degree, stack two more notes of the key, skipping one note each time: C E G on C, D F A on D, and so on. Each triad uses only the key’s notes, so its thirds come out major or minor depending on the degree. In every major key, the triads on degrees 1, 4 and 5 are major, those on 2, 3 and 6 are minor, and the one on 7 is diminished (Open Music Theory, “Triads”).

The notation

A Roman numeral names the degree of the chord’s root. Its case gives the quality: upper case for a major third above the root (major, augmented), lower case for a minor third (minor, diminished); a small circle marks a diminished triad and a plus an augmented one (Open Music Theory, “Roman Numerals”). So C major reads I ii iii IV V vi vii°, and those numerals are the same in every major key.

Each degree also has a name: tonic, supertonic, mediant, subdominant, dominant, submediant, and for degree 7 the leading tone, a half step below the tonic (Open Music Theory, “Major Scales, Scale Degrees, and Key Signatures”). The course program stacks the triads from the key’s spelled notes, names their quality from their intervals with the names GA’s chord catalog uses, and adds the degree name (Lesson6.Stack and DegreeName):

== Triads of C major: numeral, root, quality, function
numeral   course                           GA                               check
I         C major-triad Tonic              C major-triad Tonic              ok
ii        D minor-triad Supertonic         D minor-triad Supertonic         ok
iii       E minor-triad Mediant            E minor-triad Mediant            ok
IV        F major-triad Subdominant        F major-triad Subdominant        ok
V         G major-triad Dominant           G major-triad Dominant           ok
vi        A minor-triad Submediant         A minor-triad Submediant         ok
vii°      B diminished-triad LeadingTone   B diminished-triad LeadingTone   ok
symbols: C Dm Em F G Am Bdim

In GA

The GA column stacks the same thirds on Key.Major.C.Notes (lesson 5), names each chord with CanonicalChordPatternCatalog.TryFindExact, which accepts only a pattern with no missing and no extra interval, and takes the degree name from HarmonicFunctionExtensions.FromDegree. In a major key, everything matches.

Seventh chords

Stack one more third and every degree gets a seventh chord. In a major key: major seventh chords on I and IV, minor seventh chords on ii, iii and vi, a dominant seventh on V, the only one in the key, and a half-diminished seventh on vii. The numeral adds a 7, and ø marks the half-diminished chord (Open Music Theory, “Seventh Chords” and “Roman Numerals”):

== Seventh chords of C major
numeral   course                           GA                               check
I7        C major-7                        C major-7                        ok
ii7       D minor-7                        D minor-7                        ok
iii7      E minor-7                        E minor-7                        ok
IV7       F major-7                        F major-7                        ok
V7        G dominant-7                     G dominant-7                     ok
vi7       A minor-7                        A minor-7                        ok
viiø7     B half-diminished-7              B half-diminished-7              ok
symbols: Cmaj7 Dm7 Em7 Fmaj7 G7 Am7 Bm7b5

The course prints the half-diminished chord as Bm7b5, the chord symbol of lesson 3.

Minor keys and the leading tone

The idea

Stack triads on the natural minor scale and the qualities move: i, ii°, III, iv, v, VI, VII. Two degrees differ from major. The fifth-degree chord is minor, and the seventh degree is a whole step below the tonic, the subtonic, not a leading tone (Open Music Theory, “Minor Scales, Scale Degrees, and Key Signatures”). Music in minor usually raises that seventh degree when it moves to the dominant: G becomes G♯ in A minor. The v chord becomes a major V, and VII becomes a diminished vii°, the same dominant chords as in major (“Triads” and “Roman Numerals”). The notes of A minor with G♯ form the harmonic minor scale.

In GA

In natural minor, the course calls degree 7 the subtonic and GA the leading tone:

== Triads of A minor, natural minor
numeral   course                           GA                               check
i         A minor-triad Tonic              A minor-triad Tonic              ok
ii°       B diminished-triad Supertonic    B diminished-triad Supertonic    ok
III       C major-triad Mediant            C major-triad Mediant            ok
iv        D minor-triad Subdominant        D minor-triad Subdominant        ok
v         E minor-triad Dominant           E minor-triad Dominant           ok
VI        F major-triad Submediant         F major-triad Submediant         ok
VII       G major-triad Subtonic           G major-triad LeadingTone        DIFF

FromDegree(7) returns LeadingTone whatever the key, for two reasons rather than one: its only parameter is a degree number, with no key and no mode to tell a half step below the tonic from a whole step, and its return type, the HarmonicFunction enumeration, has no Subtonic member at all. GA does define one, in a separate enumeration, ScaleDegreeFunction, documented as “one whole step below tonic in natural minor scale”, which FromDegree does not use. With the raised leading tone, both sides agree; the GA column stacks GA’s Scale.HarmonicMinor:

== Triads of A minor with the raised leading tone (harmonic minor)
numeral   course                           GA                               check
i         A minor-triad Tonic              A minor-triad Tonic              ok
ii°       B diminished-triad Supertonic    B diminished-triad Supertonic    ok
III+      C augmented-triad Mediant        C augmented-triad Mediant        ok
iv        D minor-triad Subdominant        D minor-triad Subdominant        ok
V         E major-triad Dominant           E major-triad Dominant           ok
VI        F major-triad Submediant         F major-triad Submediant         ok
vii°      G# diminished-triad LeadingTone  G# diminished-triad LeadingTone  ok
symbols: Am Bdim Caug Dm E F G#dim

The mediant of harmonic minor is an augmented triad, III+: C E G♯.

The same pattern in every key

The numerals don’t depend on the tonic, but the spellings do: one letter per degree, as in lesson 5. The program stacks the triads of E♭ major, F♯ major, G♭ major and C♯ minor on its own notes, then on GA’s Key.Notes:

== The triads of other keys, stacked on the course's notes and on GA's Key.Notes
key    course                                       GA                                           check
Eb     Eb Fm Gm Ab Bb Cm Ddim                       Eb Fm Gm Ab Bb Cm Ddim                       ok
F#     F# G#m A#m B C# D#m E#dim                    F# G#m A#m B C# D#m E#dim                    ok
Gb     Gb Abm Bbm Cb Db Ebm Fdim                    Gb Abm Bbm Cb Db Ebm Fdim                    ok
C#m    C#m D#dim E F#m G#m A B                      C#m D#dim E F#m G#m A B                      ok

F♯ major’s seventh-degree chord is E♯dim, and G♭ major’s fourth-degree chord is C♭. GA’s core gets both right.

GA’s MCP tool ga_diatonic_chords does not. It is a closure of GA’s F# layer: it computes the roots by semitones from a fixed pattern, then names them from one of two 12-name tables, sharps or flats, chosen from the root’s accidental (DomainClosures.fs, diatonicChords). A table has no E♯ and no C♭. On 2026-09-15, the MCP server running on the author’s machine (its version is not reported) answered:

ga_diatonic_chords F# major  ->  F#, G#m, A#m, B, C#, D#m, Fdim
ga_diatonic_chords Gb major  ->  Gb, Abm, Bbm, B, Db, Ebm, Fdim

Fdim has the right pitch classes for F♯ major, but the key has no F; in G♭ major, B should be C♭. Another MCP tool, get_diatonic_chords, answered An error occurred invoking 'get_diatonic_chords'. for “A minor”.

Harmonic functions

The idea

Chords of a key do not all do the same job. Open Music Theory sorts them into three functions (Open Music Theory, “Introduction to Harmony, Cadences, and Phrase Endings”):

• tonic (T), stable, home: I (in minor: i);
• predominant (PD), moving away from the tonic toward the dominant; IV and ii are strong predominants, iii and vi weak ones (in minor: iv and ii° strong, VII, III and VI weak);
• dominant (D), urgent, pulling toward the tonic: V and vii°, the same in minor.

A phrase usually runs T, then PD, then D, and back to T. I, IV and V, the primary triads, cover the three functions and all seven notes of the key between them (Primary triad, Wikipedia). In C major: C E G, F A C and G B D. The next figure draws them on one bracelet of note names: three triangles, the tonic triad C E G in one colour, F A C in a second and G B D in a third. The tonic triad shares C with IV and G with V, and those two notes are ringed.

Not every textbook sorts vi and iii the same way. The Streeling module MUS-003 · How Harmony Works puts them in the tonic family, as substitutes for I that share two of its notes, a common reading in harmony textbooks; Open Music Theory keeps the tonic function for I alone.

In GA

GA has two function types. HarmonicFunction, used above, names the degree (Tonic to LeadingTone). Its service layer maps those names onto three categories, Tonic, Subdominant and Dominant, and puts the mediant and the submediant with the tonic, MUS-003’s reading (HarmonicFunctionAnalyzer.ToPrimaryCategory).

Reading the same file shows a slip that the course program cannot run, because GA.Domain.Services is too heavy to reference (to verify by a test in GA). HarmonicFunctionAnalyzer.Parse turns a name into a function by testing Contains in order, and tests “tonic” before “supertonic” and “mediant” before “submediant” (L29-L72). “Supertonic” contains “tonic”, so it should parse as Tonic, and “Submediant” as Mediant.

Which chords belong to a key?

Two questions come up all the time: does this chord fit this key, and which keys contain this chord? The course answers with the 12-bit ids of lesson 2: a chord fits when none of its bits falls outside the key’s. GA answers with PitchClassSet.IsSubsetOf and GetCompatibleKeys:

== Which triads belong to a key? (major, minor and diminished triads on 12 roots)
Key of G     course: C D Em F#dim G Am Bm
             GA:     C D Em F#dim G Am Bm
Key of Bb    course: Cm Dm Eb F Gm Adim Bb
             GA:     Cm Dm Eb F Gm Adim Bb

== Which keys contain a chord? (GA: PitchClassSet.GetCompatibleKeys)
chord  course                                             GA                                                 check
Dm     C Am F Dm Bb Gm                                    C Am F Dm Bb Gm                                    ok
G7     C Am                                               C Am                                               ok
Bdim   C Am                                               C Am                                               ok
E      A F#m E C#m B G#m Cb Abm                           A F#m E C#m B G#m Cb Abm                           ok

The answers ignore spelling. E major’s triad, E G♯ B, is in the key of C♭ major as F♭ A♭ C♭, and both sides list C♭ and A♭ minor. G7 and Bdim fit only C major and A minor: a dominant seventh chord or a diminished triad points to a single key signature, which is one reason why they matter in lesson 7.

The chords of a key plate of the Atlas des Douze builds these triads on each degree in 3D, and lets you hear them.

Exercises

1. Name the seven triads of G major.
2. Build the seventh chord on degree 5 of D minor, with the raised leading tone. What is its symbol and its numeral?
3. What is the triad on degree 3 of B♭ major, and what is the degree called?

Solutions

1. G Am Bm C D Em F♯dim: I ii iii IV V vi vii°, as in every major key.
2. D harmonic minor is D E F G A B♭ C♯. On A: A C♯ E G, a dominant seventh chord, A7, numeral V7. Its pitch classes are 1 4 7 9.
3. D F A, D minor, iii, on the mediant.

== Exercise solutions
question             course                               GA                                   check
1. G major triads    G Am Bm C D Em F#dim                 G Am Bm C D Em F#dim                 ok
2. degree 5 of D harmonic minor, stacked in four: A7 (V7)
2. A7 pitch classes  1 4 7 9                              1 4 7 9                              ok
3. iii of Bb major   D minor-triad Mediant                D minor-triad Mediant                ok

Key takeaways

• Stacking thirds on a major key gives I ii iii IV V vi vii° in every key; seventh chords add Imaj7, V7 and viiø7.
• Natural minor gives i ii° III iv v VI VII; raising the seventh degree, the leading tone, turns v into V and VII into vii°, the dominant chords of major.
• Open Music Theory’s functions are tonic (I), predominant (IV and ii strong, iii and vi weak) and dominant (V, vii°); other textbooks, MUS-003 and GA’s analyzer group vi and iii with the tonic.
• GA’s core stacks, names and places chords like the theory, except that degree 7 is always the “leading tone”. Its MCP tool spells F♯ and G♭ major with notes from a 12-name table, and its function parser tests names in an order where “Supertonic” matches “tonic”.

Sources

• Mark Gotham et al., Open Music Theory, version 2, 2023, CC BY-SA 4.0: chapters Triads, Seventh Chords, Roman Numerals, Major Scales, Scale Degrees, and Key Signatures, Minor Scales, Scale Degrees, and Key Signatures, Introduction to Harmony, Cadences, and Phrase Endings.
• GuitarAlchemist/ga at a826864: Common/GA.Domain.Core/Theory/Tonal (Key.cs, HarmonicFunction.cs, Primitives/ScaleDegreeFunction.cs), Common/GA.Domain.Core/Theory/Harmony/CanonicalChordPatternCatalog.cs, Common/GA.Domain.Core/Theory/Atonal/PitchClassSet.cs, Common/GA.Domain.Services/Tonal/HarmonicFunctionAnalyzer.cs, Common/GA.Business.DSL/Closures/BuiltinClosures/DomainClosures.fs.