Lesson 3: Chords, symbols, inversions and voicings

A chord symbol such as Cm7 is a small language: a root, a quality, extensions. A guitar chord shape such as x32010 is another one. This lesson parses both: what the symbol means, which notes it names and how to spell them, what an inversion is, and how a shape on six strings becomes a set of pitch classes and a name again. At each step it compares the textbook with Guitar Alchemist (GA), and here the two disagree more often than in the first two lessons.

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

dotnet run --project code/music-theory-ga/GaTheory -c Release -- l3

Triads and seventh chords

The idea

A triad is three notes that can be stacked in thirds: a root, a third above it and a fifth above the root. Its quality comes from those two intervals: a major third and a perfect fifth make a major triad, a minor third and a perfect fifth a minor triad; a minor third and a diminished fifth make a diminished triad, a major third and an augmented fifth an augmented one (Open Music Theory, “Triads”). The Streeling module MUS-001 · What Is a Chord? builds the major and minor triads the same way.

Stack one more third and you get a seventh chord. Five qualities are common: major seventh (major triad, major seventh), dominant seventh (major triad, minor seventh), minor seventh, half-diminished seventh (diminished triad, minor seventh) and fully diminished seventh (Open Music Theory, “Seventh Chords”). Stacking further gives ninths, elevenths and thirteenths.

The notation

A chord symbol is a root letter followed by suffixes (Open Music Theory, “Chord Symbols”):

Symbol	Meaning	Semitones above the root
C	major triad: nothing is added	0 4 7
Cm, Cdim or C°, Caug or C+	minor, diminished, augmented	0 3 7, 0 3 6, 0 4 8
Csus4 (Csus), Csus2	the third replaced by a fourth or a second	0 5 7, 0 2 7
C6	major triad plus a major sixth	0 4 7 9
C7	dominant seventh: a bare 7 is a minor seventh	0 4 7 10
Cmaj7 (CΔ7), Cm7	major seventh, minor seventh	0 4 7 11, 0 3 7 10
Cm7♭5 (Cø7), Cdim7 (C°7)	half-diminished, fully diminished	0 3 6 10, 0 3 6 9
C9	dominant seventh plus a major ninth	0 4 7 10 14
Cadd9	major triad plus a ninth, no seventh	0 4 7 14

The same chapter states the two defaults that trip beginners up: a seventh added to a triad is minor unless marked maj, and C9 implies that seventh while Cadd9 does not. A ninth is an octave plus a second, so 14 semitones is pitch class 2.

== Chord symbols on C: pitch classes
symbol   course         GA             check
C        0 4 7          0 4 7          ok
Cm       0 3 7          0 3 7          ok
Cdim     0 3 6          0 3 6          ok
Caug     0 4 8          0 4 8          ok
Csus2    0 2 7          0 2 7          ok
Csus4    0 5 7          0 5 7          ok
C6       0 4 7 9        0 4 7 9        ok
C7       0 4 7 T        0 4 7 T        ok
Cmaj7    0 4 7 E        0 4 7 E        ok
Cm7      0 3 7 T        0 3 7 T        ok
Cm7b5    0 3 6 T        0 3 6 T        ok
Cdim7    0 3 6 9        0 3 6 9        ok
C9       0 2 4 7 T      0 2 4 7 T      ok
Cadd9    0 2 4 7        0 2 4 7        ok

In GA

Going from a symbol to notes and back crosses four GA types:

flowchart LR
    S["symbol: Cm7"] -- "Chord.FromSymbol" --> F["ChordFormula: 3 7 10"]
    F -- "Chord constructor" --> N["Notes: C D# G A#"]
    N --> P["PitchClassSet: 0 3 7 T"]
    F -- "GetSymbolSuffix" --> X["suffix: m7"]
    P -- "CanonicalChordPatternCatalog" --> C["name: minor-7"]

• Chord.FromSymbol splits the root from the suffix with a regular expression, then ParseSuffix is a switch expression over the lower-cased suffix: "m7b5" or "ø7", "dim7" or "°7", and so on, one arm per chord type. Anything else throws.
• Each arm returns a ChordFormula, a named list of intervals in semitones above the root (the root itself is implicit): FromSemitones("Half Diminished 7th", 3, 6, 10).
• The parser and the course agree on all fourteen symbols.

From notes back to a symbol

The idea

Naming a chord is the reverse problem: given the notes, find the root and the suffix. The intervals above the root decide it, and the difference between two chords can be a single semitone: C7 and Cmaj7 differ only by B♭ against B.

In GA

GA has two ways back, and they do not agree.

ChordFormula.GetSymbolSuffix derives a quality, then an extension, from the formula’s intervals:

== Pitch classes back to a symbol: GA's ChordFormula.GetSymbolSuffix
symbol   course   GA       check
C        (major)  (major)  ok
Cm       m        m        ok
Cdim     dim      dim      ok
Caug     aug      aug      ok
Csus2    sus2     sus2     ok
Csus4    sus4     sus4     ok
C6       6        6        ok
C7       7        7        ok
Cmaj7    maj7     7        DIFF
Cm7      m7       m7       ok
Cm7b5    m7b5     dim7     DIFF
Cdim7    dim7     dim6     DIFF
C9       9        9        ok
Cadd9    add9     add9     ok

Three of the five seventh chords come back as the symbol of another chord. The ChordQuality enum has Major7, Minor7, HalfDiminished and Diminished7 members, but DetermineQuality never returns them: it only knows Suspended, Dominant (major third and minor seventh), Diminished, Augmented, Minor, Major and Other. Then GetSymbolSuffix concatenates a quality suffix and an extension suffix:

• Cmaj7 is Major (no minor seventh, so not dominant) plus Seventh: "" + "7", the symbol of the dominant seventh chord;
• Cm7b5 is Diminished plus Seventh: "dim" + "7", the symbol of the fully diminished chord;
• Cdim7 is Diminished, and its 9-semitone interval is read as a sixth by DetermineExtension: "dim6". Nine semitones are a major sixth or a diminished seventh; only the spelling (B𝄫 rather than A) says which, and a formula in semitones has no spelling. The next section comes back to this.

The members DetermineQuality never returns are not unused elsewhere in GA, and the right symbols are already written down twice: CadenceChordParser maps "maj7", "m7b5" and "dim7" onto Major7, HalfDiminished and Diminished7, and the F# DslCommand prints them back as maj7, m7, dim7 and ø7, the textbook’s symbols — Open Music Theory writes the half-diminished chord F∅7 and the fully diminished one Fo7 (“Seventh Chords”). The table exists in the same commit; ChordFormula just never reaches it.

CanonicalChordPatternCatalog is a hand-written list of interval patterns with names and priorities, and TryFindExact returns the first pattern, by priority, whose intervals match exactly. It names all fourteen chords correctly:

== GA's recognition catalog (CanonicalChordPatternCatalog.TryFindExact)
  C       0,4,7        major-triad
  Cm      0,3,7        minor-triad
  Cdim    0,3,6        diminished-triad
  Caug    0,4,8        augmented-triad
  Csus2   0,2,7        sus2
  Csus4   0,5,7        sus4
  C6      0,4,7,9      major-6
  C7      0,4,7,10     dominant-7
  Cmaj7   0,4,7,11     major-7
  Cm7     0,3,7,10     minor-7
  Cm7b5   0,3,6,10     half-diminished-7
  Cdim7   0,3,6,9      diminished-7
  C9      0,2,4,7,10   dominant-9
  Cadd9   0,2,4,7      add-9

Reading the catalog shows four interval sets listed twice under different names: 9-sus4 and dominant-11 (0 2 5 7 10), major-6-add-9 and 6-9, minor-6-add-9 and minor-6-9, augmented-7 and dominant-7-sharp-5 (0 4 8 10). Two names for one set of pitch classes is normal in music, but with an exact match by priority, dominant-11, 6-9, minor-6-9 and dominant-7-sharp-5 can never be returned.

Spelling chords

The idea

A chord in thirds uses every other letter: C E G, D F A, B D F. To spell a chord, write those letters from the root, then add the accidentals that give each interval its quality (Open Music Theory, “Triads”, “Spelling Triads”). So C minor is C E♭ G, never C D♯ G: D♯ is on the letter of a second, and C–D♯ is an augmented second, not a minor third (lesson 1). A diminished seventh on C needs a letter B for its seventh, lowered twice: C E♭ G♭ B𝄫.

The course spells with Theory.Spell: the letter comes from the chord degree, the accidental from the semitones.

== Spelling: one letter per chord degree
symbol   course           GA               check
Cm       C Eb G           C D# G           DIFF
Eb       Eb G Bb          Eb G A#          DIFF
Ab       Ab C Eb          Ab C D#          DIFF
F#       F# A# C#         F# A# C#         ok
Bbm7     Bb Db F Ab       Bb C# F G#       DIFF
Cdim     C Eb Gb          C D# F#          DIFF
Cdim7    C Eb Gb Bbb      C D# F# A        DIFF
Gb7      Gb Bb Db Fb      Gb A# C# E       DIFF

In GA

The Chord constructor keeps the root as written, then builds every other note from its pitch class alone:

var newPitchClassValue = (root.PitchClass.Value + interval.Interval.Semitones.Value) % 12;
var newNote = new PitchClass { Value = newPitchClassValue }.ToChromaticNote().ToAccidented();

A pitch class has no letter, and the conversion is one line: Note.Chromatic.ToAccidented is ToSharp().ToAccidented(), so every black key gets a sharp name. The pitch classes stay right, and so do chords whose correct spelling uses sharps (F♯ major), but a flat-key chord such as E♭ gets Eb G A#, mixing both, and Gb7 gets an E for its seventh. For playing, the sound is the same. For display, analysis or the interval names of lesson 1, the spelling is wrong: by its letters, E♭ to A♯ is a fourth (E F G A) seven semitones wide, not the perfect fifth of the chord.

Inversions and slash chords

The idea

The bass is the lowest note that sounds. A chord with its root in the bass is in root position; with its third in the bass, in first inversion; with its fifth, in second inversion; with a seventh, in third inversion. The root does not change: C E G, E G C and G C E are all C major (Open Music Theory, “Inversion”). This is not the intervallic inversion of lesson 1, nor the set inversion of lesson 2: the same word has three meanings.

The notation

A slash chord writes the bass after a slash: C/E is a C major triad with E in the bass (Open Music Theory, “Chord Symbols”).

== Inversions of C major
inversion  course                         GA                             check
0          C bass C inversion 0 Major     C bass C inversion 0 Major     ok
1          C/E bass E inversion 1 Major   C/E bass E inversion 1 Other   DIFF
2          C/G bass G inversion 2 Major   C/G bass G inversion 2 Major   ok

In GA

• Bass is Notes[0], and GetInversion counts where the root sits in the notes: both right.
• ToInversion rotates the notes (E G C) and calls the constructor from notes with the same root, which analyses the chord again. AnalyzeChordFormula measures every note from the root but starts with Notes.Skip(1) // Skip root: after the rotation, the skipped note is the bass. In first inversion it skips E, the third, so the formula is only G and C, with no third, and the quality is Other. In second inversion it skips G, the fifth, which DetermineQuality does not look at, so the answer is right by luck.

Voicings on the fretboard

The idea

A chord symbol names pitch classes; a voicing decides which octave each one is played in, which ones are doubled and which are left out (Open Music Theory, “Jazz Voicings”). On a guitar, a voicing is a fret, or a mute, for each string. Guitar chord charts write it as six characters from string 6 (low E) to string 1: x for a string that is not played, 0 for an open string. Open C major is x32010: C on string 5, E, G, C and E above it, three pitch classes on five strings (Guitar chord, “Triads”). The Streeling module GAA-001 · Your First Chord explains how to read a chord diagram, and GTR-002 · CAGED Geometry shows the five open major shapes and the intervals on each string.

Two measurements matter to the hand: the span, from the lowest to the highest fretted note (open strings cost nothing), and whether one finger must press several strings on the same fret, a barre, as in the F major shape 133211 (Guitar chord).

== Voicings: fret numbers from string 6 (low E) to string 1 (high E)
shape    course                           GA                               check
x32010   0-1-0-2-3-x 0 4 7 span 2         0-1-0-2-3-x 0 4 7 span 2         ok
032010   0-1-0-2-3-0 0 4 7 span 2         0-1-0-2-3-0 0 4 7 span 2 barre   DIFF
x02210   0-1-2-2-0-x 0 4 9 span 1         0-1-2-2-0-x 0 4 9 span 1         ok
022100   0-0-1-2-2-0 4 8 E span 1         0-0-1-2-2-0 4 8 E span 1 barre   DIFF
320003   3-0-0-0-2-3 2 7 E span 1         3-0-0-0-2-3 2 7 E span 1 barre   DIFF
xx0232   2-3-2-0-x-x 2 6 9 span 1         2-3-2-0-x-x 2 6 9 span 1         ok
133211   1-1-2-3-3-1 0 5 9 span 2 barre   1-1-2-3-3-1 0 5 9 span 2 barre   ok

Both columns print the diagram the way GA does, which is reversed: 0-1-0-2-3-x is x32010 read from string 1.

Drawn as chord charts draw them, string 6 on the left and the nut at the top, four of these shapes look like this. A cross marks a string that is not played, a circle an open string, and each string shows the pitch it sounds. The first two are the same C major chord: x32010 has C3 in the bass, root position, and 032010 adds the open low E, E2, which puts the chord in first inversion. 320003 is G major, and 133211 is F major with one finger barring fret 1.

C3 E3 G3 C4 E4
E2 C3 E3 G3 C4 E4
G2 B2 D3 G3 B3 G4
F2 C3 F3 A3 C4 F4

The ComfyUI course draws the same voicings as images: its lesson 11 writes GA nodes that turn x32010 into a chord diagram and a fretboard map for image generation, with the strings in chart order and GA’s order side by side.

In GA

• Voicing is a record of Positions (Position.Muted or Position.Played, another closed hierarchy) and MIDI notes. Its Diagram joins the positions in array order, and GA’s VoicingGenerator fills that array in the order of Str.Range, from string 1, “the first string (highest pitch)”, so GA’s diagrams start with the high E, the opposite of chord charts. The program builds its voicings the same way.
• FretSpan ignores open strings, like the course.
• HasBarre is true when three played strings share a fret, fret 0 included. Three open strings are not a barre, so 032010 (C with a low E), 022100 (E) and 320003 (G), all played without one, come out as barre chords. A barre is “one finger to press down multiple strings across a single fret” (Barre chord, Wikipedia), and no finger presses an open string, so all three rows are GA’s error, not a matter of definition. The follow-up named in the comment above HasBarre is already written: DetectBarreRequirement does test fret > 0 and does require the strings to be next to each other — and for that reason it answers not barre for 133211, whose fret 1 is on strings 6, 2 and 1. The course’s rule counts fretted strings without the adjacency, so it would call a shape like x1x1x1 a barre too. All three rules are approximations; only the course’s gets every row of this table right.

Naming a voicing

A shape has a bass, so naming it means finding a root among its pitch classes, possibly different from the bass. The course’s NameOf tries the bass first, then the other notes, and writes a slash chord when the root is not the bass. The GA column measures the intervals from the bass and asks TryFindExact (GaNameOf):

== Naming a voicing: course tries the bass first, GA's catalog reads from the bass
  x32010  course C      GA major-triad
  032010  course C/E    GA (none)
  x02210  course Am     GA minor-triad
  022100  course E      GA major-triad
  320003  course G      GA major-triad
  xx0232  course D      GA major-triad
  133211  course F      GA major-triad

Measured from E, the notes of 032010 are 0 3 8, a pattern that matches nothing, because the catalog’s patterns are measured from the root and E is not the root. Wikipedia’s chord chart lists 032010 as the C chord, while its text mutes string 6; whether an E in the bass is acceptable is a musical choice, but it changes the name. GA does name inversions elsewhere: its MCP tool ga_search_voicings returned a voicing 8-8-x-x-7-x labelled C/E in this session (GA MCP server, 2026-09-14, version not reported).

Exercises

1. Spell F♯dim7, one letter per chord degree.
2. Name the shapes x35543 and x02010: list their notes from low to high, then find a root and a suffix.

Solutions

1. The letters are F, A, C, E. Above F♯: A is a minor third, C a diminished fifth (6 semitones), and the diminished seventh is 9 semitones, E♭: F♯ A C E♭. GA writes D♯, the same key on a piano but the letter of a sixth.
2. x35543 is C3 G3 C4 D♯4 G4. With C as the root the intervals are 0 3 7: Cm, a barre chord on the A-string shape (spelled C E♭ G). x02010 is A2 E3 G3 C4 E4: from A, 0 3 7 10, Am7. Starting from C instead, the same notes are 0 4 7 9, C6 with A in the bass: the course tries the bass first, which is why it answers Am7.

== Exercise solutions
question         course               GA                   check
1. F#dim7        F# A C Eb            F# A C D#            DIFF
2. x35543  notes C3 G3 C4 D#4 G4  course Cm  GA minor-triad
2. x02010  notes A2 E3 G3 C4 E4  course Am7  GA minor-7

The program prints MIDI-derived note names, hence D#4 in the second line (Lesson3.cs).

What GA’s MCP server answers

GA also exposes chord tools to AI assistants through an MCP server. Asked in this session (2026-09-14; the server does not report its version, so these answers may not match commit a826864), ga_chord_intervals returned:

Symbol	MCP answer	Theory
Cm7b5	P1 m3 P5 m7	P1 m3 d5 m7
G7b9	P1 M3 P5 m7	P1 M3 P5 m7 m9
C9	P1 M3 P5 M9	P1 M3 P5 m7 M9
Cmaj9	P1 M3 P5 M9	P1 M3 P5 M7 M9

ga_parse_chord("Cm7b5") did parse the alteration ("components":["ext:7","alt:b5"]). The tool goes through a different path from Chord.FromSymbol: the F# closure domain.chordIntervals keeps only the base triad plus one interval per extension, ignores alterations, and maps 9 and maj9 to 14 semitones without their seventh. The same commit’s C# parser gets C9 right (first table of this lesson), so the two layers of GA disagree with each other.

Key takeaways

• A chord symbol is root plus quality plus extensions; a bare 7 is a minor seventh, 9 implies it, add9 does not.
• A chord in thirds takes every other letter; the spelling, not the semitones, tells a diminished seventh from a major sixth. GA spells from pitch classes, with sharps.
• Inversion is about the bass; the root does not change. Slash chords write the bass.
• A guitar voicing is a fret or a mute per string; chord charts start from the low E, GA’s diagrams from the high E.
• GA has several ways to name a chord (GetSymbolSuffix, the pattern catalog, the MCP tools) and they give different answers: test against the catalog, which matched the textbook for every root-position chord here.

Sources

• Mark Gotham et al., Open Music Theory, version 2, 2023, CC BY-SA 4.0: chapters Triads, Seventh Chords, Chord Symbols, Inversion, Jazz Voicings.
• Guitar chord, Wikipedia (open-position major chords, barre chords).
• GuitarAlchemist/ga at a826864: Theory/Harmony (Chord.cs, ChordFormula.cs, ChordQuality.cs, CanonicalChordPatternCatalog.cs), Instruments/Fretboard/Voicings/Core/Voicing.cs, GA.Domain.Services/Fretboard/Voicings/Generation/VoicingGenerator.cs, GA.Business.DSL/Closures/BuiltinClosures/DomainClosures.fs.