skills/musical/references/chords.md

# Chords

## Construction

Build every chord from its root by stacking the intervals that define it. Do not
recall chords as remembered note-groups - derive them.

Semitones above the root:

| Chord | Symbol | Intervals | Semitones |
|---|---|---|---|
| Major triad | `C` | 1 3 5 | 0 4 7 |
| Minor triad | `Cm` | 1 b3 5 | 0 3 7 |
| Diminished triad | `Cdim`, `C°` | 1 b3 b5 | 0 3 6 |
| Augmented triad | `Caug`, `C+` | 1 3 #5 | 0 4 8 |
| Suspended 2nd | `Csus2` | 1 2 5 | 0 2 7 |
| Suspended 4th | `Csus4` | 1 4 5 | 0 5 7 |
| Major 7th | `Cmaj7`, `CM7`, `C∆7` | 1 3 5 7 | 0 4 7 11 |
| Dominant 7th | `C7` | 1 3 5 b7 | 0 4 7 10 |
| Minor 7th | `Cm7` | 1 b3 5 b7 | 0 3 7 10 |
| Half-diminished | `Cm7b5`, `Cø7` | 1 b3 b5 b7 | 0 3 6 10 |
| Diminished 7th | `Cdim7`, `C°7` | 1 b3 b5 bb7 | 0 3 6 9 |
| Minor-major 7th | `CmMaj7` | 1 b3 5 7 | 0 3 7 11 |
| 6th | `C6` | 1 3 5 6 | 0 4 7 9 |
| Minor 6th | `Cm6` | 1 b3 5 6 | 0 3 7 9 |
| Dominant 9th | `C9` | 1 3 5 b7 9 | 0 4 7 10 14 |

## Spelling

A triad or seventh chord uses **every other letter name**, starting from the
root, with no letter repeated and none skipped:

- C E G, not C E F##
- D F# A, not D Gb A
- Eb G Bb, not Eb G A#

Work out the letters first from the root, then apply whatever accidentals the
intervals require. This is what makes F# major come out as F# A# C# rather than
Gb Bb Db, and it is the single most common source of wrong-but-nearly-right
answers.

Two cases where the rule forces an unusual-looking accidental, and the answer is
still correct:

- **G# major** is G# B# D#. B# is not a typo; the letters must be G, B, D.
- **Cdim7** is C Eb Gb Bbb. The double-flat is required - the seventh must be
  some kind of B.

## Inversions and slash chords

The inversion is named by which chord tone is in the bass. The notes themselves
do not change.

| Inversion | Bass note | C major |
|---|---|---|
| Root position | root | C E G |
| First inversion | third | E G C |
| Second inversion | fifth | G C E |
| Third inversion (7ths only) | seventh | Bb C E G for C7 |

Slash notation names the bass explicitly: `C/E` is C major with E in the bass.
The note after the slash need not be a chord tone - `C/D` is legitimate, and is
not the same chord as `Dm7` or `D9sus4` even though they share notes.

## Grading procedure

Given a chord symbol and the learner's answer:

1. From the symbol alone, derive the expected letters, then the expected
   accidentals. Write the expected chord down before looking at theirs.
2. Compare note by note by **letter name**, not by pitch.
3. List any note they gave that is not in the chord.
4. List any chord tone they did not give.
5. Only then decide the verdict.

Do not skip step 1. Reading their answer first makes near-misses look right.

## Worked cases

**The enharmonic trap.** Asked for the third of E major, learner says Ab.
Verdict: wrong, and specifically a spelling error. Ab and G# are the same pitch,
but E major's third must be some kind of G, because the triad is E-G-B. The
answer is G#. Say the notes are right and the spelling is not - do not mark it
simply wrong, and do not let it pass.

**The omitted fifth.** Asked for Bbmaj7, learner says Bb D A. Verdict: wrong by
omission. D and A are correct chord tones, the F is missing. Point at the gap
rather than re-listing the whole chord.

**The wrong seventh.** Asked for Am7, learner says A C E G#. Verdict: wrong.
G# is the seventh of AmMaj7; Am7 takes a minor seventh, G natural. Worth naming
the chord they actually spelled - it tells them the shape of their error.

**The right chord, wrong inversion.** Asked for C/E, learner says C E G.
Verdict: correct notes, wrong voicing. The question asked for E in the bass, so
E G C. Whether this counts as wrong depends on what was being tested - if the
exercise is about inversions, it is wrong; if it is about chord construction,
the notes are right and the inversion is a footnote. Say which you are doing.

**The legitimate ambiguity.** Asked to name the pitches C Eb Gb A, there is no
single right answer. A diminished seventh chord is symmetrical - stacked minor
thirds - so those four pitches spell `Cdim7`, `Ebdim7`, `Gbdim7` and `Adim7`
equally well, and which one it is depends on context you have not been given.
The same pitches are also `B7b9` with the root left out. The correct spelling of
the top note follows from whichever reading you take: Bbb in `Cdim7`, A in
`Adim7`. Give the readings and the reasoning. Do not pick one and present it as
the only answer.