Lesson 7: Cadences, ii–V–I and the key of a progression

Lesson 6 gave every chord of a key a numeral and a function. This lesson strings chords together: how phrases end, why V7 leads to I, what makes ii–V–I the backbone of jazz, and how to go the other way, from a list of chord symbols to the key they are in. Guitar Alchemist (GA) has a catalog of cadences, a core key finder and two MCP tools for the last question; the course program checks all three.

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

dotnet run --project code/music-theory-ga/GaTheory -c Release -- l7

Cadences

The idea

A cadence marks the end of a phrase. Open Music Theory distinguishes two main kinds (Open Music Theory, “Introduction to Harmony, Cadences, and Phrase Endings”):

• an authentic cadence goes from V to I and sounds conclusive. It is perfect (PAC) when both chords are in root position and the tonic is in the top voice over the I chord, imperfect (IAC) when V–I is there but one of those conditions is not met;
• a half cadence (HC) ends on V, from any chord, “x–V”, and sounds inconclusive.

Two more motions get their own names. IV–I, the “amen” at the end of hymns, is often called a plagal cadence; Open Music Theory prefers “plagal motion”, because in 18th- and 19th-century music it mostly prolongs a tonic that has already arrived (Open Music Theory, “Plagal Motion as a Form of Prolongation”). V(7)–vi, where the ear expects I and gets vi, is deceptive motion (Open Music Theory, “La in the Bass at Beginnings, Middles, and Endings”). The Streeling module MUS-003 · How Harmony Works presents the same five endings, with guitar examples in G major.

In GA

GA lists 21 cadences and common endings in a configuration file, Cadences.yaml, from the PAC to jazz, rock, metal, flamenco and neo-soul patterns. Each entry gives Roman numerals, a key and the chords in that key. The program reads the file, computes the numerals from the chords and the key, with the rules of lesson 6 plus a flat or a sharp when a root is not in the key’s scale (Lesson7.Numeral), and compares them with the file’s, ignoring the chord-quality suffixes:

== GA's Cadences.yaml: its Roman numerals, and the course's from the chords and the key
cadence                            course           GA               check
Perfect Authentic Cadence (PAC)    V I              V I              ok
Imperfect Authentic Cadence (IAC)  V I              V I              ok
Plagal Cadence                     IV I             IV I             ok
Half Cadence                       I V              I V              ok
Deceptive Cadence                  V vi             V vi             ok
Major ii-V-I                       ii V I           ii V I           ok
Minor ii-V-i                       ii V i           ii V i           ok
Tritone Substitution               ii bII I         ii bII I         ok
Backdoor Cadence                   iv bVII I        iv bVII I        ok
Tadd Dameron Turnaround            I bIII bVI bII   I bIII bVI bII   ok
Coltrane Turnaround                I bIII bVI bII I I bIII bVI bII I ok
Rock Authentic Cadence             V I              V I              ok
Mixolydian Cadence (bVII-I)        bVII I           bVII I           ok
Aeolian Cadence (Mario Cadence)    bVI bVII I       bVI bVII I       ok
Double Plagal                      bVII IV I        bVII IV I        ok
Phrygian Half Cadence              bII i            bII i            ok
Chromatic Mediant (Metal)          i iii            i biii           DIFF
Andalusian Cadence                 iv III II I      iv III II I      ok
Neo-Soul ii-V                      ii V I           ii V I           ok
Minor Plagal (Neo-Soul)            iv I             iv I             ok
Tritone Sub ii-V (Neo-Soul)        ii bII I         ii bII I         ok

The chords and the numerals agree, except in one row. “Chromatic Mediant (Metal)” is Em–Gm in E minor. G is the third degree of E minor, so the course writes iii; the file writes ♭iii, measuring from E major, whose third degree is G♯. The same file analyses the Andalusian cadence, Am G F E, in E Phrygian (“iv III II I”), from the notes of the mode. This is not a clash of conventions: Open Music Theory numbers chords in a minor key from the minor scale, “iii and vi in major keys; III and VI in minor keys” (“Mediants”), and so does GA’s own F# analyser, whose minor numerals are i, ii°, III, iv, v, VI, VII (DomainClosures). Read in E minor, ♭iii points at G♭, a note the chord does not contain. Only Cadences.yaml measures from the parallel major, and only in this row.

One name is also misleading. The file’s “Phrygian Half Cadence” is ♭II–i, F to Em, which ends on the tonic. In classical theory, a Phrygian half cadence is iv⁶–V in minor, a half cadence named after the half step in its bass (Cadence, Wikipedia, “Phrygian half cadence”).

V7 and the pull toward I

The idea

Add a seventh to V and it pulls harder toward I (Open Music Theory, “Strengthening Endings with V7”). In C major, V7 is G7, G B D F. Its active notes have tendencies: the leading tone B rises to C, the seventh F falls to E, and D moves to C (same chapter). B and F are six semitones apart, a tritone, the only one in the key; the C major triad has none. The program moves each note of G7 to the nearest note of C, then counts tritones with the interval-class vectors of lesson 4:

== Resolving V7 to I: each tone of G7 to the nearest tone of C
  G (root)                 -> G (0)
  B (third, leading tone)  -> C (+1)
  D (fifth)                -> C (-2) or E (+2)
  F (seventh)              -> E (-1)
tritone        course       GA           check
in G7          1            1            ok
in C           0            0            ok

The next figure draws G7 and C on one bracelet of note names: a quadrilateral G B D F and a triangle C E G, which share only G, ringed. Read around the circle, B sits one step below C and F one step above E: the two resolutions are the two shortest moves on the clock.

Chords that share notes connect smoothly, and the program lists the common tones of a few pairs. The outputs are pitch classes, as in lesson 2, and the program writes 10 and 11 as T and E: here E is pitch class 11, B, not the note E.

== Common tones between neighbouring chords (GA: PitchClassSet.Intersect)
chords         course       GA           check
G7 C           7            7            ok
Dm7 G7         2 5          2 5          ok
G7 Cmaj7       7 E          7 E          ok
C Am           0 4          0 4          ok
C F#           (none)       (none)       ok

The GA column is PitchClassSet.Intersect; it agrees. G7 and Cmaj7 share G and B, and C and F♯ share nothing.

ii–V–I

The idea

In jazz, the most important progression is ii–V–I: ii7–V7–Imaj7 in major, iiø7–V7–i7 in minor (Open Music Theory, “ii–V–I”). Each root is a fifth above the next, D, G, C, and the chord qualities follow a fixed pattern: minor seventh, dominant seventh, major seventh in major; half-diminished seventh, dominant seventh, minor seventh in minor. The V chord is dominant in both modes (same chapter). A ii–V even works without its I, and whole ii–V–Is move from key to key inside a tune. The Streeling module MUS-003 analyses it in G major (section “Functional Analysis in Practice”), and MUS-005 · Jazz Harmony for Guitar in C major, with shell and drop 2 voicings.

The program stacks degrees 2, 5 and 1 of C major, then of C minor with the raised leading tone for V, and names the qualities; the GA column names the chords parsed with Chord.FromSymbol through the catalog of lesson 6:

== ii-V-I: chord qualities from the key (major, then minor with the raised leading tone)
progression        course                                   GA                                       check
Dm7 G7 Cmaj7       minor-7 dominant-7 major-7               minor-7 dominant-7 major-7               ok
  numerals in C major: ii V I; roots D G C, each a fifth below the one before
Dm7b5 G7 Cm7       half-diminished-7 dominant-7 minor-7     half-diminished-7 dominant-7 minor-7     ok
  numerals in C minor: ii V i; roots D G C, each a fifth below the one before

The figure below gives one set of open-position shapes for ii–V–I in C major, as chord grids, string 6 on the left. Dm7 is xx0211: strings 6 and 5 muted, open D, then A, C and F. G7 is 320001: G, B, open D, G and B, and F on the first fret of string 1. Cmaj7 is x32000: C, E, then open G, B and E. From G7 to Cmaj7, the F on string 1 falls a half step to the open E, and the B on string 2 stays; the leading tone does not always rise on a guitar.

D3 A3 C4 F4
G2 B2 D3 G3 B3 F4
C3 E3 G3 B3 E4

Finding the key of a progression

The idea

Given only chord symbols, which key are they in? A first answer is: the key whose notes contain all the chords’ notes. Lesson 6 showed that one chord often fits several keys, and so can a whole progression. C F G C uses exactly the notes of C major, which are also the notes of A minor; in every such tie, a second clue decides. A phrase usually starts on the tonic, and a conclusive ending, an authentic cadence, returns to it (Open Music Theory, “Introduction to Harmony, Cadences, and Phrase Endings”). The course’s rule (Lesson7.CourseKey) is:

1. keep the keys that contain the most chords, with, for minor keys, the natural minor scale plus the raised leading tone;
2. among them, prefer the key whose tonic chord ends the progression, then the one whose tonic chord starts it; print a tie as “X or Y”.

This is a heuristic, not a theory: real pieces modulate, borrow chords and end on V. It is enough for short progressions.

In GA

GA’s core has PitchClassSet.ClosestDiatonicKey. It counts, for each of the 30 keys, how many of the key’s notes are in the set, and breaks ties with an expected mode: minor if the set’s normal form contains pitch class 3, E♭, and major otherwise (FindClosestDiatonicKey2). The program gives it the union of the chords’ notes:

== Finding the key: the course, then GA's PitchClassSet.ClosestDiatonicKey on all the notes
progression    course             GA                 check
C F G C        Key of C           Key of Am          DIFF
  notes 0 2 4 5 7 9 E, GA normal form 0 1 3 5 6 8 T; GA compatible keys: C, Am; roots only: C major 4/4
G D Em C       Key of G           Key of Em          DIFF
  notes 0 2 4 6 7 9 E, GA normal form 0 1 3 5 6 8 T; GA compatible keys: G, Em; roots only: G major 4/4
Dm7 G7 Cmaj7   Key of C           Key of Am          DIFF
  notes 0 2 4 5 7 9 E, GA normal form 0 1 3 5 6 8 T; GA compatible keys: C, Am; roots only: D minor 3/3
Am Dm E7 Am    Key of Am          Key of Am          ok
  notes 0 2 4 5 8 9 E, GA normal form 0 1 3 4 6 8 9; GA compatible keys: (none); roots only: A major 4/4
Am F C G       Key of Am          Key of Am          ok
  notes 0 2 4 5 7 9 E, GA normal form 0 1 3 5 6 8 T; GA compatible keys: C, Am; roots only: A minor 4/4
C Am F G7      Key of C           Key of Am          DIFF
  notes 0 2 4 5 7 9 E, GA normal form 0 1 3 5 6 8 T; GA compatible keys: C, Am; roots only: C major 4/4

The normal form is a transposition of the set, starting on 0, not the set seen from its tonic, so “contains 3” says nothing about the mode. And it is not two unlucky examples: every seven-note diatonic collection has the same normal form, 0 1 3 5 6 8 T, which contains 3. The flag is always raised, the tie always breaks toward minor, and since every major key has a relative minor with the same seven pitch classes, the method can never answer a major key for a complete major scale. A progression that starts and ends on C comes out in A minor. Nothing in the method looks at which chord is first or last, and after the mode flag there is no tie-break left either: for sets of fewer than seven notes the answer is whichever key comes first in GA’s list, which starts at C♭ major. The “notes” line also shows GetCompatibleKeys, which is right but can’t choose, and nothing fits Am Dm E7 Am’s G♯ except the course’s minor collection.

GA’s MCP server has two tools for the same question, ga_key_from_progression and ga_analyze_progression. Both score each major key and natural minor key by counting the chord roots in its scale, not the chords’ notes, and prefer the key on the first chord’s root (GaKeyFromProgressionTool, analyzeProgression). The numerals then come from a fixed list per mode. The “roots only” part of each line above is that rule, rewritten in the course program (Lesson7.RootsOnlyKey). On 2026-09-15, the MCP server running on the author’s machine (its version is not reported) answered:

ga_key_from_progression ["Am","F","C","G"]   ->  bestGuess "A minor" (candidates A minor, C major, D minor, all 4/4)
ga_key_from_progression ["Dm7","G7","Cmaj7"] ->  bestGuess "D minor" (candidates D minor, C major, C minor, all 3/3)
ga_analyze_progression "Am Dm E7 Am"         ->  Key: A major  (confidence 4/4)   I IV V I
ga_analyze_progression "Dm7 G7 Cmaj7"        ->  Key: D minor  (confidence 3/3)   i iv VII

The first tool’s own description promises “Am F C G → best guess C major”. A minor is a defensible answer for a progression that starts on Am; C minor, a candidate for Dm7 G7 Cmaj7, is not: Cmaj7’s E and B are not in C minor, but the tool only counts the roots D, G and C, which are. Am Dm E7 Am is in A minor: A major has C♯, F♯ and G♯, and the progression’s C and F rule it out.

The cadence machine plate of the Atlas des Douze plays these cadences and shows where GA’s ClosestDiatonicKey disagrees with the course.

Exercises

1. Give the numerals of F Gm C7 F in F major. Are all four chords in the key?
2. What key is Em Am B7 Em in?
3. Which note do A7 and D have in common, and why does it matter?

Solutions

1. I ii V7 I. F major has one flat, B♭: F A C, G B♭ D, C E G B♭ and F A C are all inside. The progression is a PD–D–T ending, a V7–I authentic cadence.
2. E minor. B7 contains D♯, the raised leading tone of E minor, and the progression starts and ends on Em. The roots-only rule answers E major: E, A and B are all in E major’s scale, and the minor thirds of Em and Am are not counted.
3. A, pitch class 9: the root of A7 is the fifth of D. A7–D is V7–I in D major, and that shared note stays while C♯ rises to D and G falls to F♯.

== Exercise solutions
question             course         GA             check
1. F Gm C7 F in F major: I ii V I
1. all in F major    True           True           ok
2. Em Am B7 Em       Key of Em      Key of Em      ok
   roots only: E major 4/4
3. common A7 D       9              9              ok

The first line prints the course’s numerals without the 7 of C7: lesson 6’s numerals mark the quality, the chord symbol already says it is a seventh chord.

Key takeaways

• An authentic cadence is V–I, perfect with both chords in root position and the tonic on top; a half cadence ends on V. IV–I is mostly plagal motion, V–vi deceptive motion.
• V7 holds the key’s only tritone, B–F in C; the leading tone rises and the seventh falls, to C and E.
• ii–V–I is minor seventh, dominant seventh, major seventh in major, and half-diminished, dominant, minor seventh in minor, with roots moving down by fifths.
• The key of a progression needs the chords’ notes and a tie-break on the tonic chord. GA’s core breaks ties on a normal form that doesn’t encode the mode, so it answers the relative minor for every complete major scale; its MCP tools count roots only; neither looks at the last chord.
• GA’s cadence catalog is right chord by chord, but mixes two numeral conventions for minor and names ♭II–i a “Phrygian half cadence”.

Sources

• Mark Gotham et al., Open Music Theory, version 2, 2023, CC BY-SA 4.0: chapters Introduction to Harmony, Cadences, and Phrase Endings, Plagal Motion as a Form of Prolongation, La in the Bass at Beginnings, Middles, and Endings, Strengthening Endings with V7, ii–V–I, Roman Numerals.
• Cadence, Wikipedia.
• GuitarAlchemist/ga at a826864: Common/GA.Business.Config/Cadences.yaml, Common/GA.Domain.Core/Theory/Atonal/PitchClassSet.cs, GaMcpServer/Tools/GuitaristProblemTools.cs, Common/GA.Business.DSL/Closures/BuiltinClosures/DomainClosures.fs.