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<h1>Circle of Fifths</h1>
<p>The circle of fifths is a map of all twelve keys arranged so each adjacent pair is a perfect fifth apart. It's a reference tool: once you understand how to read it, it immediately tells you key signatures, relative minors, chord relationships, and which keys are closely related. It also explains why the ii-V-I progression is so fundamental — it's movement around the circle by fifths, the most harmonically satisfying direction of motion.</p>

<h2>The Circle</h2>
<pre>
                 C (0)
           F           G
         (1♭)           (1#)
       B♭                   D
     (2♭)                   (2#)
       E♭                   A
         (3♭)           (3#)
           A♭         E
             (4♭)   (4#)
           D♭/C# (5♭ / 5#)
               G♭/F#
              (6♭ / 6#)

Clockwise:        add one sharp per step
Counter-clockwise: add one flat per step
</pre>
<p>Moving clockwise goes up a perfect fifth (C→G→D→A→E→B→F#). Moving counter-clockwise goes up a perfect fourth (C→F→B♭→E♭→A♭→D♭→G♭). At the bottom, enharmonic equivalents (D♭=C#, G♭=F#) meet — both spellings work, one with flats and one with sharps.</p>

<h2>Key Signatures at a Glance</h2>

<table>
  <thead><tr><th>Key</th><th>Accidentals</th><th>Notes added</th><th>Relative minor</th></tr></thead>
  <tbody>
    <tr><td>C major</td><td>None</td><td>—</td><td>A minor</td></tr>
    <tr><td>G major</td><td>1#</td><td>F#</td><td>E minor</td></tr>
    <tr><td>D major</td><td>2#</td><td>F#, C#</td><td>B minor</td></tr>
    <tr><td>A major</td><td>3#</td><td>F#, C#, G#</td><td>F# minor</td></tr>
    <tr><td>E major</td><td>4#</td><td>F#, C#, G#, D#</td><td>C# minor</td></tr>
    <tr><td>B major</td><td>5#</td><td>F#, C#, G#, D#, A#</td><td>G# minor</td></tr>
    <tr><td>F# major</td><td>6#</td><td>F#, C#, G#, D#, A#, E#</td><td>D# minor</td></tr>
    <tr><td>F major</td><td>1♭</td><td>B♭</td><td>D minor</td></tr>
    <tr><td>B♭ major</td><td>2♭</td><td>B♭, E♭</td><td>G minor</td></tr>
    <tr><td>E♭ major</td><td>3♭</td><td>B♭, E♭, A♭</td><td>C minor</td></tr>
    <tr><td>A♭ major</td><td>4♭</td><td>B♭, E♭, A♭, D♭</td><td>F minor</td></tr>
    <tr><td>D♭ major</td><td>5♭</td><td>B♭, E♭, A♭, D♭, G♭</td><td>B♭ minor</td></tr>
    <tr><td>G♭ major</td><td>6♭</td><td>B♭, E♭, A♭, D♭, G♭, C♭</td><td>E♭ minor</td></tr>
  </tbody>
</table>

<h2>Reading Key Signatures</h2>
<pre>
Sharps: the key is one half step above the last sharp added.
  G major: 1# (F#) → one half step above F# = G ✓
  D major: 2# (F#, C#) → one half step above C# = D ✓

Flats: the key is the second-to-last flat.
  B♭ major: 2♭ (B♭, E♭) → second-to-last flat = B♭ ✓
  E♭ major: 3♭ (B♭, E♭, A♭) → second-to-last flat = E♭ ✓

Special cases: C major (no accidentals), F major (only 1♭ — the key IS the last flat)
</pre>

<h2>Relative Minor Keys</h2>
<p>Every major key has a relative minor that shares its key signature. The relative minor starts on the 6th degree of the major scale (or equivalently, a minor third below the major root):</p>
<pre>
C major ↔ A minor     (same key signature, different tonal centre)
G major ↔ E minor
D major ↔ B minor
A major ↔ F# minor
E major ↔ C# minor
B major ↔ G# minor
F major ↔ D minor
B♭ major ↔ G minor
E♭ major ↔ C minor
A♭ major ↔ F minor
D♭ major ↔ B♭ minor

Trick: the relative minor is always 3 semitones below the major key's root.
C → down 3 semitones → A  ✓
G → down 3 semitones → E  ✓
</pre>

<h2>Related Keys and Modulation</h2>
<p>Keys that are adjacent on the circle share six of their seven notes and one accidental. Modulating between adjacent keys sounds smooth and natural. Keys on opposite sides of the circle (six steps apart) share only one note — moving between them creates maximum harmonic contrast:</p>

<table>
  <thead><tr><th>Distance on circle</th><th>Notes in common</th><th>Modulation character</th><th>Example (from C major)</th></tr></thead>
  <tbody>
    <tr><td>1 step (adjacent)</td><td>6 of 7</td><td>Smooth, natural</td><td>C → G or C → F</td></tr>
    <tr><td>2 steps</td><td>5 of 7</td><td>Clear but easy</td><td>C → D or C → B♭</td></tr>
    <tr><td>3 steps (quarter-circle)</td><td>4 of 7</td><td>Noticeable change</td><td>C → A or C → E♭</td></tr>
    <tr><td>6 steps (opposite)</td><td>1 of 7 (just the tritone)</td><td>Dramatic, jarring</td><td>C → F# (tritone key)</td></tr>
  </tbody>
</table>

<h2>The Circle and ii-V-I</h2>
<p>The ii-V-I progression moves counter-clockwise around the circle by fifths — each chord's root is a fourth above (or fifth below) the next:</p>
<pre>
Dm7 (ii) → G7 (V) → Cmaj7 (I)
  D  →  G  →  C    (each root a 5th below the previous)

This is the most harmonically purposeful direction of motion —
the circle of fifths is a gravity map, and music falls toward the tonic.

A sequence of ii-V-Is through different keys traces a path around the circle:
  Dm7-G7-Cmaj7  (landing on C)
  Gm7-C7-Fmaj7  (landing on F)
  Cm7-F7-B♭maj7 (landing on B♭)
  ...descending by whole steps through the cycle — a common jazz reharmonisation
</pre>

<h2>The Tritone Substitution Connection</h2>
<p>The tritone substitution replaces a dominant seventh chord with the dominant seventh built on the opposite side of the circle — exactly six steps away, which is a tritone:</p>
<pre>
G7 (dominant of C) → D♭7 (tritone sub, opposite G on the circle)

Both chords share the same tritone (B/F), just with 3rd and 7th swapped.
D♭7 → C: bass moves by descending half step (smoother than G→C fifth).

The circle of fifths makes this substitution intuitive —
diametrically opposite keys have the same resolution energy.
</pre>

<h2>References</h2>
<ul>
  <li>Levine — <em>The Jazz Theory Book</em></li>
  <li>Rawlins &amp; Bahha — <em>Jazzology</em></li>
  <li><a href="https://www.musictheory.net/lessons/25" target="_blank" rel="noopener">musictheory.net — Circle of Fifths ↗</a></li>
</ul>
