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<h1>Diminished Scale</h1>
<p>The diminished scale alternates half steps and whole steps (or whole and half, depending on where you start). It's a symmetrical eight-note scale with a distinctive tense, dissonant sound. Jazz musicians use it over diminished seventh chords and dominant seventh chords with alterations — it simultaneously contains multiple chord tones and multiple tension notes, making it harmonically dense in a way no seven-note scale can match.</p>

<h2>Two Forms</h2>
<pre>
Half-Whole Diminished (used over dominant 7th chords):
C  D♭  E♭  E  F#  G  A  B♭  C
H  W   H   W  H   W  H  W

Formula: 1 — ♭2 — ♭3 — 3 — #4 — 5 — 6 — ♭7

Whole-Half Diminished (used over diminished 7th chords):
C  D  E♭  F  G♭  A♭  A  B  C
W  H  W   H  W   H   W  H

Formula: 1 — 2 — ♭3 — 4 — ♭5 — ♭6 — 6 — 7
</pre>

<table>
  <thead><tr><th>Form</th><th>Pattern</th><th>Used over</th><th>Key tensions it provides</th></tr></thead>
  <tbody>
    <tr><td>Half-Whole</td><td>H-W alternating, starting H</td><td>Dominant 7th chords (V7, 7♭9, 7♯9)</td><td>♭9, ♯9, ♯11, 13</td></tr>
    <tr><td>Whole-Half</td><td>W-H alternating, starting W</td><td>Diminished 7th chords (°7)</td><td>Root, ♭3, ♭5, 6 (all dim7 chord tones)</td></tr>
  </tbody>
</table>

<h2>Symmetry — Three Distinct Scales</h2>
<p>The diminished scale repeats its pattern every minor third (3 semitones). Transposing it up a minor third gives the same set of pitches. This means there are only three distinct diminished scales:</p>
<pre>
Group 1: C, E♭, G♭, A    (all four roots yield the same 8 pitches)
  C dim scale: C D♭ E♭ E F# G A B♭
  E♭ dim scale: E♭ E F# G A B♭ C D♭  (same notes, rotated)

Group 2: C#, E, G, B♭
Group 3: D, F, A♭, B

Consequence: a lick or pattern that works in C will also work
unchanged in E♭, G♭, and A — four keys for the price of one.
</pre>

<h2>Half-Whole Over Dominant 7th Chords</h2>
<pre>
C7 chord tones: C  E  G  B♭
C half-whole:   C  D♭  E♭  E  F#  G  A  B♭

Chord tones in scale:
  C = root ✓
  E = major 3rd ✓
  G = 5th ✓
  B♭ = minor 7th ✓

Available tensions:
  D♭ = ♭9  (creates harsh dominant tension)
  E♭ = #9  (the "Hendrix chord" note)
  F# = #11  (tritone extension — Lydian dominant colour)
  A  = 13th (natural 13th — bright)

The half-whole diminished simultaneously provides ♭9, ♯9, ♯11, and 13 —
a richer set of alterations than any other single scale over a dominant chord.
</pre>

<h2>Characteristic Sound</h2>
<p>The combination of ♭9 and ♯9 over a dominant chord is the most characteristically "tense" dominant sound in jazz. It's neither fully altered (which would have ♭13) nor fully clean (which would be Mixolydian). The half-whole diminished sits in between — providing two chromatic tensions (♭9 and ♯9) while keeping the natural 5th and 13th. Coltrane, McCoy Tyner, and Thelonious Monk all used it heavily.</p>

<h2>Whole-Half Over Diminished Seventh Chords</h2>
<pre>
Cdim7 chord tones: C  E♭  G♭  B♭♭(=A)
C whole-half:      C  D  E♭  F  G♭  A♭  A  B

Chord tones in scale:
  C  = root ✓
  E♭ = ♭3 ✓
  G♭ = ♭5 ✓
  A  = 6/double-flat 7th (= ♭♭7 = diminished 7th) ✓

The whole-half form is designed for the fully-diminished chord.
It adds passing tones (D, F, A♭, B) that connect the chord tones.
</pre>

<h2>The Diminished 7th Chord's Symmetry</h2>
<pre>
Cdim7: C E♭ G♭ A   (all minor thirds)

This chord sounds identical in any inversion:
  Cdim7   = C  E♭  G♭  A
  E♭dim7  = E♭ G♭  A   C    (same chord, 1st inversion)
  G♭dim7  = G♭ A   C   E♭   (same chord, 2nd inversion)
  Adim7   = A  C   E♭  G♭   (same chord, 3rd inversion)

So Bdim7 substitutes for G7♭9 (shares B, D, F, A♭ with G7♭9).
And Ddim7 also substitutes for G7♭9 — all four are the same chord.
This explains why diminished 7ths appear as passing and substitute chords
in jazz and classical music — they're dominant 7♭9 chords in disguise.
</pre>

<h2>References</h2>
<ul>
  <li>Levine — <em>The Jazz Theory Book</em></li>
  <li>Liebman — <em>A Chromatic Approach to Jazz Harmony and Melody</em></li>
  <li><a href="https://www.jazzguitar.be/blog/diminished-scale/" target="_blank" rel="noopener">jazzguitar.be — Diminished Scale ↗</a></li>
</ul>
