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<h1>Intervals</h1>
<p>An interval is the distance between two notes. Intervals are the atomic vocabulary of music theory — chords are stacks of intervals, scales are a sequence of intervals, and melodies move through intervals. More practically: if you can hear intervals, you can figure out melodies by ear and understand why chords have the character they do.</p>

<h2>All Twelve Intervals</h2>

<table>
  <thead><tr><th>Semitones</th><th>Name</th><th>Abbreviation</th><th>Example from C</th><th>Sound</th></tr></thead>
  <tbody>
    <tr><td>0</td><td>Unison</td><td>P1</td><td>C → C</td><td>Same pitch — perfect consonance</td></tr>
    <tr><td>1</td><td>Minor 2nd</td><td>m2</td><td>C → D♭</td><td>Strongly dissonant, unstable — half step</td></tr>
    <tr><td>2</td><td>Major 2nd</td><td>M2</td><td>C → D</td><td>Mildly dissonant — whole step, scale motion</td></tr>
    <tr><td>3</td><td>Minor 3rd</td><td>m3</td><td>C → E♭</td><td>Soft consonance — minor quality</td></tr>
    <tr><td>4</td><td>Major 3rd</td><td>M3</td><td>C → E</td><td>Bright consonance — major quality</td></tr>
    <tr><td>5</td><td>Perfect 4th</td><td>P4</td><td>C → F</td><td>Pure, stable — though somewhat hollow</td></tr>
    <tr><td>6</td><td>Tritone</td><td>TT / A4 / d5</td><td>C → F#/G♭</td><td>Maximum dissonance — exactly half an octave</td></tr>
    <tr><td>7</td><td>Perfect 5th</td><td>P5</td><td>C → G</td><td>Strongest consonance after unison/octave — open, powerful</td></tr>
    <tr><td>8</td><td>Minor 6th</td><td>m6</td><td>C → A♭</td><td>Soft, bittersweet consonance</td></tr>
    <tr><td>9</td><td>Major 6th</td><td>M6</td><td>C → A</td><td>Warm, open consonance</td></tr>
    <tr><td>10</td><td>Minor 7th</td><td>m7</td><td>C → B♭</td><td>Mild dissonance — the jazz colour note</td></tr>
    <tr><td>11</td><td>Major 7th</td><td>M7</td><td>C → B</td><td>Sharp dissonance — one step below octave, creates tension</td></tr>
    <tr><td>12</td><td>Octave</td><td>P8</td><td>C → C</td><td>Perfect consonance — same pitch class</td></tr>
  </tbody>
</table>

<h2>Interval Quality</h2>
<p>Quality precisely specifies the interval size:</p>
<ul>
  <li><strong>Perfect</strong> — applies to unison, 4th, 5th, octave. These don't have a major/minor version — they're "perfect" in both major and minor scale contexts. A perfect 5th is always 7 semitones.</li>
  <li><strong>Major / Minor</strong> — applies to 2nds, 3rds, 6ths, 7ths. Major is always one semitone larger than minor.</li>
  <li><strong>Augmented</strong> — one semitone larger than perfect or major. Augmented 4th = 6 semitones (tritone).</li>
  <li><strong>Diminished</strong> — one semitone smaller than perfect or minor. Diminished 5th = 6 semitones (also a tritone).</li>
</ul>

<h2>The Tritone</h2>
<p>Six semitones — exactly halfway through the octave. Augmented 4th and diminished 5th are enharmonically the same interval. It's the most dissonant interval in Western music, and the engine of tonal harmony:</p>
<pre>
In G7: B (major 3rd of G7) and F (minor 7th of G7) are a tritone apart.
This tritone wants to resolve:
  B resolves up by half step → C (tonic's root)
  F resolves down by half step → E (tonic's major 3rd)

Inward resolution: F↓E, B↑C  → resolves to Cmaj (I chord)

The tritone substitution works because D♭7 has the same tritone (C♭=B, F)
just with 3rd and 7th swapped — same resolution energy, different bass.
</pre>

<h2>Consonance and Dissonance</h2>

<table>
  <thead><tr><th>Category</th><th>Intervals</th><th>Character</th></tr></thead>
  <tbody>
    <tr><td>Perfect consonance</td><td>Unison, octave, perfect 5th</td><td>Completely stable, open, powerful</td></tr>
    <tr><td>Imperfect consonance</td><td>Major/minor 3rd, major/minor 6th</td><td>Stable but warm, characterful — the sounds of chords</td></tr>
    <tr><td>Mild dissonance</td><td>Major 2nd, minor 7th, perfect 4th (in some contexts)</td><td>Some tension, but often acceptable as resting points</td></tr>
    <tr><td>Sharp dissonance</td><td>Minor 2nd, major 7th</td><td>Strong tension, wants immediate resolution</td></tr>
    <tr><td>Maximum dissonance</td><td>Tritone</td><td>Most tense interval — drives harmonic resolution</td></tr>
  </tbody>
</table>

<h2>Compound Intervals</h2>
<p>Intervals larger than an octave are compound intervals — the same class as their simple equivalent plus an octave:</p>
<pre>
9th  = octave + 2nd   (same pitch class as 2nd)
10th = octave + 3rd
11th = octave + 4th   (same pitch class as 4th)
12th = octave + 5th
13th = octave + 6th   (same pitch class as 6th)

Jazz chords (maj9, 7#11, 13) are named for compound intervals.
Cmaj9 = C E G B D — the D is a major 9th (= major 2nd an octave higher) above C.
C13 = C E G B♭ D A — the A is a major 13th (= major 6th up an octave) above C.
</pre>

<h2>Inversion of Intervals</h2>
<pre>
Invert an interval by moving the lower note up an octave (or upper note down):
  Perfect 5th → Perfect 4th       (5 + 4 = 9, perfect stays perfect)
  Major 3rd   → Minor 6th         (3 + 6 = 9, major ↔ minor)
  Major 7th   → Minor 2nd         (7 + 2 = 9)
  Tritone     → Tritone            (6 + 6 = 12 = octave — tritone inverts to itself)

Formula: semitones of interval + semitones of its inversion = 12
Quality: major ↔ minor, perfect ↔ perfect, augmented ↔ diminished
</pre>

<h2>Ear Training Reference</h2>
<p>Associate each interval with a song you know well — hearing-first learning internalises intervals much faster than drill:</p>

<table>
  <thead><tr><th>Interval</th><th>Ascending reference</th><th>Descending reference</th></tr></thead>
  <tbody>
    <tr><td>m2</td><td>"Jaws" theme, "Pink Panther" (first two notes)</td><td>"Joy to the World" (first drop)</td></tr>
    <tr><td>M2</td><td>"Happy Birthday" (first two notes), "Frère Jacques"</td><td>"Mary Had a Little Lamb" (first drop)</td></tr>
    <tr><td>m3</td><td>"Smoke on the Water" (riff), "Greensleeves"</td><td>"Hey Joe" (first two notes)</td></tr>
    <tr><td>M3</td><td>"When the Saints Go Marching In" (first leap), "E.T." (flying theme)</td><td>"Norwegian Wood" (opening)</td></tr>
    <tr><td>P4</td><td>"Here Comes the Bride," "Amazing Grace"</td><td>"Eine Kleine Nachtmusik" (first leap)</td></tr>
    <tr><td>Tritone</td><td>"The Simpsons" theme, "Maria" (West Side Story, opening leap)</td><td>"Black Sabbath" (title riff)</td></tr>
    <tr><td>P5</td><td>"Star Wars" theme (opening brass), "Twinkle Twinkle Little Star"</td><td>"Feelings" (opening)</td></tr>
    <tr><td>m6</td><td>"The Entertainer" (opening), "Because" (Beatles)</td><td>"The Entertainer" (second phrase)</td></tr>
    <tr><td>M6</td><td>"My Bonnie Lies Over the Ocean" (first leap), NBC chimes</td><td>"Nobody Knows the Trouble I've Seen"</td></tr>
    <tr><td>m7</td><td>"Star Trek" original theme (first leap), "Somewhere" (West Side Story)</td><td>"Watermelon Man" (opening)</td></tr>
    <tr><td>M7</td><td>"Take On Me" (A-ha, opening leap), "Don't Know Why" (Norah Jones)</td><td>Rare — most songs avoid descending M7</td></tr>
    <tr><td>Octave</td><td>"Somewhere Over the Rainbow" (first two syllables)</td><td>"Willow Weep for Me"</td></tr>
  </tbody>
</table>

<h2>References</h2>
<ul>
  <li>Levine — <em>The Jazz Theory Book</em></li>
  <li>Aldwell &amp; Schachter — <em>Harmony and Voice Leading</em></li>
  <li><a href="https://musictheory.net/lessons/31" target="_blank" rel="noopener">musictheory.net — Intervals ↗</a></li>
  <li><a href="https://tonedear.com" target="_blank" rel="noopener">ToneDear — ear training for intervals ↗</a></li>
</ul>
