Every chord symbol in a jazz fake book — C9, Dm11, G13 — is built on compound intervals. Master the vocabulary of extended harmony and modern chord notation makes complete sense.
Interval Mastery Series
The interval guides up to this point have stayed within a single octave. But intervals can keep going — two notes can be more than an octave apart, earning new names: 9th, 10th, 11th, 12th, 13th. These are called compound intervals, and they're where music theory meets the real-world vocabulary of jazz and pop chord notation.
Open any jazz fake book and you'll see C9, Dm11, G13, Fmaj7♯11. Understanding compound intervals is the key to understanding modern chord notation. For pianists, compound intervals matter practically: extended chords (9ths, 11ths, 13ths) can pile up 6–7 notes — too many for comfortable hand position without careful voicing choices.
A compound interval is any interval larger than an octave. The simplest case is the 9th — one octave plus one whole step. The numerical naming continues past 8, following the same letter-counting logic:
| Compound interval | = Simple + octave | Number rule |
|---|---|---|
| 9th | 2nd + octave | simple number + 7 |
| 10th | 3rd + octave | simple number + 7 |
| 11th | 4th + octave | simple number + 7 |
| 12th | 5th + octave | simple number + 7 |
| 13th | 6th + octave | simple number + 7 |
| 14th | 7th + octave | simple number + 7 |
| 15th (double octave) | octave + octave | simple number + 7 |
Qualities (major, minor, perfect, augmented, diminished) carry over unchanged. A major 2nd plus an octave = a major 9th. A perfect 4th plus an octave = a perfect 11th.
In practice, only certain compound intervals appear regularly: 9ths, 11ths, and 13ths. The 10th, 12th, and 14th appear occasionally but are usually written as basic 3rd, 5th, or 7th. The reason is the chord-extension system: a basic 7th chord stacks degrees 1–3–5–7. Extensions continue the odd-numbered pattern: 1–3–5–7–9–11–13. The 9th, 11th, and 13th represent odd-numbered scale degrees not already in a 7th chord — so they get their own names.
The intervals below are the complete working vocabulary. Each carries the same quality as its simple-interval parent.
| Name | Abbr. | Semitones | = Simple + octave | Notes |
|---|---|---|---|---|
| Minor 9th | m9 | 13 | m2 + octave | |
| Major 9th | M9 | 14 | M2 + octave | |
| Augmented 9th | A9 | 15 | A2 + octave | the ♯9 in jazz alterations |
| Minor 10th | m10 | 15 | m3 + octave | |
| Major 10th | M10 | 16 | M3 + octave | |
| Perfect 11th | P11 | 17 | P4 + octave | |
| Augmented 11th | A11 | 18 | A4 + octave | the ♯11 — Lydian sound in jazz |
| Perfect 12th | P12 | 19 | P5 + octave | |
| Minor 13th | m13 | 20 | m6 + octave | the ♭13 in altered chords |
| Major 13th | M13 | 21 | M6 + octave |
Mental shortcut: subtract the octave
To identify any compound interval, subtract an octave from the upper note. Example: C4 to A5 — subtract an octave from A5 to get A4. C to A = major 6th. Add the octave back = major 13th.
This subtract-add-back technique is faster than counting all the way through from root to a note two octaves up.
Select a compound interval. Toggle "Reduce" to see and hear the equivalent simple interval (upper note dropped one octave).
Compound Name
Major 9th
14 semitones
Simple Equivalent
Major 2nd
M quality preserved
Formula
M2 + octave
= Major 9th
Compound Interval Calculator
Pick two notes more than an octave apart. See the compound name and the simple interval it's built from.
The most important property of compound intervals: they function almost identically to their simple-interval equivalents. A major 9th and a major 2nd produce the same harmonic flavor — the "major-second sound" — just with the upper note voiced an octave higher. The compound version sounds more open (registral spread) but retains the same underlying quality.
This has a direct practical consequence. When you build a chord with a 9th extension, the 9th provides essentially the same harmonic flavor as if you'd added a 2nd to the chord — just voiced an octave higher to avoid clashing with the root in the same register. Adding a 2nd directly to a C major triad creates an acoustically congested cluster (C–D–E–G). Voicing the D up an octave (C–E–G–D) thins out the cluster and produces a clean, open sound. Same harmonic content — cleaner result.
The same logic applies to 11ths and 13ths. An 11th voices the 4th up an octave; a 13th voices the 6th up an octave. This is why chord notation distinguishes "add9" from "add2" and "sus2": Cadd9 has D voiced above (C–E–G–D), while Csus2 has D in the same register as C (C–D–G). Related but different effects — Cadd9 is lush and open, Csus2 is suspended and compressed.
Cadd2 (or Csus2)
C – D – E – G
D in same register. Dense, compressed.
Cadd9
C – E – G – D (upper octave)
D voiced up an octave. Spacious, open.
Chord symbols indicate the root, quality, and the highest extension used. Lower extensions are implied. "C9" means C dominant 9th: the 7th is implied because the chord-stack logic runs 1–3–5–7–9 — you cannot have a 9th without the 7th below it. C9 contains: C–E–G–B♭–D.
| Symbol | Full name | Notes | 7th quality |
|---|---|---|---|
| C9 | Dominant 9th | C–E–G–B♭–D | minor 7th (implied) |
| Cmaj9 | Major 9th | C–E–G–B–D | major 7th (implied) |
| Cm9 | Minor 9th | C–E♭–G–B♭–D | minor 7th (implied) |
| C11 | Dominant 11th | C–E–G–B♭–D–F | minor 7th + 9th (implied) |
| Cm11 | Minor 11th | C–E♭–G–B♭–D–F | minor 7th + 9th (implied) |
| C13 | Dominant 13th | C–E–G–B♭–D–F–A | minor 7th + 9th + 11th (implied) |
| Cmaj13 | Major 13th | C–E–G–B–D–F–A | major 7th + 9th + 11th (implied) |
Alterations are written after the base symbol: C7♯9 (the "Hendrix chord"), C7♭9, C7♯11, C7♭13. Multiple alterations stack: C13♭9, C9♯11.
Omissions are indicated with "no": C9(no5), C13(no11). Pianists routinely omit the 5th from extended chords — it's the least harmonically informative note and its omission opens up voicing space. When an omission is unmarked, experienced players infer it from context.
Chord Symbol Decoder
Type any chord symbol. See its full note content and a practical piano voicing.
The 9th is the most common extension. It adds richness without significantly complicating the chord's basic harmonic identity.
C–E–G–B–D
Lush, spacious, cool jazz sensibility. Opens many ballads and bossa nova progressions. Signature voicing: left hand C in bass, right hand E–G–B–D.
C–E–G–B♭–D
Bluesy and jazzy — the heart of soul, funk, and R&B. The tritone between E (3rd) and B♭ (7th) is the dominant engine; D (9th) sits on top as color. Signature voicing: left hand C, right hand E–B♭–D.
C–E♭–G–B♭–D
Moody and introspective. One of the most commonly-used extended chords in modern songwriting and neo-soul. Signature voicing: left hand C, right hand E♭–G–B♭–D.
C–E–G–B♭–D♯
Contains both the major 3rd (E) and the ♯9 (D♯), which is enharmonically the minor 3rd — giving it a dual-quality, ambiguous character. Immortalized in Jimi Hendrix's "Purple Haze." Particularly powerful in rock-funk contexts. The ♯9 should generally be voiced above the 3rd.
C–E–G–B♭–D♭
More sinister and tense than the regular dominant 9th. Resolves with exceptional force to a minor tonic. Common in tango, flamenco, and dramatic jazz cadences.
Practical note
The 9th should generally be voiced above the 3rd and 7th. Placing the 9th below the 3rd creates a harsh minor 9th interval (13 semitones) that clashes with itself. Voicing order from bottom to top in the right hand: 3rd → 7th → 9th, or 7th → 9th → 3rd (shell voicing above).
The 11th requires more delicate handling. The interval between the 3rd and the natural 11th is a minor 9th — a sharp dissonance when voiced in the same register. This shapes how different chord qualities interact with the 11th.
C–E♭–G–B♭–D–F
Much more usable than the major 11th. The interval between the minor 3rd (E♭) and the 11th (F) is a whole step — smooth and open, not a half-step clash. Spacious, contemplative, harmonically rich. A workhorse chord in jazz and neo-soul.
C–E–G–B–D–F
Theoretically complete but rarely used in full voicing. The F (natural 11th) creates a harsh half-step clash with E (major 3rd). In practice: either omit the 3rd (creating a sus4-like feel) or raise the 11th to ♯11 (see below). The complete 6-note voicing almost never appears as written.
C–E–G–B♭–D–F
Same 3rd-vs-11th problem as the major 11th. On a chord chart, "C11" is usually interpreted by knowledgeable pianists as C7sus — the 3rd is omitted, leaving a sus4 quality above a dominant 7th. This is the standard professional interpretation.
C–E–G–B♭–D–F♯
When the 11th is raised a half-step, the clash disappears. F♯ forms a tritone above C (the Lydian characteristic note) but doesn't clash with E. This is the Lydian dominant sound — jazz fusion, progressive rock, modern film scoring. "Mysterious and expansive" quality. One of the most sophisticated and beautiful chord colors in jazz.
Practical principle for 11th chords
In major and dominant chords: omit the 3rd (creating a sus4 feel) or raise the 11th to ♯11 (Lydian feel). In minor chords: keep both the 3rd and the 11th — they're a whole step apart and sound clean together. This single principle resolves nearly every voicing question that comes up with 11th chords.
The 13th is the most expressive extension and the most harmonically loaded. A theoretical C13 has seven notes (C–E–G–B♭–D–F–A) — impossible to play as a complete voicing with two hands. It is always voiced selectively, using only the most harmonically essential notes.
Root, 3rd, 7th, 13th (practical)
Theoretical: C–E–G–B♭–D–F–A (theoretical)
Common voicing: left hand C, right hand E–B♭–D–A or E–B♭–A. The 13th (A) gives a warm, full sound. Common in big-band jazz, R&B, and gospel. Typically omits the 11th to avoid the 3rd-vs-11th clash.
Root, 3rd, 7th, 13th (practical)
Theoretical: C–E–G–B–D–F–A (theoretical)
Dreamy and sophisticated — bossa nova, ballads, sophisticated pop. Common voicing: left hand C, right hand E–B–D–A or B–D–A (omitting 3rd for a more open sound).
Root, ♭3rd, 7th, 13th (practical)
Theoretical: C–E♭–G–B♭–D–F–A (theoretical)
The natural A (major 6th above C) introduces Dorian mode flavor rather than natural minor's A♭. Less common but appears in jazz and progressive contexts. The Dorian ♭3 + natural 6 combination is characteristic.
C–E–G–B♭–A♭
Darker and more tense than a standard dominant 13th. The ♭13 (A♭) sits a half-step above the 5th, creating an altered sound that resolves powerfully to a minor tonic. Part of the "altered dominant" family: C7alt.
Practical voicing formula for 13th chords
Include: root (in bass), 3rd (essential for quality), 7th (essential for family), 13th (the named extension). Omit: 5th (redundant) and 11th (clashes with 3rd). Result: 4-note right-hand voicing + root in bass = 5 notes total. Clean, expressive, playable.
The fundamental challenge: not every note can be played simultaneously without producing harmonic mud. Extended chords require deliberate selection.
Seven voicing principles
Root in the bass
Exception: rootless voicings in jazz combo contexts, where the bassist covers it.
3rd is essential
Determines major vs. minor quality. Never omit unless you want an ambiguous or sus quality.
7th is essential for any chord above a triad
Distinguishes major 7th, dominant 7th, and minor 7th families. The foundation of extended harmony.
5th is most expendable
The 5th adds little harmonic information beyond the root and 3rd. Omit it to create space for extensions.
Named extension is essential
The 9th, 11th, or 13th is the defining feature of the chord. It must be present.
Lower extensions can be omitted
In a 13th chord, the 11th is usually omitted (clashes with 3rd). The 9th is optional but adds richness.
Avoid close intervals in low register
Bass register needs space. Close intervals (2nds, 3rds) in the bass register create mud. Wide spacing below, closer intervals above.
Two standard approaches
Rootless voicing
Jazz combo / band context
Root omitted — the bassist plays it. Right hand plays 3–7–9 or 3–7–9–13. Sophisticated, modern-sounding. Common 4-note patterns: 3–7–9–13 or 7–9–3–5 (up an octave). This is the foundation of jazz piano comp technique.
Root-position voicing
Solo piano / pop / classical
Root in bass (left hand), right hand plays 3–7 minimum plus extensions. Standard for pop, classical, solo jazz. Left hand can play root + 7th (shell), then right hand adds 9th and 13th above.
Voice Any Extended Chord Cleanly
Pick a chord. See multiple voicing options. Hear how each one sits on the piano.
Use a 9th when
Use an 11th when
Use a 13th when
Use altered extensions (♯9, ♭9, ♯11, ♭13) when
Use simple chords (no extensions) when
Hear the Same Progression with Different Extensions
Same chord changes. Different complexity levels. Hear how extensions transform the texture.
One last thought
Compound intervals are where music theory connects to the practical vocabulary of jazz, pop, and modern songwriting. Every chord chart in a fake book uses them. Once you can read 9, 11, and 13 chord symbols fluently — knowing which notes they imply, which to voice, which to omit, and which to alter — the entire landscape of contemporary harmony opens up.
The journey from "what's a major 9th?" to "I can voice this Cm11 in three different ways" takes practice, but it's a finite journey. With consistent attention to voicing principles, the vocabulary becomes yours. From there, jazz, R&B, gospel, and modern pop harmony is at your fingertips.
Interval Mastery Series