The Architectural Formulas Behind Major, Minor, 7th, 9th, 11th, 13th, and Altered Jazz Chords

Executive Summary

Discover how all guitar chords are built from mathematical interval formulas. Learn how stacking 3rds constructs everything from simple triads to lush 13th chords, and master Drop 2 and rootless comping shell grips used by professional guitarists.

1. The Core Triads (The 3-Note Foundation)

All harmonic structures in Western music start as three-note chords called Triads, formed by stacking major and minor thirds:

Triad Type	Formula (Intervals)	Example (Root C)	Musical Character
Major	1 - 3 - 5 (Root, Major 3rd, Perfect 5th)	C - E - G	Bright, stable, consonant
Minor	1 - ♭ 3 - 5 (Root, Minor 3rd, Perfect 5th)	C - E♭ - G	Dark, melancholic, reflective
Diminished	1 - ♭ 3 - ♭ 5 (Root, Minor 3rd, Diminished 5th)	C - E♭ - G♭	Tense, unstable, resolving
Augmented	1 - 3 - ♯ 5 (Root, Major 3rd, Augmented 5th)	C - E - G♯	Dreamy, floating, suspenseful
Sus4 / Sus2	1 - 4 - 5 or 1 - 2 - 5	C - F - G / C - D - G	Open, unresolved, modern

2. The Four Essential Seventh Chords

Adding a fourth note (a third above the 5th) creates 7th chords:

Major 7th (1 - 3 - 5 - 7): (Cmaj7: C - E - G - B) — Lush, sophisticated, romantic. Signature sound of jazz and bossa nova.
Dominant 7th (1 - 3 - 5 - ♭ 7): (C7: C - E - G - B♭) — Bluesy, driving, contains the tritone between 3 and ♭ 7.
Minor 7th (1 - ♭ 3 - 5 - ♭ 7): (Cm7: C - E♭ - G - B♭) — Mellow, soulful, funky.
Minor 7th Flat 5 / Half-Diminished (1 - ♭ 3 - ♭ 5 - ♭ 7): (Cm7♭5: C - E♭ - G♭ - B♭) — The essential ii chord in minor key cadences.

3. Extensions: 9ths, 11ths, 13ths, and Alterations

When chords extend beyond the octave, we add upper tensions:

9th Chords (1 - 3 - 5 - ♭ 7 - 9): Adds color without clutter. In guitar voicings, the 5th is commonly omitted to accommodate 4-finger grips.
The Hendrix Chord (7♯9): (1 - 3 - 5 - ♭ 7 - ♯ 9) (E7♯9: E - G♯ - D - G) — Blends the major third and sharp ninth (enharmonic minor third) simultaneously for raw psychedelic grit.
Drop 2 Voicings: Arranges chord tones by taking the second-highest voice from a closed-position chord and dropping it down an octave. This distributes notes across adjacent 4-string sets (e.g., strings 4-3-2-1 or 5-4-3-2), creating the definitive jazz comping grip.

These formulas are the raw material for the progressions in Understanding Harmonic Progressions, Roman Numerals & Cadences, and they are the shapes you will read on paper once you learn How to Read Guitar & Bass Chord Diagrams.

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Frequently Asked Questions

Q: Why do guitarists omit the 5th in extended chords?

A: The perfect 5th provides acoustic weight but zero harmonic quality (it does not define whether a chord is major, minor, or dominant). Omitting the 5th frees up fretting fingers to articulate expressive color tones like the 9th, 11th, or 13th.

Q: What is a rootless voicing?

A: In band or ensemble settings where a bass player covers root frequencies, guitarists play rootless voicings consisting purely of 3rds, 7ths, and upper extensions to avoid sonic mud in the low register.

Guitar Chord Theory: From Triads to Extended & Altered Voicings