Working with Chords

A chord is two or more tones sounding simultaneously. Chords are the vertical dimension of music — while melody moves horizontally through time, harmony stacks tones on top of each other.

Chord Construction

Chords are built by stacking intervals above a root note. The most common chord type is the triad — three notes built from alternating scale degrees (root, 3rd, 5th).

The four triad types:

Major       root + major 3rd (4) + perfect 5th (7)    Bright, stable
Minor       root + minor 3rd (3) + perfect 5th (7)    Dark, sad
Diminished  root + minor 3rd (3) + diminished 5th (6) Tense, unstable
Augmented   root + major 3rd (4) + augmented 5th (8)  Eerie, unresolved

Adding a 7th creates a seventh chord — the foundation of jazz harmony:

Dominant 7th   root + 4 + 7 + 10   Bluesy, wants to resolve (G7)
Major 7th      root + 4 + 7 + 11   Dreamy, sophisticated (Cmaj7)
Minor 7th      root + 3 + 7 + 10   Warm, mellow (Am7)
Diminished 7th root + 3 + 6 + 9    Dramatic, symmetrical

Inversions

A chord is in root position when the root is the lowest note. When a different chord tone is in the bass, the chord is inverted:

  • Root position: C E G (root in bass)

  • First inversion: E G C (3rd in bass) — notated C/E

  • Second inversion: G C E (5th in bass) — notated C/G

Inversions change the color and weight of a chord without changing its identity. First inversion sounds lighter; second inversion sounds suspended, often used as a passing chord.

For seventh chords, there’s also third inversion (7th in bass):

  • G7 in third inversion: F G B D (notated G7/F)

>>> from pytheory import Chord, Tone

>>> root   = Chord([Tone.from_string(n, system="western") for n in ["C4", "E4", "G4"]])
>>> first  = Chord([Tone.from_string(n, system="western") for n in ["E3", "G3", "C4"]])
>>> second = Chord([Tone.from_string(n, system="western") for n in ["G3", "C4", "E4"]])

>>> root.identify()
'C major'
>>> first.identify()
'C major'
>>> second.identify()
'C major'

Extended Chords

Beyond seventh chords, jazz harmony builds extended chords by continuing to stack thirds:

  • 9th chord: adds the 9th (= 2nd, one octave up)

  • 11th chord: adds the 9th and 11th (= 4th)

  • 13th chord: adds the 9th, 11th, and 13th (= 6th)

A full 13th chord contains all 7 notes of the scale! In practice, tones are usually omitted — the 5th is typically dropped first, then the 11th (which clashes with the 3rd in dominant chords).

>>> from pytheory import TonedScale

>>> scale = TonedScale(tonic="C4")["major"]

>>> cmaj9 = scale.chord(0, 2, 4, 6, 8)
>>> c13 = scale.chord(0, 2, 4, 6, 8, 10, 12)

Using the Chord Chart

PyTheory includes 144 pre-built chords (12 roots x 12 qualities):

>>> from pytheory import Fretboard

>>> fb = Fretboard.guitar()
>>> fb.chord("C")
Fingering(e=0, B=1, G=0, D=2, A=3, E=x)
>>> fb.chord("Am")
Fingering(e=0, B=1, G=2, D=2, A=0, E=x)
>>> fb.chord("G7")
Fingering(e=1, B=0, G=0, D=0, A=2, E=3)

You can also build chords directly with Chord.from_name():

>>> from pytheory import Chord

>>> Chord.from_name("G7").identify()
'G dominant 7th'
>>> Chord.from_name("Ddim").identify()
'D diminished'

Available qualities:

Quality

Intervals

Example tones (from C)

""

4, 7

C E G (major triad)

"maj"

4, 7

C E G (explicit major)

"m"

3, 7

C Eb G (minor triad)

"5"

7

C G (power chord)

"7"

4, 7, 10

C E G Bb (dominant 7th)

"9"

4, 7, 10, 14

C E G Bb D (dominant 9th)

"dim"

3, 6

C Eb Gb (diminished)

"m6"

3, 7, 9

C Eb G A (minor 6th)

"m7"

3, 7, 10

C Eb G Bb (minor 7th)

"m9"

3, 7, 10, 14

C Eb G Bb D (minor 9th)

"maj7"

4, 7, 11

C E G B (major 7th)

"maj9"

4, 7, 11, 14

C E G B D (major 9th)

 
>>> from pytheory import CHARTS
>>> chart = CHARTS["western"]

>>> chart["C"].acceptable_tone_names
('C', 'E', 'G')

>>> chart["Cm7"].acceptable_tone_names
('C', 'Eb', 'G', 'Bb')

Building Chords

Several convenience constructors make chord creation concise:

>>> from pytheory import Chord

>>> Chord.from_tones("C", "E", "G").identify()
'C major'
>>> Chord.from_tones("A", "C", "E").identify()
'A minor'

>>> Chord.from_name("Am7").identify()
'A minor 7th'
>>> Chord.from_name("G7").identify()
'G dominant 7th'

>>> Chord.from_intervals("C", 4, 7).identify()
'C major'
>>> Chord.from_intervals("G", 4, 7, 10).identify()
'G dominant 7th'

>>> Chord.from_midi_message(60, 64, 67).identify()
'C major'

>>> len(Chord.from_name("C"))
3
>>> "C" in Chord.from_name("C")
True

Intervals

The intervals property returns semitone distances between adjacent tones — these are musically meaningful and octave-invariant:

>>> Chord.from_tones("C", "E", "G").intervals
[4, 3]

>>> Chord.from_tones("C", "Eb", "G").intervals
[3, 4]

Consonance and Dissonance

Consonance is the perception of stability and “pleasantness” when tones sound together. Dissonance is the perception of tension and roughness. Neither is inherently good or bad — music needs both.

Harmony Score

The harmony property measures consonance using frequency ratio simplicity. The insight dates back to Pythagoras (6th century BC): intervals whose frequencies form simple integer ratios sound consonant.

Interval

Ratio

Why it sounds “good”

Octave

2:1

Every 2nd wave aligns

Perfect 5th

3:2

Every 3rd wave aligns

Perfect 4th

4:3

Every 4th wave aligns

Major 3rd

5:4

Every 5th wave aligns

Minor 3rd

6:5

Every 6th wave aligns

Tritone

45:32

Waves rarely align

 
>>> from pytheory import Chord, Tone
>>> C4 = Tone.from_string("C4", system="western")
>>> G4 = Tone.from_string("G4", system="western")

>>> fifth = Chord([C4, G4])
>>> tritone = Chord([C4, C4 + 6])
>>> fifth.harmony > tritone.harmony
True

Dissonance Score

The dissonance property uses the Plomp-Levelt roughness model (1965). When two frequencies are close together, their sound waves interfere and produce rapid amplitude fluctuations called beating. This beating is perceived as roughness — the physiological basis of dissonance.

The roughness depends on the frequency difference relative to the critical bandwidth of the human ear (~25% of the frequency at that register). Maximum roughness occurs when the difference equals the critical bandwidth.

>>> E4 = Tone.from_string("E4", system="western")
>>> octave = Chord([C4, C4 + 12])
>>> third = Chord([C4, E4])
>>> octave.dissonance < third.dissonance
True

Beat Frequencies

When two tones with slightly different frequencies are played together, you hear a pulsing at the beat frequency: |f1 - f2| Hz.

  • < 1 Hz: Slow pulsing, used for tuning instruments

  • 1–15 Hz: Audible rhythmic beating

  • 15–30 Hz: Perceived as buzzing/roughness

  • > 30 Hz: No longer beating — becomes part of the timbre

>>> A4 = Tone.from_string("A4", system="western")
>>> chord = Chord([A4, A4 + 7, A4 + 12])

>>> chord.beat_frequencies
[...]

>>> round(chord.beat_pulse, 1)
219.3

Transposition

Shift an entire chord up or down by any number of semitones:

>>> Chord.from_name("C").transpose(7).identify()
'G major'

>>> Chord.from_name("Am7").transpose(-2).identify()
'G minor 7th'

Chord Manipulation

Add or remove individual tones from a chord:

>>> from pytheory import Chord, Tone

>>> c_major = Chord.from_tones("C", "E", "G")

>>> b4 = Tone.from_string("B4", system="western")
>>> cmaj7 = c_major.add_tone(b4)
>>> cmaj7.identify()
'C major 7th'

>>> c_again = cmaj7.remove_tone("B")
>>> c_again.identify()
'C major'

Chord Identification

Give PyTheory any set of tones and it will tell you what chord it is. It tries every tone as a potential root and matches the interval pattern against 17 known chord types (triads, 7ths, 9ths, sus, power chords).

>>> from pytheory import Chord

>>> Chord.from_tones("A", "C", "E").identify()
'A minor'
>>> Chord.from_tones("G", "B", "D", "F").identify()
'G dominant 7th'

>>> Chord.from_tones("E", "G", "C").identify()
'C major'

>>> Chord.from_tones("Bb", "D", "F").identify()
'Bb major'

Enharmonic spellings are fully supported — Cb, Fb, E#, B#, double sharps/flats, and unicode symbols (see Working with Tones for details):

>>> Chord.from_tones("Cb", "Eb", "Gb").identify()
'B minor'

You can also access the root and quality separately:

>>> chord = Chord.from_name("Am7")
>>> chord.root
<Tone A4>
>>> chord.quality
'minor 7th'

Harmonic Analysis

Roman numeral analysis labels each chord by its function within a key. This is how musicians describe chord progressions independent of key — “I-IV-V” means the same thing in C major (C-F-G) as in G major (G-C-D).

>>> from pytheory import Chord, Tone

>>> C4 = Tone.from_string("C4", system="western")
>>> E4 = Tone.from_string("E4", system="western")
>>> G4 = Tone.from_string("G4", system="western")

>>> Chord([C4, E4, G4]).analyze("C")
'I'
>>> Chord.from_tones("D", "F", "A").analyze("C")
'ii'
>>> Chord([G4, G4+4, G4+7]).analyze("C")
'V'
>>> Chord([G4, G4+4, G4+7, G4+10]).analyze("C")
'V7'

Tension and Resolution

Tension is what makes music move forward. Without it, there’s no desire to resolve — no drama, no narrative. The tension property quantifies this based on:

  • Tritones (6 semitones): the most unstable interval. The tritone between the 3rd and 7th of a dominant chord (e.g. B and F in G7) creates the strongest pull toward resolution.

  • Minor 2nds: semitone clashes that add bite and urgency.

  • Dominant function: the specific combination of a major 3rd and minor 7th above the root — the hallmark of the V7 chord.

>>> c_major = Chord([C4, E4, G4])
>>> c_major.tension['score']
0.0
>>> c_major.tension['tritones']
0

>>> g7 = Chord([G4, G4+4, G4+7, G4+10])
>>> g7.tension['score']
0.6
>>> g7.tension['tritones']
1
>>> g7.tension['has_dominant_function']
True

Voice Leading

Voice leading is the art of connecting chords smoothly. Instead of jumping all voices to new positions, good voice leading moves each note the minimum distance to reach the next chord. Bach’s chorales are the gold standard — every voice moves by step whenever possible.

>>> c_maj = Chord.from_tones("C", "E", "G")
>>> f_maj = Chord.from_tones("F", "A", "C")

>>> for src, dst, motion in c_maj.voice_leading(f_maj):
...     print(f"{src} -> {dst}  ({motion:+d} semitones)")
G4 -> A4  (+2 semitones)
E4 -> F4  (+1 semitones)
C4 -> C4  (+0 semitones)

Tritone Substitution

In jazz harmony, any dominant chord can be replaced by the dominant chord a tritone (6 semitones) away. This works because the two chords share the same tritone interval — the 3rd and 7th simply swap roles.

Common tritone subs: G7 <-> Db7, C7 <-> F#7, D7 <-> Ab7.

>>> from pytheory import Chord

>>> g7 = Chord.from_name("G7")
>>> sub = g7.tritone_sub()
>>> sub.identify()
'C# dominant 7th'

The Overtone Series

Every musical tone is actually a stack of frequencies — the fundamental plus its overtones (harmonics). The overtone series is nature’s chord: it contains the octave, perfect fifth, perfect fourth, major third, and more, in that order.

This is why consonance exists. When you play C and G together, the overtones of C already contain G. The two tones share acoustic energy, reinforcing each other. A dissonant interval like C and C# shares almost no overtones — the waves clash.

>>> from pytheory import Tone

>>> a4 = Tone.from_string("A4", system="western")
>>> [round(f, 1) for f in a4.overtones(8)]
[440.0, 880.0, 1320.0, 1760.0, 2200.0, 2640.0, 3080.0, 3520.0]

Chord Symbols

The symbol property returns compact lead-sheet notation, while from_symbol() parses any standard chord symbol — no lookup table needed:

>>> Chord.from_tones("C", "E", "G").symbol
'C'
>>> Chord.from_name("Am7").symbol
'Am7'
>>> Chord.from_symbol("F#m7b5").identify()
'F# half-diminished 7th'
>>> Chord.from_symbol("Bbmaj9").symbol
'Bbmaj9'

Slash Chords

Slash chords place a specific note in the bass below the chord. They’re written as Chord/Bass in lead sheets:

>>> c = Chord.from_symbol("C")
>>> c_over_g = c.slash("G")
>>> c_over_g.slash_name
'C/G'
>>> c.slash("E").slash_name
'C/E'

Drop Voicings

Drop voicings are standard arranging techniques for spreading chord tones across registers:

  • Close voicing — all tones packed within one octave

  • Open voicing — alternating tones raised an octave for wider spacing

  • Drop 2 — second-highest voice dropped an octave (standard jazz guitar)

  • Drop 3 — third-highest voice dropped an octave

>>> cmaj7 = Chord.from_symbol("Cmaj7")
>>> cmaj7.close_voicing()
<Chord C major 7th>
>>> cmaj7.open_voicing()
<Chord C major 7th>
>>> cmaj7.drop2()
<Chord C major 7th>

open_voicing() takes the close voicing and raises every other non-root tone by an octave, spreading the chord across two octaves. The result is a wider, more spacious sound — common in orchestral writing and piano ballads where you want the harmony to breathe.

Chord Extensions

The extensions() method suggests available extensions (9th, 11th, 13th) that don’t clash with existing chord tones:

>>> from pytheory import Chord, TonedScale
>>> cm = Chord.from_symbol("C")
>>> cm.extensions()
[...]

>>> # Filter extensions against a scale for diatonic correctness:
>>> scale = TonedScale(tonic="C4")["major"]
>>> cm.extensions(scale=scale)
[...]

Borrowed Chord Analysis

analyze() now recognizes chromatic chords from modal interchange, labeling them with flat-degree prefixes:

>>> Chord.from_symbol("Ab").analyze("C", "major")
'bVI'
>>> Chord.from_symbol("Bb").analyze("C", "major")
'bVII'

Figured Bass

Figured bass is the classical notation for chord inversions — numbers below the bass note describing the intervals above it. It’s how Bach, Handel, and every Baroque composer communicated harmony.

>>> from pytheory import Chord, Tone

>>> root = Chord([Tone.from_string("C4"), Tone.from_string("E4"), Tone.from_string("G4")])
>>> root.figured_bass
''

>>> first_inv = Chord([Tone.from_string("E3"), Tone.from_string("G3"), Tone.from_string("C4")])
>>> first_inv.figured_bass
'6'

>>> second_inv = Chord([Tone.from_string("G3"), Tone.from_string("C4"), Tone.from_string("E4")])
>>> second_inv.figured_bass
'6/4'

For seventh chords: root position → "7", first inversion → "6/5", second inversion → "4/3", third inversion → "2".

Combine with Roman numeral analysis using analyze_figured():

>>> first_inv.analyze_figured("C")
'I6'

Pitch Class Sets

Pitch class set theory is the framework for analyzing atonal and post-tonal music. It reduces any collection of notes to abstract pitch classes (0–11, where C=0), finds the most compact form, and catalogs it with a Forte number.

If you’re studying Schoenberg, Webern, Bartók, or any 20th-century music that doesn’t follow traditional harmony, this is the tool.

>>> Chord.from_tones("C", "E", "G").pitch_classes
{0, 4, 7}

>>> Chord.from_tones("C", "E", "G").prime_form
(0, 3, 7)

>>> Chord.from_tones("A", "C", "E").prime_form
(0, 3, 7)

Major and minor triads share the same prime form — they’re inversions of each other in pitch class space.

The normal form is the intermediate step — the most compact ascending arrangement of pitch classes before transposition. It preserves the actual pitch classes (not transposed to 0), so it tells you which specific notes are in the set:

>>> Chord.from_tones("C", "E", "G").normal_form
(0, 4, 7)

>>> Chord.from_tones("A", "C", "E").normal_form
(9, 0, 4)

Normal form keeps the original pitch classes; prime form transposes to 0 for comparison. Use normal_form when you care about which notes, prime_form when you care about the abstract shape.

>>> Chord.from_tones("C", "E", "G").forte_number
'3-11'

>>> Chord.from_tones("C", "E", "G", "B").forte_number
'4-20'

>>> Chord.from_tones("C", "E", "G#").forte_number
'3-12'

Chords are the vertical dimension of music – melody tells you where you’re going, but harmony tells you how it feels to be there. Between construction, identification, voice leading, tension analysis, and pitch class sets, you’ve got tools to look at any chord from every angle. Pick a song you love, grab its chords, and start asking questions.