Music Theory Fundamentals¶

This page covers the essential concepts of music theory — the framework behind everything PyTheory does.

Sound and Pitch¶

All sound is vibration. When an object vibrates, it pushes air molecules back and forth, creating pressure waves that travel to your ears. The speed of this vibration — measured in cycles per second (Hertz, Hz) — determines the pitch you hear.

20 Hz: the lowest pitch most humans can hear
60–250 Hz: the range of the human voice (speaking)
261.63 Hz: middle C (C4)
440 Hz: the concert pitch tuning standard A (A4)
4186 Hz: the highest C on a piano (C8)
20,000 Hz: the upper limit of human hearing

The relationship between pitch and frequency is logarithmic — each octave doubles the frequency. This means the distance from A3 (220 Hz) to A4 (440 Hz) is 220 Hz, but the distance from A4 to A5 (880 Hz) is 440 Hz. Both sound like “one octave” to our ears.

Why Twelve Notes?¶

The Western chromatic scale has 12 notes per octave. This isn’t arbitrary — it emerges from the physics of vibrating strings and air columns.

The harmonic series is the sequence of frequencies produced when a string vibrates: f, 2f, 3f, 4f, 5f… The relationships between these harmonics create the intervals we perceive as consonant:

2:1 = octave (the most fundamental)
3:2 = perfect fifth
4:3 = perfect fourth
5:4 = major third
6:5 = minor third

If you stack perfect fifths (multiply by 3/2 repeatedly) and reduce to within one octave, you get 12 roughly evenly-spaced notes before the cycle almost closes. The tiny gap where it doesn’t close perfectly is the Pythagorean comma — the reason we need temperament.

from pytheory import Tone

# Walk the circle of fifths — all 12 notes
c = Tone.from_string("C4", system="western")
[t.name for t in c.circle_of_fifths()]
# ['C', 'G', 'D', 'A', 'E', 'B', 'F#', 'C#', 'G#', 'D#', 'A#', 'F']

Other cultures divide the octave differently: Indonesian gamelan uses 5 or 7 unequal divisions; Indian classical music theoretically has 22 shrutis (microtones); Arabic maqam uses quarter-tones.

Intervals: The Atoms of Music¶

An interval is the distance between two pitches. Intervals are the building blocks of everything — melodies are sequences of intervals, chords are stacks of intervals, and scales are patterns of intervals.

Every interval has two properties:

Size (how many scale steps):

Unison → 2nd → 3rd → 4th → 5th → 6th → 7th → Octave

Quality (exact number of semitones):

Perfect:     unison (0), 4th (5), 5th (7), octave (12)
Major:       2nd (2), 3rd (4), 6th (9), 7th (11)
Minor:       2nd (1), 3rd (3), 6th (8), 7th (10)
Augmented:   one semitone larger than perfect or major
Diminished:  one semitone smaller than perfect or minor

The “perfect” intervals (unison, 4th, 5th, octave) are called perfect because they appear in both major AND minor scales unchanged. They’ve been considered consonant across virtually all musical cultures throughout history.

The tritone (augmented 4th / diminished 5th = 6 semitones) divides the octave exactly in half. Medieval theorists called it diabolus in musica (“the devil in music”) because of its extreme instability. Today it’s the foundation of dominant harmony and the blues.

Keys and Key Signatures¶

A key is a group of notes that form the tonal center of a piece. The key of C major uses only the white keys on the piano: C D E F G A B. The key of G major uses the same notes except F becomes F#.

Key signatures tell you which notes are sharped or flatted throughout a piece. They follow the circle of fifths:

Sharp keys (add one sharp per step clockwise):

C major:  no sharps or flats
G major:  F#
D major:  F# C#
A major:  F# C# G#
E major:  F# C# G# D#
B major:  F# C# G# D# A#

Flat keys (add one flat per step counter-clockwise):

C major:  no sharps or flats
F major:  Bb
Bb major: Bb Eb
Eb major: Bb Eb Ab
Ab major: Bb Eb Ab Db
Db major: Bb Eb Ab Db Gb

The order of sharps is always F C G D A E B (Father Charles Goes Down And Ends Battle). The order of flats is the reverse: B E A D G C F.

Harmony: How Chords Work¶

Harmony is the art of combining tones simultaneously. While melody is horizontal (tones in sequence), harmony is vertical (tones stacked).

The simplest harmony is the triad — three notes built by stacking thirds. The quality of each third determines the chord type:

Major triad = major 3rd + minor 3rd (e.g. C-E-G)
Minor triad = minor 3rd + major 3rd (e.g. C-Eb-G)
Diminished triad = minor 3rd + minor 3rd (e.g. B-D-F)
Augmented triad = major 3rd + major 3rd (e.g. C-E-G#)

In any major key, the triads built on each scale degree always follow the same pattern:

Degree   Quality        Function
I        Major          Tonic (home)
ii       Minor          Pre-dominant
iii      Minor          Tonic substitute
IV       Major          Subdominant (departure)
V        Major          Dominant (tension, wants to go home)
vi       Minor          Tonic substitute, relative minor
vii°     Diminished     Dominant substitute (leading tone chord)

This pattern is the DNA of Western harmony. Pop songs, classical sonatas, jazz standards, and church hymns all derive from it.

Functional Harmony¶

Chords don’t just have names — they have functions:

Tonic function (I, iii, vi): stability, rest, home
Subdominant function (ii, IV): motion away from home
Dominant function (V, vii°): tension, desire to return home

The most fundamental progression in Western music is T → S → D → T (tonic → subdominant → dominant → tonic). The classic I-IV-V-I is exactly this pattern. Every “Louie Louie” and every Bach chorale follows this basic tonal gravity.

from pytheory import TonedScale

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

# The I-IV-V-I progression
I  = scale.triad(0)   # C major — home
IV = scale.triad(3)   # F major — departure
V  = scale.triad(4)   # G major — tension
# I again               # C major — resolution

The Dominant Seventh¶

The most important chord in tonal music is the dominant seventh — the V7 chord. In C major, this is G-B-D-F. It contains:

A leading tone (B) that pulls up to the tonic (C) by half step
A tritone (B-F) that wants to resolve inward (B→C, F→E)
The dominant note (G) that falls to the tonic by a fifth

This combination creates the strongest possible pull toward resolution. When you hear V7→I, you feel arrival.

from pytheory import Chord, Tone

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

g7 = Chord([G4, G4+4, G4+7, G4+10])   # G B D F
g7.identify()                           # 'G dominant 7th'
g7.tension['has_dominant_function']      # True
g7.tension['tritones']                  # 1

c_major = Chord([C4, C4+4, C4+7])      # C E G
c_major.tension['score']                # 0.0 — fully resolved

Rhythm and Meter¶

While PyTheory focuses on pitch, rhythm is the other half of music.

Rhythm is the pattern of durations. Meter is the recurring pattern of strong and weak beats that organizes rhythm.

4/4 time: the most common meter. Strong-weak-medium-weak. Used in rock, pop, hip-hop, most Western music.
3/4 time: waltz time. Strong-weak-weak. A lilting, circular feel.
6/8 time: compound duple. Two groups of three. Irish jigs, many ballads.
12/8 time: compound quadruple. Four groups of three. Slow blues, doo-wop, gospel. Has a triplet feel over a 4/4 pulse — the shuffle groove of “Stormy Monday” and “Oh! Darling.”
5/4 time: asymmetric. “Take Five” by Dave Brubeck. Creates constant forward momentum because it never fully settles.
7/8 time: common in Balkan folk music. Often felt as 2+2+3 or 3+2+2.

The Physics of Consonance¶

Why do some intervals sound “good” and others “bad”? The answer lies in the physics of sound waves and the Plomp-Levelt model of sensory dissonance.

When two frequencies are related by a simple ratio (like 3:2 for a perfect fifth), their waveforms align regularly. The combined wave is smooth and periodic — the brain perceives this as consonant.

When two frequencies are related by a complex ratio (like 45:32 for a tritone), their waveforms rarely align. The combined wave is irregular and the brain perceives roughness — dissonance.

But consonance and dissonance are also cultural. The major third (5:4) was considered dissonant in medieval European music but consonant since the Renaissance. The tritone was forbidden in church music but is the foundation of blues and jazz. Indonesian gamelan embraces beating between paired instruments as a core aesthetic.

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")

# The overtone series — the fifth is "built into" every tone
C4.overtones(6)
# [261.63, 523.25, 784.88, 1046.50, 1308.13, 1569.75]
# 3rd harmonic (784.88) ≈ G5 (783.99) — a perfect fifth

# Consonance: simple frequency ratios score high
fifth = Chord([C4, G4])           # 3:2 ratio
tritone = Chord([C4, C4 + 6])     # 45:32 ratio
fifth.harmony > tritone.harmony   # True

# Dissonance: Plomp-Levelt roughness model
# An octave has low roughness (frequencies far apart)
# A major 3rd has more roughness (closer frequencies)
octave = Chord([C4, C4 + 12])
third = Chord([C4, E4])
octave.dissonance < third.dissonance   # True

# Tension: tritones and dominant function
c_major = Chord([C4, E4, G4])
c_major.tension['score']              # 0.0 — fully resolved

g7 = Chord([G4, G4+4, G4+7, G4+10])  # G dominant 7th
g7.tension['score']                   # 0.6 — wants to resolve
g7.tension['tritones']                # 1 (B-F)
g7.tension['has_dominant_function']    # True

# Beat frequencies — the pulsing between close pitches
g7.beat_frequencies
# [(tone_a, tone_b, hz), ...] sorted by frequency