This page is a spine, not a finished book. Parts I–V cover the fundamentals you need before anything else makes sense: pitch, rhythm, chords, and what chords do. The Roadmap at the bottom lists the sections we haven't written yet — each is a placeholder with a stable anchor, so new material drops in without reshuffling anything above it.
Sound is on. Every ▶ plays through the Web Audio API — a plain synth tone, no samples, nothing downloaded. Click a note name, a chord row, or a key on the keyboard to hear it. Nothing plays until you ask it to.
IWhat tonal harmony is
The one paragraph that defines the whole subject, and the four properties that make Bach and a pop song members of the same family.
Harmony is the sound that results when two or more pitches are performed simultaneously. It is the vertical aspect of music, produced by combining the components of the horizontal aspect (melody and rhythm).
Much of today's popular music is based on tonal harmony, just as Bach's music was — which means the two have a good deal in common. Four properties do the work:
Property 1
A tonal center
A pitch class that provides a center of gravity — the note the music leans toward and comes to rest on. Everything else is heard in relation to it.
Property 2
Major and minor scales
Both bodies of music use these two scale families almost exclusively. Seven notes, a fixed pattern of whole and half steps.
Property 3
Tertian chords
Tertian means "built of thirds." C–E–G is tertian; C–F–B is not. Stacking thirds is the default chord-building move of the entire system.
Property 4
Function
Chords built on the various scale degrees relate to each other and to the tonal center in fairly complex ways. Each chord has a more or less standard role within a key — hence functional harmony.
Tonal harmony refers to music with a tonal center, based on major and/or minor scales, and using tertian chords that are related to one another and to the tonal center in various ways.
A pitch class is a note name regardless of octave: every C on the piano — low, middle, high — belongs to pitch class C. There are twelve pitch classes, which is why the keyboard repeats every twelve keys.
IIPitch
Scales, key signatures, and intervals — the raw material. If you only ever internalize one thing here, make it the major scale's step pattern.
The keyboard & pitch class
The smallest distance between two adjacent keys — black or white — is a half step (semitone). Two half steps make a whole step. The white keys are the letters A–G; the black keys are named relative to their neighbours, so the same black key is C♯ (C raised) or D♭ (D lowered). Those two names sound identical on the piano — they are enharmonic equivalents.
Two octaves from C4 (middle C). Notice the black-key groups of two and three — that pattern is how you find any note by eye: C sits immediately left of a group of two.
There is no black key between E–F or between B–C. Those two pairs are half steps on the white keys, and that single fact generates every key signature in the system.
The major scale
The major scale is a fixed pattern of whole (W) and half (H) steps:
Half steps in the major scale occur only between scale degrees 3̂ and 4̂, and between 7̂ and 1̂. Everything else is a whole step. Because the pattern is fixed, transposing it to a different starting note forces you to add sharps or flats — which is exactly what a key signature is.
Tetrachords
The major scale can be thought of as two identical four-note patterns separated by a whole step. Those four-note patterns are called tetrachords: each is W–W–H.
The upper tetrachord of one major scale is the lower tetrachord of the scale a fifth above it. Start on G, reuse C major's upper tetrachord (G–A–B–C), continue the pattern, and you need F♯. That is the circle of fifths in one sentence.
Key signatures
A key signature is a pattern of sharps or flats at the beginning of a staff indicating that certain notes are consistently raised or lowered. There are seven sharp signatures and seven flat signatures.
Naming rule — sharps
The name of the major key is found by going up a half step from the last sharp in the signature.
Last sharp F♯ → G major. Last sharp C♯ → D major.
Naming rule — flats
The name of the major key is the same as the next-to-last flat in the signature. The exception is F major, which has one flat.
Flats B♭ E♭ A♭ → next-to-last is E♭ → E♭ major.
Order of sharps: F C G D A E B ·
Order of flats: B E A D G C F — the same letters backwards.
Say them aloud until they are automatic; nearly every key-signature question reduces to them.
Accidentals in standard treble-clef placement. Note the pattern shifts at the fifth sharp — that is conventional, not arbitrary; it keeps the signs on the staff.
Enharmonic key pairs
Three pairs of major keys sound exactly the same — the very same keys of the piano:
B major = C♭ major
5 sharps / 7 flats
F♯ major = G♭ major
6 sharps / 6 flats
C♯ major = D♭ major
7 sharps / 5 flats
The circle of fifths
A diagram shaped like a clock face. Reading clockwise, each new key begins on 5̂ (the fifth scale degree) of the previous key and adds a sharp. Reading counterclockwise, each new key begins on 4̂ of the previous one and adds a flat. The inner ring gives the relative minor of each major key — they share a key signature.
Click any wedge to load that key into the Key Explorer below.
The minor scales
Minor comes in three forms. All three share the same first five notes — the minor pentachord, W–H–W–W. Only 6̂ and 7̂ vary.
| Form | Built from major by… | Built from natural minor by… | Degrees | |
|---|---|---|---|---|
| Natural minor | lowering 3̂, 6̂, 7̂ | — | 1 2 ♭3 4 5 ♭6 ♭7 | |
| Harmonic minor | lowering 3̂ and 6̂ | raising 7̂ | 1 2 ♭3 4 5 ♭6 7 | |
| Melodic minor (ascending) | lowering 3̂ | raising 6̂ and 7̂ | 1 2 ♭3 4 5 6 7 | |
| Melodic minor (descending) | identical to natural minor | 1 2 ♭3 4 5 ♭6 ♭7 | ||
Relative and parallel keys
Minor key signatures conform to the natural minor scale, no matter which minor form is actually in use — the raised 6̂ and 7̂ appear as written accidentals in the music.
Relative keys
Share the same key signature. The relative major of any minor key starts on 3̂ of the minor scale; the relative minor of any major key begins on 6̂ of the major scale.
c minor ↔ E♭ major (3 flats).
Parallel keys
Share the same 1̂. C major and c minor are parallel keys; C major is the parallel major of c minor.
Same tonic, different signature.
To find any minor key signature other than g♯, d♯, or a♯: start with the key signature of the parallel major and add three flats and/or subtract three sharps. C major has 0 → c minor has 3 flats. D major has 2 sharps → d minor has 1 flat.
Convention used on this page: uppercase letters for major keys (E♭ major), lowercase for minor (c minor).
Scale-degree names
Every degree has a traditional name. These names are the vocabulary of harmonic analysis — "dominant" and "subdominant" will appear constantly from Part V onward.
| Degree | Name | What it means | In C major |
|---|---|---|---|
| 1̂ | Tonic | The tonal center. Home. | C |
| 2̂ | Supertonic | "Above the tonic." | D |
| 3̂ | Mediant | Halfway between tonic and dominant. | E |
| 4̂ | Subdominant | A fifth below the tonic, mirroring the dominant. | F |
| 5̂ | Dominant | The strongest pull back to tonic. | G |
| 6̂ | Submediant | Halfway between tonic and subdominant. | A |
| 7̂ | Leading tone | A half step below tonic; leads to it. | B |
| ♭7̂ | Subtonic | In minor, when 7̂ is not raised — a whole step below tonic. | B♭ (in c minor) |
The mediant lies halfway between tonic and dominant; the sub-mediant lies halfway between tonic and sub-dominant. The "sub-" names are not "one below" — they are the mirror image, measured downward from the tonic. That is why 4̂ is called the subdominant even though it sits just below 5̂.
Intervals
An interval is the measurement of distance in pitch between two notes. A harmonic interval sounds both notes at once; a melodic interval sounds them one after the other.
Every interval name has two parts: a numerical name and a modifier preceding it. Count letter names inclusively — C up to E spans C, D, E, so it is a 3rd, whatever the accidentals. We say unison instead of 1 and octave (8ve) instead of 8. Intervals smaller than an octave are simple; those an octave or larger are compound (a 9th, an 11th).
Perfect, major, minor
The easiest way to learn the modifiers is to relate them to the major scale — specifically the intervals from 1̂ up to each other scale degree. Once learned, apply it in any context, whether or not a major scale is in play.
- Perfect (P) is used only for unisons, 4ths, 5ths, octaves, and their compounds. P1, P4, P5 and P8 all appear using 1̂ of the major scale as the bottom note.
- Major (M) and minor (m) are used only for 2nds, 3rds, 6ths, 7ths, and their compounds. The intervals 1̂–2̂, 1̂–3̂, 1̂–6̂, 1̂–7̂ in the major scale are all major intervals.
- Make a major interval a half step smaller without changing its numerical name and it becomes minor. You can shrink an interval either by lowering the top note or raising the bottom.
Augmented and diminished
- A perfect or major interval made a half step larger (same numerical name) becomes augmented (+ or A).
- A perfect or minor interval made a half step smaller (same numerical name) becomes diminished (° or d).
- There is no such thing as a diminished unison. Doubly augmented and doubly diminished intervals exist but are rare.
- Tritone is the term for the +4 or its enharmonic equivalent, the °5 — three whole steps, the interval that splits the octave exactly in half.
The interval table
Every simple interval, measured in half steps from the lower note. Click ▶ to hear it melodically then harmonically, from C4.
| Interval | Abbrev. | Half steps | From C4 | Consonant? | Hear |
|---|
Inversion of intervals
Descending intervals, especially large ones, are often easier to spell and identify through inversion: put the bottom pitch above the top one, or the reverse. D up to A inverts to A up to D.
The rule of 9
The new numerical name is always 9 minus the old one.
2↔7 · 3↔6 · 4↔5 · 1↔8
The modifier flips too
M↔m · +↔° · P stays P. A M3 inverts to a m6; a P5 inverts to a P4.
What note lies a m6 below G3? Invert the m6 down into a M3 up: G3 up a M3 is B3. Transpose B3 down an octave and the answer is B2.
Consonance and dissonance
Roughly, consonant means pleasing to the ear and dissonant means not — but these are heavily dependent on context. Some of the most exciting moments in tonal music are dissonances, which are hardly displeasing there; they resolve eventually to the consonances that give them meaning.
Consonant
Major and minor 3rds and 6ths; perfect 5ths and octaves (and the unison).
Dissonant
All other harmonic intervals, including every augmented and diminished interval. Exception: the P4 is dissonant in tonal music only when it occurs above the lowest voice (the bass).
IIIRhythm
Rhythm is the general term for the time aspect of music, as contrasted with the pitch aspect.
Durational symbols
Durations are notated with symbols organised so that each symbol is twice the duration of the next shorter one and half the duration of the next longer one. That is the entire system — a binary tree.
Ties
A tie is a curved line connecting two notes of the same pitch, creating a new duration equal to their sum. It is how you write a duration the symbols alone can't express — and how you carry a note across a bar line.
A tie joins identical pitches; a slur connects different ones and means "play smoothly." Different things.
Dots
A dot following a symbol always adds one-half the value of the note, rest, or dot that precedes it. So a dotted half = half + quarter. A double-dotted half = half + quarter + eighth.
Notation detail: a dot is never placed on a staff line. If the notehead is on a line, the dot goes to the right of the note but in the space above it.
Beat, tempo, meter, time signatures
Beat
The basic pulse of a passage. To find it, tap your foot, or imagine the conductor's arm movement.
Tempo
The rate at which the beats occur.
Meter
Beats group into patterns consistent throughout a passage. The pattern of beats is the meter; the groups are measures (m. / mm.), separated by bar lines.
The two questions a meter answers
Meter is classified along two axes, and a time signature encodes both:
- How many beats per measure? Two = duple, three = triple, four = quadruple.
- How does the beat divide? Into two = simple, into three = compound.
| Type | Beat divides into | Duple | Triple | Quadruple | Beat note |
|---|---|---|---|---|---|
| Simple | 2 | 2/4 · 2/2 · 2/8 | 3/4 · 3/2 · 3/8 | 4/4 · 4/2 · 4/8 | The lower number names it: 4 = quarter |
| Compound | 3 | 6/8 · 6/4 · 6/16 | 9/8 · 9/4 · 9/16 | 12/8 · 12/4 · 12/16 | A dotted note: 6/8 → dotted quarter |
In compound meter the top number counts divisions, not beats. 6/8 is not six beats — it is two dotted-quarter beats, each dividing into three eighths. Divide the top number by 3 to get the real beat count.
Notating rhythm clearly
When rhythms are notated, it is customary to use rests, beams, ties, and dots so that the metric accents and the individual beats are emphasised rather than obscured. If a reader has to decode where beat 3 falls, the notation has failed.
Syncopations — rhythmic figures that stress normally weak beats or divisions — are frequently notated that way deliberately, with ties and rests that make the displacement visible rather than hiding it inside a longer symbol.
IVTriads and seventh chords
Stack thirds. Everything in tonal harmony is stacked thirds, and the whole vocabulary of chord types falls out of how you stack them.
Triads
The fundamental tertian sonority is the triad: a three-note chord consisting of a 5th divided into two superimposed 3rds. There are exactly four ways to combine major and minor 3rds, and therefore exactly four triad types.
| Triad | Symbol | Lower 3rd | Upper 3rd | Outer 5th | Half steps | On C | Hear |
|---|---|---|---|---|---|---|---|
| Major | C | major | minor | P5 | 0–4–7 | C E G | |
| Minor | Cm | minor | major | P5 | 0–3–7 | C E♭ G | |
| Diminished | C° | minor | minor | °5 | 0–3–6 | C E♭ G♭ | |
| Augmented | C+ | major | major | +5 | 0–4–8 | C E G♯ |
The bottom note of the stack is the root; the others are the third and the fifth, named for their interval above the root. These names stay attached to the notes no matter how the chord is rearranged.
Diatonic triads: the same pattern in every major key
Build a triad on each degree of a major scale using only notes of that scale, and the qualities always come out in the same order. This is why Roman numerals are portable across keys:
| Degree | 1̂ | 2̂ | 3̂ | 4̂ | 5̂ | 6̂ | 7̂ |
|---|---|---|---|---|---|---|---|
| Major key | I | ii | iii | IV | V | vi | vii° |
| Quality | major | minor | minor | major | major | minor | dim. |
| Natural minor | i | ii° | III | iv | v | VI | VII |
| Harmonic minor | i | ii° | III+ | iv | V | VI | vii° |
Uppercase numeral = major, lowercase = minor, ° = diminished, + = augmented. In minor, the raised 7̂ of harmonic minor is what turns v into V and VII into vii° — and that is almost always what composers actually use, because only a major V has a leading tone.
Seventh chords
Extend a tertian triad by adding another 3rd on top of the 5th and you get a four-note chord. The interval between the added note and the root is some kind of 7th — hence seventh chords. Tonal harmony commonly uses only five types.
| Seventh chord | Symbol | Triad + 7th | Half steps | On C | Typical home | Hear |
|---|---|---|---|---|---|---|
| Major seventh | Cmaj7 / CM7 | major + M7 | 0–4–7–11 | C E G B | I, IV | |
| Dominant seventh (major–minor) | C7 | major + m7 | 0–4–7–10 | C E G B♭ | V | |
| Minor seventh | Cm7 | minor + m7 | 0–3–7–10 | C E♭ G B♭ | ii, iii, vi | |
| Half-diminished seventh | Cø7 / Cm7♭5 | diminished + m7 | 0–3–6–10 | C E♭ G♭ B♭ | viiø7 in major, iiø7 in minor | |
| Fully diminished seventh | C°7 | diminished + °7 | 0–3–6–9 | C E♭ G♭ B𝄫 | vii°7 in minor |
Stack four notes (1–3–5–7) on each degree of any major scale and the qualities are always: Imaj7 · ii7 · iii7 · IVmaj7 · V7 · vi7 · viiø7. The V7 is the only dominant-seventh chord that occurs naturally in the key — which is precisely why it points home so hard. Use the Key Explorer to see this in all 15 keys.
Quality reference
- maj7 — major triad + major 7th. Bright, jazzy, at rest.
- m7 — minor triad + minor 7th. Smooth, mellow.
- 7 — major triad + minor 7th. Bluesy, strong tendency to resolve.
- ø7 / m7♭5 — diminished triad + minor 7th. Tense, unstable.
- °7 — diminished triad + diminished 7th. Maximum instability; symmetrical.
Why V7 resolves
A dominant seventh contains a tritone between 7̂ (the leading tone) and 4̂. The leading tone wants to rise a half step to 1̂; 4̂ wants to fall a half step to 3̂. Both resolve into the tonic triad at once. That single tritone is the engine of tonal harmony's forward motion.
Inversions and figured bass
The inversion of a triad or seventh chord is determined only by which chord member is in the bass. Any chord member may appear in the soprano or in the other voices without changing the inversion.
| Chord | Bass note | Name | Figured bass | Roman numeral | C major example | Hear |
|---|---|---|---|---|---|---|
| Triad | root | root position | (5/3) | I | C E G | |
| third | first inversion | 6 | I6 | E G C | ||
| fifth | second inversion | 6/4 | I6/4 | G C E | ||
| Seventh chord | root | root position | 7 | V7 | G B D F | |
| third | first inversion | 6/5 | V6/5 | B D F G | ||
| fifth | second inversion | 4/3 | V4/3 | D F G B | ||
| seventh | third inversion | 4/2 (or 2) | V4/2 | F G B D |
Figured-bass numerals are intervals above the bass note, reduced to their simplest useful form. A first-inversion triad (E G C) has a 3rd and a 6th above the bass — the 3rd is assumed, so only the 6 is written. A second-inversion triad (G C E) has a 4th and a 6th, neither assumed, so both are written: 6/4. Learn the derivation once and you never have to memorise the table.
A triad has two inversions (three bass positions). A seventh chord works the same way except that three inversions — four bass positions — are possible.
Key Explorer
Everything above, computed live for any key. Pick a tonic and a mode; the page spells the scale, draws the key signature, lights up the keyboard, and builds the diatonic triads and sevenths.
| Degree | Name | Numeral | Triad | Notes | 7th chord | Notes | Hear |
|---|
Spellings are generated by letter, not by enharmonic shortcut — so you'll see genuine double sharps and double flats in the remote keys, exactly as they'd appear on the page.
VHarmony in motion
The bridge from "here are the chords" to "here is why they go in that order." This part is deliberately brief — it is the seam where most of the later material will attach.
Harmonic function
Chords in a key sort into three functional families. Most tonal phrases are a trip through them in one direction, with the return to tonic as the point of arrival.
Tonic
I · vi · iii
Repose. The place the music is heard as leaving from and returning to. vi and iii share notes with I and can stand in for it.
Predominant
IV · ii · ii7
Departure. Prepares the dominant — it contains 4̂ or 6̂ and moves naturally into V.
Dominant
V · V7 · vii°
Tension. Contains the leading tone, and in V7 the tritone. Wants tonic, strongly.
T → PD → D → T. Moving forward along that chain sounds normal; skipping ahead sounds normal; moving backward (V → IV) sounds like a deliberate effect, which is exactly why blues and rock use it so much.
Progressions to have in your hands
| Progression | In C | Function path | Where you hear it | Hear |
|---|
Cadences
A cadence is the harmonic punctuation at the end of a phrase. What the phrase means structurally depends almost entirely on which cadence closes it.
| Cadence | Motion | Conditions | Effect | Hear (C major) |
|---|---|---|---|---|
| Perfect authentic (PAC) | V → I | Both in root position, 1̂ in the soprano | Full stop. The strongest close available. | |
| Imperfect authentic (IAC) | V → I | Anything weaker — inversion, or 3̂/5̂ on top | Closes, but keeps moving. | |
| Half (HC) | … → V | Ends on the dominant | A comma. Demands continuation. | |
| Deceptive (DC) | V → vi | Dominant resolves anywhere but I | The rug pulled. Extends the phrase. | |
| Plagal (PC) | IV → I | Usually after an authentic cadence | The "Amen." A gentle confirmation. |
Voice-leading basics
Tonal harmony is written as independent melodic lines that happen to make chords — not as a stack of chords that happen to have a top note. Four rules of thumb cover most of it:
- Move as little as possible. Keep common tones in the same voice; move the others by step where you can.
- Resolve tendency tones. The leading tone (7̂) rises to 1̂. The seventh of a seventh chord falls by step. These are near-absolute.
- No parallel fifths or octaves. Two voices a P5 or P8 apart moving to another P5 or P8 collapses their independence — the ear hears one voice, not two.
- Watch the outer voices. Soprano and bass are what listeners actually track. Contrary motion between them is the safest default.
In four-part writing, keep adjacent upper voices (soprano–alto, alto–tenor) within an octave of each other. The bass may sit further away — wide spacing at the bottom of a chord sounds normal, because that is how the overtone series is arranged.
→Roadmap — what goes here next
Placeholders with stable anchors. Each becomes a full section with the same structure as the ones above — prose, a table, a diagram, and something playable.
Near-term (direct continuations of Part V)
Non-chord tones
Passing tones, neighbours, suspensions, appoggiaturas, retardations, anticipations, pedal point — how dissonance actually appears in real textures.
Phrase structure & period forms
Antecedent/consequent, parallel and contrasting periods, sentence structure, phrase extension.
Part writing in detail
Doubling rules, root-position and inverted triads in four parts, the cadential 6/4, common part-writing errors.
Chromatic harmony
Secondary functions
V/V, V7/IV, vii°/V — tonicising a chord that is not the tonic. The single most useful chromatic device.
Modulation
Common-chord (pivot) modulation, chromatic modulation, sequential modulation; closely related keys.
Mode mixture & the Neapolitan
Borrowed chords from the parallel key, ♭VI, ♭III, the Picardy third, and the N6.
Augmented sixth chords
Italian, French, German — where they come from and where they go.
Beyond common practice
The modes
Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian — and why modal harmony behaves differently from functional harmony.
Jazz harmony
Extensions (9ths, 11ths, 13ths), tritone substitution, ii–V–I in all keys, voicings and rootless shells.
Twentieth-century harmony
Quartal harmony, pandiatonicism, set theory basics, planing — the companion vault note already exists for this.
Cross-reference: Carnatic theory
Swaras against scale degrees, the 72 melakarta as a systematic scale generator, gamaka — and where the two systems genuinely map onto each other and where they don't.
Practice tools to build into this page
Ear-training drills
Interval identification, chord-quality identification, cadence identification — the audio engine is already here.
Key-signature and spelling quiz
Random prompts, scored, with a "copy my weak spots" export.
Progression builder
Click Roman numerals into a sequence, hear it in any key, export as chord symbols.
§Glossary
| Term | Definition |
|---|---|
| Bar line | Vertical line through the staff marking the end of a measure. |
| Beat | The basic pulse of a passage. |
| Compound interval | An interval of an octave or larger. |
| Consonance | M/m 3rds and 6ths, P5, P8. Heard as stable. |
| Diatonic | Belonging to the notes of the prevailing key, without added accidentals. |
| Dissonance | Everything else, including all + and ° intervals; the P4 above the bass. |
| Enharmonic | Different spellings of the same sounding pitch (C♯ / D♭). |
| Figured bass | Numerals showing intervals above the bass note; encodes inversion. |
| Functional harmony | Harmony in which each chord has a standard role relative to the tonic. |
| Harmony | The sound of two or more pitches performed simultaneously; the vertical aspect. |
| Interval | The distance in pitch between two notes. |
| Inversion (chord) | Determined solely by which chord member is in the bass. |
| Inversion (interval) | Moving the lower note above the upper; numerical names sum to 9. |
| Key signature | Sharps or flats at the start of a staff, applied consistently. |
| Meter | The consistent pattern into which beats group. |
| Minor pentachord | Degrees 1̂–5̂ of any minor scale: W–H–W–W. Identical in all three forms. |
| Parallel keys | Major and minor sharing the same tonic (C major / c minor). |
| Pitch class | A note name irrespective of octave. There are twelve. |
| Relative keys | Major and minor sharing the same key signature (E♭ major / c minor). |
| Rhythm | The time aspect of music, as opposed to the pitch aspect. |
| Simple interval | An interval smaller than an octave. |
| Syncopation | A rhythmic figure stressing normally weak beats or divisions. |
| Tempo | The rate at which beats occur. |
| Tertian | Built of thirds. C–E–G is tertian; C–F–B is not. |
| Tetrachord | A four-note pattern; the major scale is two identical ones (W–W–H) a whole step apart. |
| Tie | Curved line joining two notes of the same pitch into one duration. |
| Tonal center | The pitch class acting as the music's center of gravity. |
| Triad | A three-note tertian chord: a 5th divided into two superimposed 3rds. |
| Tritone | The +4 or its enharmonic equivalent the °5; three whole steps. |