Sing "Frère Jacques" with a friend, and have them start when you have sung the first two phrases. Nothing was arranged for two people. One tune, copied and delayed, fits against itself, and every beat the two of you land on sounds like a chord. A thousand years of European music grew out of that trick: independent lines that fit.
The course has two halves that look different and are one. Counterpoint is the craft of writing lines that are each worth singing and, together, sound right. Form is what happens when lines and chords are stretched across minutes: a theme, its return, a contrast, a journey and a homecoming. This piece takes six ideas, plays them with you, and keeps a text readout of every result so it works with the sound off. (The tune used in the canon card is a traditional French round; all other music here is written for this page.) Sound is not available in this reader, so the cards show notes and numbers only.
- Who takes it
- Music majors, usually in year 2, in conservatories and universities (the US accrediting body NASM, the UK music subject benchmark and Berklee's core all include counterpoint and analysis of form), and composers, arrangers and film-score writers at any level.
- What it is for
- Writing lines that stand on their own, and recognising the large shapes in which music is built.
- Comes after
- Harmony & Voice Leading (35/65), where parallel fifths first appear.
- Leads to
- Composition, Arranging & Orchestration (35/72), and it deepens Western Music History: Chant to Now (35/70).
- What this piece gives you
- You will not be able to write a four-voice fugue. You will see what Fux's rules are protecting, why a suspension is a dissonance with permission, why a round works at one entry distance and clatters at another, what the parts of a fugue are, and how a sonata is a plan for keys. Easier first reads: How to Actually Listen to Music (09/02) and Music Theory Accelerators vol. 2 (09/08), which has the twelve-bar blues.
Independence: two lines, one harmony
A single melody is easy to follow. Two melodies at once only work if you can still hear each as itself, and that is what counterpoint (from the Latin punctus contra punctum, point against point) trains. The ideal is that each line is a good tune alone and the two together never hide each other: they move in different directions, at different moments, and avoid the intervals that blend them into one sound (the parallel fifths and octaves of the previous course).
Counterpoint was not only European. Many singing traditions are polyphonic in their own terms: Georgian three-part song, the interlocking yodelled lines of Aka (BaAka) forest music in the Central African Republic and Congo, Balinese interlocking parts in gamelan, call and response across West Africa. This piece teaches one well-documented system, the one that European conservatories used from the 18th century on, and does not claim it is the only way two lines can fit.
Species counterpoint: Fux's rules as a game
In 1725 the Austrian composer and theorist Johann Joseph Fux published Gradus ad Parnassum, a Latin dialogue between a master and a pupil, modelled on the vocal style of Palestrina (c. 1525–1594). It introduced species counterpoint: a graded course in which you add one line to a fixed melody, the cantus firmus (Latin, "fixed song"), with rules that start severe and loosen. Haydn is said to have worked through it, and it is still the standard way to begin. The five species are in the table.
| Species | Notes against one | What it adds |
|---|---|---|
| 1st | 1 : 1 | Only consonances; learn independence. |
| 2nd | 2 : 1 | Passing tones on weak beats. |
| 3rd | 4 : 1 | More passing tones, small figures. |
| 4th | syncopated | Suspensions: the dissonance with a licence. |
| 5th | florid | All of the above, free rhythm. |
First species is a puzzle with checkable rules. In two voices the consonances are the unison and octave, thirds, the perfect fifth and sixths; seconds, sevenths, the tritone and (against the lowest voice) the perfect fourth count as dissonant. Begin on an octave, unison or fifth; end on an octave or unison; avoid parallel perfect intervals; avoid unisons in the middle; and approach the last note by step from a major sixth, which in this mode needs a raised leading tone. Write the line above a cantus firmus modelled on Fux's own: eleven notes in D Dorian, the mode of D to D on the white keys.
The lower notes (orange) are the cantus firmus; yours are blue, and a note's interval over the cantus is in the readout. The start is deliberately wrong: every note is an octave above the cantus, so it breaks the parallel-octaves rule ten times. Errors turn the columns red. The checker implements these rules only, not taste.
The model line the button shows was found by a computer search over the same rules (it passes with no errors and no warnings), but it is one of many: when the rules are strict, the space of answers is small but not empty. The rules do not make a good melody. They keep a bad one from happening.
A dissonance with a licence
If dissonance is forbidden, why is the sharpest tension in Renaissance music a dissonance? Because the rule is not "never"; it is "only like this". The suspension is the canonical licence, and it is the whole reason the fourth species exists. It has three steps: prepare a note while it is consonant, suspend it by tying it over as the other voice moves, so it is now a dissonance on the strong beat, and resolve it by stepping down to a consonance. The dissonance is "legal" because it is prepared and resolved in exactly that way.
Upper voice C 5 is held while the lower voice moves F 3 to G 3, then the upper voice steps down to B 4. The interval names are counted modulo an octave; the 4th is dissonant against the lower voice in two parts.
Play it and listen for the middle step. The ear hears the held note go sour and wait; the bass is "right" and the upper voice is "late". That small delay, repeated down a line of four or five voices, is where a great deal of Renaissance and Baroque expressiveness comes from.
Canon: why a round works
Imitation means one voice repeats another. When the repeat is exact and continuous, it is a canon, and a round is a canon that loops. The earliest famous English example is Sumer is icumen in, from the mid-13th century. The trick is that the melody must fit against a delayed copy of itself. Whether it does depends only on the delay.
Take the French round on the card. It is 32 beats long. Delay a copy by one beat and many beats clash; delay it by four, or by eight, and the notes you land on almost always fit one chord. Count how many beats have everything sounding inside a single major or minor triad of C:
Traditional French round, in C major, 32 beats. Each later voice sounds an octave lower than the one before it. A beat "fits" when all notes sounding on it lie in one major or minor triad of the key (C, F, G, Dm, Em, Am); passing notes on half beats are not counted. Red ticks mark the other beats. The measure is a rough one, made for this card.
Two things show up when you drag. First, no entry distance is perfect: the tune was not written with a two-voice canon at one beat in mind. Second, more voices make fitting harder, because every extra copy adds a note to the same beat. Round-writers choose a tune whose own harmony, repeated, lands on the same few chords: that is composing in advance for the delay.
Fugue: one subject, many voices
A fugue is a canon that can think. It is built from a subject, a short, memorable tune that appears in one voice alone. When a second voice enters, it plays the subject again, usually at the fifth (the answer), while the first voice moves on to an accompanying line, the countersubject. Each later voice enters the same way until all are in: this opening is the exposition. After that the composer is free: episodes that wander, further entries in other keys, a stretto where the entries crowd closer, and a final return.
The canonical collection is Bach's Well-Tempered Clavier: Book I (1722) has 24 preludes and fugues, one in every major and minor key, in two to five voices. "Well-tempered" did not necessarily mean the equal temperament used on pianos today; scholars still debate which tunings Bach intended, and the title says only that all keys were usable.
An original eight-note subject in C. The answer is the subject moved up a perfect fifth, note for note (a "real" answer; Bach often adjusts it, a "tonal" answer). The countersubject was found by a computer search under first-species rules: every beat consonant against the answer, no parallel fifths or octaves.
The countersubject is the quiet hero. It has to be good enough to repeat, because in a full fugue it returns every time the subject does. A good countersubject is built so that it still works when the voices swap places, a feature called invertible counterpoint.
Form: the shape of a whole piece
A form is a plan for time. The shapes are few, because a listener needs to recognise a return to know that anything has returned. Binary (AB) is two halves; the first leaves home, the second comes back (the dance movements of Bach's suites). Ternary (ABA) is statement, contrast, return (a minuet and trio). Rondo (ABACA) keeps coming back to the same refrain between contrasts: Beethoven's "Für Elise" is the famous one. Theme and variations keeps the skeleton and changes everything else: Bach's Goldberg Variations (1741), Beethoven's Diabelli Variations (1823).
The most elaborate is sonata form. Its logic is about keys: in the exposition a first theme in the home key I is followed by a second in a new key (in a major piece, usually the dominant V); the development wanders through other keys, breaking the themes into fragments; the recapitulation brings both themes back, now both in the home key. The "problem" set up in the exposition (two keys) is solved in the recapitulation (one key). Mozart's piano sonata in C, K. 545 (1788), is the textbook model.
Two cautions. "Sonata form" as a named template was codified by 19th-century theorists after the fact, and Charles Rosen (Sonata Forms) and, later, James Hepokoski and Warren Darcy (Elements of Sonata Theory, 2006) argue it is better described as a family of strategies than a mould. And Western forms are one tradition: raga uses the alap, jor and jhala movement from free to pulsed; the Ghanaian drum ensemble builds from a bell timeline; gamelan has its cyclic gong structure (see Music Traditions of the World, 35/69).
Four-beat original themes, written for this page, with a low drone on the key's home note. A is in C; B is in G (the dominant). The miniature is a sketch of the plan, not a real binary or sonata movement.
Songs have forms too: twelve bars and thirty-two
Popular music runs on the same principle: states you recognise, and a plan. The twelve-bar blues, which grew in African American communities in the southern United States around the turn of the twentieth century and was widely recorded by the 1920s, is a three-line form: the same line twice over I, then an answering line, with the chords I, IV and V in a fixed place. The AABA or 32-bar song, the standard form of Broadway and Tin Pan Alley, states a tune (A), repeats it, leaves for a contrasting bridge (B), then returns to A: George Gershwin's "I Got Rhythm" (1930) is the classic, and its chords are the basis of countless jazz tunes. Both are binary-and-ternary thinking in a 4/4 groove (35/71 takes them further).
Blues in C with dominant sevenths and a simple walking bass. The AABA miniature has four-bar sections instead of the usual eight, and uses a plain I–vi–ii–V pattern for A and a chain of dominant sevenths (the usual bridge idea) for B; chord choices are illustrative.
What a graduate just sees
- Every good second line is a tune in its own right, not just a harmony part.
- Contrary motion and different rhythm keep voices independent; parallel perfect intervals merge them.
- A dissonance is fine if it is prepared, on the beat, and resolved by step: that is a suspension.
- A round works because its melody keeps landing on the same few chords when it is delayed.
- In a fugue: subject, answer a fifth away, countersubject, episodes, then a return.
- Almost every long piece is a few simple shapes: AB, ABA, ABACA, variations, sonata.
- Sonata form is a plan for keys: leave home, wander, come back and bring everything with you.
- 12 bars or 32, a song form is a chord plan the listener memorises without knowing it.
Where it shows up next
- Composition, Arranging & Orchestration (35/72): every inner part in an arrangement is counterpoint, and every album has a form.
- Western Music History (35/70) tells how the forms rose and changed: the motet, the fugue, the sonata, the symphonic poem.
- Jazz, Popular Music & the Recording Age (35/71): chorus forms, call and response, and what a bridge does in a song.
- In the world: a film composer building a theme that returns changed, a DJ structuring a set, a game composer layering parts that enter one by one.
If you want to go deeper
- 📘 Textbook. Kent Kennan, Counterpoint (Pearson). A standard classroom text on 16th- and 18th-century counterpoint.
- 📘 Primary source. Johann Joseph Fux, Gradus ad Parnassum (1725), in Alfred Mann's translation as The Study of Counterpoint (Norton). The master-and-pupil dialogue itself.
- 📘 Form. William Caplin, Classical Form (Oxford), and Charles Rosen, Sonata Forms (Norton). Rigorous and readable; Hepokoski and Darcy's Elements of Sonata Theory (Oxford, 2006) is the modern challenger.
- 🌐 Gentler, free. musictheory.net for the basics, and the open textbook Open Music Theory (search the title) for counterpoint and form.
- 🌐 Free course. MIT OpenCourseWare 21M.301 Harmony and Counterpoint I (Spring 2005), which covers two-part counterpoint.
- 🌐 IMSLP: look up Bach's Well-Tempered Clavier Book I and trace one fugue's subject through its voices.
Counterpoint is the discipline of making two lines both worth hearing, and its rules are the rules that keep them separate. Dissonance is allowed when it is prepared and resolved. Canon and fugue are what imitation does when it fits; form is the plan that lets minutes hold together. Learn six shapes and you can follow most of what you ever hear.
Try this: pick any two-part round you know and sing it with someone, starting a different number of beats apart. Notice which entry distances feel like chords and which feel like a collision. You have just measured what the canon card computes.
Sources. Species counterpoint, its rules and its date (Gradus ad Parnassum, 1725) were checked against an encyclopaedia summary and the standard texts (Fux in Mann's translation; Kennan). The cantus firmus here is original in the manner of Fux, not a transcription. That Haydn studied the book is a common biographical statement, not independently re-verified. Bach, Well-Tempered Clavier Book I (1722); the debate over its temperament is described, not settled. Sonata-form critiques: Rosen, Hepokoski and Darcy. Gershwin's "I Got Rhythm" (1930), the Goldberg Variations (published 1741) and the Diabelli Variations (1823) are standard facts. The French round is traditional; Sumer is icumen in is described as "mid-13th century" because scholars vary. The model line, countersubject and every percentage on the cards were computed in node. Whether the 12-bar blues began earlier than its first recordings is debated.
- 🌐 MIT OCW 21M.301
- 🌐 IMSLP
- 📘 Fux, The Study of Counterpoint (Mann)
- 📘 Caplin, Classical Form