Counterpoint and fugue

Counterpoint is the art of making several independent melodies sing at once (from punctus contra punctum, “note against note”). Where harmony thinks in vertical chords, counterpoint thinks in horizontal lines. At the piano it's a fascinating challenge: your ten fingers become a little choir.

The four kinds of motion

• Parallel: both voices move the same way, at a constant interval;
• Similar: same direction, different intervals;
• Contrary: opposite directions — the most independent;
• Oblique: one voice moves, the other stays.

Consonance and dissonance

Perfect consonances: unison, fifth, octave. Imperfect consonances: thirds and sixths. Dissonances: seconds, sevenths, the tritone (and the fourth above the bass). Classical counterpoint alternates consonance and dissonance according to precise rules.

The five species (Fux, 1725)

In his treatise Gradus ad Parnassum, Johann Joseph Fux teaches counterpoint in five stages, over a given melody in whole notes, the cantus firmus: Haydn, Mozart and Beethoven all learned from this book.

1. First species: note against note, consonances only.
2. Second species: two notes against one; dissonant passing notes on weak beats.
3. Third species: four notes against one; neighbour and passing notes.
4. Fourth species: syncopation; suspensions (preparation, dissonance, resolution).
5. Fifth species: florid counterpoint, a free mix of the others.

A first-species example: begin and end on the octave, favour contrary motion, avoid parallel fifths and octaves, and let the penultimate note form a sixth that opens onto the octave.

M:4/4
L:1/4
%%staves {CP CF}
V:CP clef=treble
V:CF clef=bass
K:C
[V:CP] C4 | G4 | A4 | G4 | c4 | B4 | c4 |]
[V:CF] "_8"C,4 | "_10"E,4 | "_10"F,4 | "_8"G,4 | "_6"E,4 | "_6"D,4 | "_8"C,4 |]

Invertible counterpoint

Two voices are in invertible counterpoint at the octave if they can swap places (the upper goes below) and still be correct. The trap: an inverted fifth becomes a fourth, dissonant above the bass. Bach uses this everywhere: in his inventions, the hands constantly trade material.

The fugue

The fugue is the summit of contrapuntal writing. Its elements:

• Subject: the theme, stated alone by the first voice;
• Answer: the subject transposed to the dominant by the second voice — real if transposed exactly, tonal if slightly adjusted to stay in the key;
• Countersubject: the melody accompanying the answer, often in invertible counterpoint;
• Exposition: each voice enters in turn with the subject or answer;
• Episodes: passages without the full subject, often sequential, that modulate;
• Middle entries in related keys; stretto: overlapping entries; a final pedal.