Definition
Triple counterpoint is a set of three lines written so that each can occupy any position (high, middle, low) and remain correct: there are six possible combinations. Quadruple counterpoint extends the principle to four lines, with twenty-four theoretical arrangements. It is the extension of double counterpoint at the octave (or at the fifteenth) to several voices, and it is the basis of fugues with several regular countersubjects.
The problem of the fifth
At the octave, intervals invert as follows: 1↔8, 2↔7, 3↔6, 4↔5. Thirds and sixths remain consonances; the fifth becomes a fourth. In two-voice double counterpoint it is enough to treat fifths as if they were fourths (that is, prepared or passing). In triple counterpoint the problem is more serious: any voice can become the bass, so every fifth between two lines can become a fourth against the bass, that is, a dissonance.
Practical rules
- Between every pair of lines use mainly thirds, sixths, octaves.
- Fifths appear only as passing notes or on weak beats, or at points where the third line, whatever the arrangement, forms a chord with the other two that justifies them.
- Avoid the fourth between two lines on the strong beat: inverted it becomes a fifth, but before the inversion it is dissonant if one of the two is in the bass.
- The lines must not exceed an octave of distance between them in the starting arrangement (or you reason at the fifteenth).
- Complementary rhythms: one line in long values (often with suspensions), one in medium values, one in short values. This way the ear recognizes each line even when it changes position.
- Incomplete chords: the absence of the fifth is often accepted, because the third and octave are safe in every arrangement.
Examples in the repertoire
- Bach, The Well-Tempered Clavier I, Fugue in C# minor BWV 849: five voices, three themes. The first is the subject C# B# E D# C# in long values; the second, in eighth notes, enters at bar 36; the third at bar 49. From bars 49–88 the three themes are combined in invertible counterpoint, in different arrangements.
- Bach, The Well-Tempered Clavier II, Fugue in F# minor BWV 883: subject and two countersubjects, first presented separately and then superimposed in all the voices in the final section.
- Bach, The Art of Fugue, Contrapunctus 8 and 11: triple fugues, the second with the three subjects of the first in inversion.
- Sinfonias (three-part inventions) by Bach: several are built on three ideas permuted in triple counterpoint.
How to practice
- Write three four-bar lines in C major: one in half notes with a 7–6 suspension, one in quarter notes by step, one in eighth notes with arpeggios.
- Verify all six arrangements (ABC, ACB, BAC, BCA, CAB, CBA), transposing the lines by an octave when needed.
- Mark every fifth on a strong beat: it is a weak point to be corrected.
- Play each arrangement in a DAW with three different timbres: the ear immediately discovers the exposed fourths against the bass.
Typical errors
- Verifying only the convenient arrangements.
- Three lines with the same rhythm: the inversion becomes unreadable.
- Basing the harmonic stability on the fifth: in the inversion the chord becomes a 6/4.
Modern application
- Soundtracks: three motifs for different characters written in triple counterpoint can be swapped in register (cellos, horns, violins) depending on who dominates the scene, without reharmonizing.
- Pop and production: a bass riff, a synth riff and a vocal hook that work in any octave are easier to reshuffle in remixes and live.
- Choirs: in finales with three different phrases superimposed (the quodlibet technique in musicals), triple counterpoint guarantees that every section of the choir can sing any phrase.
Related topics
Invertible counterpoint at the octave, tenth and twelfth, double and triple fugues, countersubject, The Art of Fugue.