counterpoint

Counterpoint

Counterpoint is the art of combining independent melodic lines that sound together harmonically. Where harmonic analysis reads music vertically (chords at a moment in time), contrapuntal analysis reads music horizontally (melodies across time). The word derives from Latin punctus contra punctum — "note against note." Counterpoint is simultaneously the oldest and most rigorous discipline in Western music theory, with an unbroken pedagogical tradition from the Renaissance through the present day.

Agent affinity: rameau (harmonic dimension of counterpoint), bach (fugue, invention, chorale)

Concept IDs: harmony, chord-progressions, scales-intervals

Part I — Species Counterpoint (Fux's Gradus)

Johann Joseph Fux codified species counterpoint in Gradus ad Parnassum (1725), structuring the discipline into five progressive species. The method starts with the simplest rhythmic relationship (note against note) and systematically introduces complexity. Nearly three centuries later, this remains the standard pedagogical framework.

The Cantus Firmus

Every species exercise begins with a cantus firmus (CF) — a fixed melody, typically 8-12 notes, all whole notes, in a church mode. The student writes a counterpoint line against it.

Properties of a good cantus firmus:

• Mostly stepwise motion (conjunct), with no more than two consecutive leaps
• Range within a tenth
• A single climax (highest note), reached by step or small leap, not repeated
• Begins and ends on the modal final (tonic)
• No repeated notes (maintains rhythmic momentum)
• No augmented or diminished intervals (melodic)
• No chromatic motion

Example cantus firmus in D Dorian:

D - F - E - D - G - F - A - G - F - E - D

This CF moves mostly by step, has a single climax on A, spans an octave (D to D), and begins and ends on D.

First Species — Note Against Note

One note of counterpoint for each note of the cantus firmus. All intervals between the two voices are consonances.

Consonance table:

| Interval | Type | Treatment | |---|---|---| | Unison | Perfect consonance | Only at beginning and end | | Third | Imperfect consonance | Freely used, backbone of counterpoint | | Fifth | Perfect consonance | Approached by contrary or oblique motion | | Sixth | Imperfect consonance | Freely used | | Octave | Perfect consonance | Approached by contrary or oblique motion |

Rules:

1. Begin and end on a perfect consonance (unison, fifth, or octave).
2. No parallel fifths or parallel octaves between successive intervals.
3. No direct (hidden) fifths or octaves — two voices moving in the same direction to a perfect consonance by leap in the upper voice.
4. Prefer contrary motion (voices move in opposite directions) and oblique motion (one voice holds while the other moves).
5. Imperfect consonances (thirds and sixths) should predominate. A string of perfect consonances sounds hollow.

Worked example. Counterpoint above the CF in D Dorian:

CF:    D  | F  | E  | D  | G  | F  | A  | G  | F  | E  | D
CP:    A  | A  | G  | F  | B  | A  | C  | B  | A  | G  | D (8ve below: octave)
Int:   5  | 3  | 3  | 3  | 3  | 3  | 3  | 3  | 3  | 3  | 8

The counterpoint begins on a fifth and ends on an octave. Motion between every pair of notes is either contrary or oblique. Imperfect consonances dominate. No parallel fifths or octaves appear.

Second Species — Two Notes Against One

Two half notes in the counterpoint against each whole note of the cantus firmus. Dissonance enters the picture: the second half note in each bar may be a passing tone (dissonance approached and left by step in the same direction).

Rules beyond first species:

1. Downbeats must be consonant with the CF.
2. Upbeats may be dissonant only as passing tones (stepwise approach, stepwise departure, same direction).
3. No unisons on upbeats (weakens independence).
4. Begin with a half rest followed by a consonance.

Third Species — Four Notes Against One

Four quarter notes against each whole note. Dissonance may appear on beats 2, 3, or 4 as passing tones or neighbor tones (step away from a consonance and step back to the same note). The first beat of each bar must be consonant.

Neighbor tone (auxiliary): A dissonance reached by step from a consonance and returning by step to the same consonance. The nota cambiata (changing tone) pattern — a descending step from a consonance to a dissonance, then a descending leap of a third to a consonance — is also permitted in third species.

Fourth Species — Syncopated (Suspensions)

Tied whole notes across the barline create suspensions — the most expressive dissonance treatment in tonal counterpoint.

Suspension mechanics:

1. Preparation: The note is consonant on a weak beat.
2. Suspension: The note is held (tied) into the next strong beat, where it becomes dissonant against the CF.
3. Resolution: The suspended note moves down by step to a consonance.

Standard suspension types:

| Suspension | Intervals (against bass) | Resolution | |---|---|---| | 7-6 | Seventh resolves to sixth | Most common upper-voice suspension | | 4-3 | Fourth resolves to third | Common in cadences | | 9-8 | Ninth resolves to octave | Really a 2-1 in a higher octave | | 2-3 (bass) | Second resolves to third | Bass suspension (resolves DOWN) |

Rule: If no suspension is possible on a given beat, tie a consonance across the barline instead (a consonant suspension, or "consonant syncopation"). Never break the syncopated rhythm arbitrarily.

Fifth Species — Florid Counterpoint

Free combination of all previous species. The counterpoint mixes whole notes, half notes, quarter notes, and suspensions in a musically convincing line. This is the closest species approach to free composition.

Guidelines:

• Maintain rhythmic variety; do not settle into one species for too long
• The florid line should be singable and well-shaped as an independent melody
• Dissonance treatment follows the rules of whichever species is locally active
• Suspensions should be used at cadence points for maximum effect

Part II — Consonance and Dissonance Treatment

The treatment of dissonance is what separates counterpoint from mere chord stacking. In strict counterpoint, every dissonance must be prepared, approached, and resolved according to specific rules. The rules are not arbitrary — they emerged from centuries of vocal practice and reflect what singers can tune reliably in ensemble.

Hierarchy of Dissonance Treatment

| Treatment | Description | Species | |---|---|---| | Passing tone | Step-step, same direction | 2nd and above | | Neighbor tone | Step away, step back to same note | 3rd and above | | Suspension | Tied over barline, resolves down by step | 4th and above | | Anticipation | Consonance arrived early, on weak beat | 5th (florid) | | Escape tone | Step from consonance, leap away (rare) | 5th (florid) | | Appoggiatura | Leap to dissonance, step resolution (free comp.) | Free composition |

Critical principle: In strict counterpoint, dissonances on strong beats must be suspensions. Dissonances on weak beats must be passing or neighbor tones. The appoggiatura (accented, non-prepared dissonance) belongs to free composition, not strict species work.

Part III — Invertible Counterpoint

Two contrapuntal lines are in invertible counterpoint if they can swap registral positions — the upper voice becomes the lower voice and vice versa — and the result remains correct counterpoint.

Invertible Counterpoint at the Octave

When lines swap by an octave, intervals transform: a third becomes a sixth, a fifth becomes a fourth, a sixth becomes a third. The critical issue: a fifth (consonant) becomes a fourth (dissonant in two-part writing). Therefore, invertible counterpoint at the octave avoids fifths between the voices.

Interval transformation table (at the octave):

| Original | Inversion | Status | |---|---|