From Johann Joseph Fux’s Gradus Ad Parnassum, translated and edited by Alfred Mann.

Introduction

  • Study of counterpoint could be compared to the study of perspective; both important developments in Renaissance art; rise of three-dimensional thought.
  • Discantus (two voices) was predecessor to counterpoint.
  • Published in 1725.
  • Based on Palestrina; not Bach.
  • Studied by Haydn, Mozart.
  • Beethoven had hoped to study with Mozart. Mozart died. Haydn was second choice. Beethoven “demanded a more systematic instruction than Haydn was, by age and temperament, disposed to give.”
  • Beethoven’s studies were published; list of subscribers to French edition included “Cherubini, Berlioz, Meyerbeer, Chopin, Rossini, Auber, Paganini, Moscheles, Hummel, Liszt”
  • Chain proceeds to Schubert, Bruckner, Brahms; and finally to Strauss and Hindemith.

Counterpoint

In earlier times, instead of our modern notes, dots or points were used. Thus one used to call a composition in which point was set against or counter to point, counterpoint.

A composition which is written strictly according to technical rules.

Consonance

  Type of interval Type of consonance
Unison Consonance Perfect
2nd Dissonance  
3rd Consonance Imperfect
4th Dissonance; consonance if the higher note is fundamental  
5th Consonance Perfect
6th Consonance Imperfect
7th Dissonance  
8ve Consonance Perfect
  • Purpose of harmony: pleasure.
  • Pleasure is awakened by variety of sounds.
  • Variety: result of progression from one interval to another.
  • Progression is achieved by motion.

Thus it remains to examine the nature of motion.

Motion

Direct

Two or more parts ascend or descend in the same direction by step or skip (don’t have to be the same in all cases).

Contrary

One part ascends and the other descends.

Oblique

One part moves while the other remains stationary.

4 Fundamental Rules

(Perfect; Imperfect; Consonance.)

Motions that are allowed:

  1. PC -> PC – contrary or oblique motion.
  2. PC -> IC – any motion.
  3. IC -> PC – contrary or oblique.
  4. IC -> IC – any motion.

TL;DR – Direct motion into a perfect consonance is NOT allowed.

First Species of Counterpoint

  • Simplest composition of two or more voices.

    • Notes of equal length.
    • Only consonances.

  • Can be composed on all modes whose tonic note is part of P5 (i.e. not Locrian) – cannot be the final of a mode.

  • Adding an accidental to create a 5th: natural mode -> transposed mode.

    • Not considering transposed modes for now.

Cantus Firmus (cf)

Melody.

  • Possibility of vocal performance.
  • No augmented, diminished, chromatic intervals.
  • No intervals larger than 5th, except 8ve and minor sixth (latter only in upward direction).
  • Avoid too high/too low registers; skips following each other in same direction; skips not subsequently compensated.
  • Avoid tritone progression in one voice: devil in music.

Counterpoint (cp)

  • Each note in cf receives suitable consonance in voice above or below.
  • Contrary or oblique motion used as much as possible.
  • Greater care needed when using direct motion.
  • More imperfect than perfect consonances should be used.
    • Imperfect -> more harmonious (reason not yet given).
  • Begin (perfection) and end (relaxation) with perfect consonances.
    • Imperfect lack perfection; cannot express relaxation.
  • Voice crossing: allowed, if it improves voice leading.
  • Unison: only allowed at beginning and end.
  • Octave: avoid from larger intervals (Italian: “battuta”, Greek: “thesis”).
  • Accidentals allowed in special cases, more or less to accommodate leading tones.

When counterpoint above

  • Next to last bar: M6

When counterpoint below

  • 5th should not be used to begin if it takes the melody out of the mode. 8ve instead.
  • Next to last bar: m3