Counterpoint is the relationship between two or more voices that are independent in contour and rhythm, and interdependent in harmony. It has been most commonly used in Western music, developing strongly in the Renaissance, and also dominant in much of the common practice period, especially in Baroque music. The term comes from the Latin punctus contra punctum ("point against point", where "point" may mean "note").
Strict, or "species", counterpoint, provides a clear framework for the study of voice leading: the relationship between two or more voices. In 16th to 18th century strict counterpoint, we move from the simplest rhythmic relationship of note-against-note in the first species to the florid style of the fifth and final species, and through this orderly and systematic progression we develop an intimate knowledge of the processes by which voices or lines interact in the most natural, purposeful, and satisfying way, complementing and contrasting with one another. <ref>'An Introduction to Species Counterpoint', ListeningArts.com (Accessed 19/2/2009), <http://www.listeningarts.com/music/general_theory/species/menu.htm></ref> The following sections should provide the student with the fundamental concepts involved in writing the 18th century species counterpoint which can be used as a tool in composing styles of more contemporary genres.
When first studying counterpoint, the student will always begin with a cantus firmus. A cantus firmus is a melody to which one or more contrapuntal parts are added. Since this melody is the sine qua non of a satisfying contrapuntal exercise, particular care must be taken to craft it beautifully. Here are the requirements for an effective Fuxian cantus firmus:
- The cantus firmus is traditionally written in alto clef, a member of the movable clef family known as "C" clefs. C clefs include the alto clef, the tenor clef (both still in use today), and the soprano clef. In all C clefs, middle C is located where the arms of the clef meet
- The cantus firmus begins and ends on the tonic of the key or the final of the mode. The penultimate note should be the note a step above the tonic or final (the second tonal or modal degree).
- All notes are of equal length; the whole note is the traditional value.
- Notes are usually not repeated immediately (although, in treatises of the 16th and 17th centuries, examples can be found which contradict this).
- The range of the melody is generally limited to an octave. This range is occasionally stretched as far as a 10th. Most cantus firmi move within a much smaller range; some are confined to a 6th or even just a 5th above the tonic.
- Only diatonic notes are used in the cantus firmus. No accidentals may be applied within a key.
- The melody consists of from eight to thirteen notes
- Conjunct (stepwise; by a diatonic interval of a second) movement should predominate, interspersed by three or four judiciously employed leaps. If the leap is greater than a 3rd, it must be followed immediately by motion, preferably by step, in the opposite direction to that of the leap. This opposite conjunct motion is called "recovering the leap."
- The melody should be conceived in terms of what can be sung easily by the average musician
- The following melodic intervals are permitted in the cantus firmus: major and minor 2nds, major and minor 3rds, perfect 4ths, perfect 5ths, minor 6ths (ascending only), and perfect 8ves.
- Two successive leaps in the same direction are to be avoided, since they suggest an empty space in the line
- Repetition of groups of notes (a), and sequences (b), are generally forbidden
- The cantus firmus should have a climax on a high note, which should be melodically consonant with the first and final notes (i.e. at a distance of a major or minor 3rd, perfect 4th or 5th, major or minor 6th, perfect 8ve, or major or minor 10th). Do not repeat this climactic note, since this detracts from its commanding effect
- There should be a good balance between ascending and descending motion; the cantus firmus should possess a pleasing shape and should change direction several times.
The first species of counterpoint involves writing a new melody above or below the cantus firmus, with one note in the counterpoint for every note of the cantus firmus, or melody. The line of counterpoint should be written for an adjacent voice; that is, if the cantus firmus is in the alto, the line of counterpoint can be written for either soprano or tenor. The line of counterpoint must be compatible with the cantus firmus, but it must also maintain its independence from it.
Unlike the cantus firmus, an occasional tied note can be used in the line of counterpoint. Be aware, though, that you should avoid tying the same note twice in the same exercise; too much pitch-centeredness may result otherwise. The following harmonic (vertical) intervals are permitted between the two lines: P1 (restricted to the first and last interval of the exercise (lower counterpoint only), P5, P8, P12 (rare), M3, m3, M6, m6, and M10 and m10. Under exceptional circumstances, the compound 6th (M13 and m13) may be used.
There are four types of motion by which one may progress from one interval to another: parallel, similar, oblique, and contrary. In parallel motion, both voices move in the same direction by the same melodic interval. In similar motion, both voices simply move in the same direction. In oblique motion, one voice moves while the other is stationary. In contrary motion, the voices move in opposite directions. Of these four types of motion, contrary motion is best, since it fosters the greatest sense of independence between lines, and should be used more often than the other three types.
- Parallel perfect intervals (P1s, P5s, P8s, and P12s) are forbidden at all times. These intervallic progressions destroy utterly any sense of independence between lines. Parallel unisons are the worst offenders in this regard, followed closely by parallel 8ves and parallel 5ths.
- Direct or hidden 5ths and 8ves (approaching a 5th or 8ve by similar motion) are also forbidden, because of the "hidden" 5ths or 8ves that they produce (these progressions will be permitted in textures of more than 2 voices, however).
- Perfect consonances must be approached either by oblique motion, or by contrary motion. In the latter instance, one voice (usually the upper voice) will generally move by step .
- When both voices move by contrary motion and by leap into a perfect consonance, they create so-called "beaten" 5ths or 8ves (quinta battuta, ottava battuta) . These progressions should be avoided.
- Avoid simultaneous leaps (leaps in both the cantus firmus and the line of counterpoint at the same time) in the same direction when possible, especially large leaps (equal to or greater than a P4).
- Voice crossing , a situation in which the higher voice drops below the lower voice, or vice versa, is also undesirable, as is overlapping . In overlapping, the lower voice moves to a pitch higher than the previous location of the upper voice, or vice versa.
Most of the principles of first species still apply here. In writing two notes in the line of counterpoint against each whole note in the cantus firmus, the first half note in each measure must be consonant. Now, however, it is possible to introduce dissonance in the second half of the measure in the form of passing tones, and thereby the energy and tension of the line of counterpoint can be greatly enhanced.
The counterpoint will end on the note an octave or unison away from the final note in the cantus firmus, as in first species, and the penultimate note must be the leading tone or seventh modal degree. Occasionally a whole note may be used in the line of counterpoint in the penultimate measure; in this case it will always be the leading tone or seventh modal degree. It is still necessary to begin on a perfect consonance, but now the line of counterpoint may begin either on the first beat, or it may begin with a half-note rest. Beginning the line of counterpoint with a half-note rest encourages the sense of independence between the two lines.
- The counterpoint will end on the note an octave or unison away from the final note in the cantus firmus, as in first species, and the penultimate note must be the leading tone or seventh modal degree. Occasionally a whole note may be used in the line of counterpoint in the penultimate measure; in this case it will always be the leading tone or seventh modal degree.
- Dissonant intervals, such as the 2nd, 4th, 7th, 9th, and tritone, can be used on the second half note of a measure if the line of counterpoint moves up or down by step from one consonance to another. Such dissonances are called passing tones (P or PT). A passing tone is a stepwise connection between two other tones a third apart. The neighbor note (N -- a stepwise connection between two instances of the same note) may be used if and only if it is consonant with the cantus firmus.
- Never let the line of counterpoint leap into or out of a dissonant interval, since this effectively leaves the dissonance hanging, unfulfilled and unresolved.
- Leaps greater than a 3rd which cross over the barline (from 2nd beat to the subsequent strong beat) are more problematic than leaps within the measure. The goal of the leap sounds as though it had been shot out of a cannon. If you must leap over the barline, follow the leap with an immediate change of direction -- this will mitigate that dangling sensation.
- Large leaps within the measure should also be followed with a change of direction ("recovered"). This keeps the line of counterpoint from extending its register too low or too high. A large melodic interval (P5, major or minor 6th, or P8) can be broken into two smaller leaps; this is a useful technique for slowing down registral expansion.
- A P5 can be traversed by using two consecutive leaps of a 3rd; a 6th can be traversed by using two consecutive leaps of a 3rd and a P4, or a P4 and a 3rd; and finally, a P8 can be traversed by leaping a P5 and a P4, or a P4 and a P5. Any other combination of consecutive leaps (P4 + P4, 3rd + P5, for example) will result in a dissonant outline, and is therefore forbidden. As always with leaps, all intervals formed against the cantus firmus must be consonant. When you use two smaller leaps to gain a larger interval, the first of the two leaps should not be recovered. Once the goal is reached, however, an immediate change of direction should occur.
- Directly adjacent (parallel) P5s and P8s are, as before, forbidden under any circumstances. P5s and P8s on successive strong beats, mitigated by only one intervening half note, are to be strenuously avoided. P5s and P8s on successive weak beats, however, are acceptable, as long as they do not form a sequence.
- The unison can now be used, but only on the second half of the measure, in the middle of an exercise, and it must be left by step in the opposite direction from its approach. If you leave it with a skip, it is as though the line of counterpoint has fallen temporarily into a black hole.
As in 1st species counterpoint, the 2nd species counterpoint line should achieve a unique (unrepeated) climax which is not simultaneous with the high point of the cantus firmus.
The third species extends possibilities further than the second in that it allows for four notes of countermelody for each note in the cantus firmus. In third species, the line of counterpoint may either begin on the first quarter note of the measure, or it may begin with a quarter note rest. In either case, the first interval must be one of the same perfect consonances used in both first and second species.
Here is a summary of the categories of dissonance and embellishment which are now available to you:
- Passing tone (P): a three-note stepwise figure spanning the melodic interval of a 3rd, moving completely in one direction; first and last intervals must be consonant..
- Double passing tones (P P): a four-note stepwise figure spanning the melodic interval of a 4th, moving completely in one direction; first and last intervals must be consonant.
- Complete upper and lower neighbor note (UN, LN): a three-note stepwise figure leaving and returning to the same note, moving up by step, then returning down by step (UN), or the reverse -- moving down by step, then returning up by step (LN); first and last intervals must be consonant.
- Double neighbor notes (DN): a four-note figure which begins and ends on the same note. Profile: stable tone, UN, LN, return to stable tone. LN may precede UN in this figure, but this is less common than UN followed by LN. There is a leap of a 3rd between the two neighbors.
- Nota cambiata (n.c.): a five-note figure embellishing a stepwise progression from beat 1 (or 3) to the next beat 1 (or 3). The n.c.'s normal contour moves down a step, down a 3rd, up a step, up another step. If the n.c. is inverted (rare), the contour is up a step, up a 3rd, down a step, down another step. 1st, 3rd, and 5th intervals in this five-note idiom must be consonant.
- Embellishing tones and consonant skips: embellishing tones are consonant skips which leave and return to the same note (similar to neighbor notes in that regard). Consonant skips can be used to relieve stepwise motion and to break up larger skips. The skip itself, and the intervals formed at the beginning and the end of the skip, must all be consonant.
"Counterpoint is an easy way to write music; imagine only poles around which you will wrap your melodie by adding non-harmonic notes like "passing notes (Fr. notes de passage) anticipation, auxiliary note (Fr. broderie; Ger. Hilfsnote), échappée or cambiata and appoggiaturas (Fr. appoggiature; Ger. Vorschlag). You may read and listen to my recent music to better understand what counterpoint means. For my part, counterpoint is very easy way of composing music. Enjoy it and tell me soon how you liked it. A bientôt! (bye!)"
- François-Xavier JEAN, French composer. 
Dissonant counterpoint was first theororized by Charles Seeger's, who formulated it as counterpoint but with all the rules reversed. First species counterpoint is required to be all dissonances, and consonances are "resolved" through a skip, not step. Seeger was not the first to employ dissonant counterpoint, but was the first to theorize and promote it. Other composers who have used dissonant counterpoint, if not in the exact manner prescribed by Charles Seeger, include Ruth Crawford-Seeger, Carl Ruggles, Dhane Rudhyar, and Arnold Schoenberg.
Examples of Dissonant Counterpoint
- Concerto for Orchestra, II - Bela Bartok
- Sonata for Bass, Tuba, and Piano, III - Paul Hindemith
- Symphony No. 3 - Ligeti
- Trio for Violin, Cello, and Piano, Op. 67, No. 2, I (opening of second section) - Dmitri Shostakovich