In music theory, counterpoint refers to the way two or more musical lines, also called voices, work together in harmony. These lines are independent in rhythm and the shape of their melodies but rely on each other for harmony. The term comes from the Latin phrase punctus contra punctum, meaning "note against note."

John Rahn explains that counterpoint is most closely linked to the European classical tradition. It became more common during the Renaissance and was widely used during the Baroque period, which is part of the common practice period in Western music. In music education, counterpoint is taught using a system called species, as described below.

There are different types of counterpoint, such as imitative counterpoint and free counterpoint. Imitative counterpoint happens when a main melody is repeated in different vocal parts, sometimes with changes. Free counterpoint often includes unusual harmonies, chords, and sounds like chromaticism and dissonance.

General principles

The word "counterpoint" refers to a single musical line or an entire piece of music. Counterpoint focuses on how melodies interact with each other, and it is less about the harmonies that result from this interaction.

Guerino Mazzola (born in 1947) began work that gave counterpoint theory a mathematical basis. His model explains rules, such as why certain intervals like fifths and dissonant fourths are avoided, using structure rather than psychology. Octavio Agustin later adapted this model to include music with very small pitch differences. Another theorist, Sergei Taneyev (1856–1915), also used math to study counterpoint. Inspired by Spinoza, Taneyev created a theory that explains many complex counterpoint techniques, including a type called "invertible counterpoint," which he described using his own special terms linked to simple math rules.

In counterpoint, each musical line must remain independent. When this independence is broken, it creates effects that are not allowed in counterpoint. On an organ, pressing one key can activate multiple sounds at once, causing melodies to move together in parallel. This makes the separate voices blend into one, creating a new sound. In counterpoint, parallel movement is not allowed because it reduces the variety of sounds in the music, making it less distinct when voices merge or separate.

Development

Some examples of related compositional techniques include: the round (common in folk music), the canon, and the fugue, which is the most complex type of contrapuntal writing. All of these are forms of imitative counterpoint.

Examples from the repertoire

There are many examples of songs with melodies that work well together in harmony but have different rhythms and shapes. For instance, "Frère Jacques" and "Three Blind Mice" sound pleasant when sung together. Many popular songs that use the same chord pattern can also be sung together as counterpoint. A famous example is "My Way" combined with "Life on Mars?".

Johann Sebastian Bach is considered one of the greatest experts in counterpoint. For example, the harmony in the opening melody of the Fugue in G♯ minor from Book II of the Well-Tempered Clavier is heard again in a subtle way when a second voice is added. "The counterpoint in bars 5-8… sheds an unexpected light on the tonality of the Subject.":

Bach's 3-part Invention in F minor combines three separate melodies:

According to pianist András Schiff, Bach's use of counterpoint influenced the music of both Mozart and Beethoven. In the development section of the opening movement of Beethoven's Piano Sonata in E minor, Beethoven shows this influence by adding "a wonderful counterpoint" to one of the main themes.

Another example of flexible counterpoint in Beethoven's later work can be found in the first orchestral variation of the "Ode to Joy" theme in the final movement of Beethoven's Symphony No. 9, bars 116–123. The famous melody is played by violas and cellos, while "the basses add a bass-line that seems to be played on the spot. At the same time, a solo bassoon adds a counterpoint that also feels spontaneous."

In the Prelude to Richard Wagner's opera Die Meistersinger von Nürnberg, three themes from the opera are played at the same time. According to Gordon Jacob, "This is universally and justly acclaimed as an extraordinary feat of skill." However, Donald Tovey notes that here "the combination of themes… unlike classical counterpoint, really do not of themselves combine into complete or pleasant harmony."

One impressive example of 5-voice counterpoint is found in the finale of Mozart's Symphony No. 41 ("Jupiter" Symphony). Here, five different melodies are played together in "a rich tapestry of dialogue":

See also Invertible counterpoint.

Species counterpoint

Species counterpoint is a teaching method used in music education. Students learn by progressing through different levels, or "species," each with increasing difficulty. One part, called the cantus firmus (Latin for "fixed melody"), remains simple and unchanged throughout the exercises. Species counterpoint gives composers fewer choices compared to other types of counterpoint, so it is called "strict" counterpoint. Over time, students learn to write "free" counterpoint, which has fewer rules and does not always use a cantus firmus. This method dates back at least to 1532, when Giovanni Maria Lanfranco described a similar idea in his book Scintille di musica. Later, in the 16th century, Zarlino expanded on the concept in Le institutioni harmoniche. The first detailed version of species counterpoint was written in 1619 by Lodovico Zacconi in Prattica di musica. Zacconi included extra techniques, such as invertible counterpoint, which allows melodies to be rearranged or flipped.

In 1725, Johann Joseph Fux wrote Gradus ad Parnassum ("Steps to Parnassus"), describing five species:
1. Note against note;
2. Two notes against one;
3. Four notes against one;
4. Notes offset against each other (as suspensions);
5. All the first four species combined, called "florid" counterpoint.

Later theorists followed Fux’s work closely, often making small changes to the rules. Many of Fux’s rules about melody construction came from solfeggio, a method for teaching singing. During the common practice era, rules were adjusted to support harmonic structures, such as figured bass, which uses numbers to indicate chord notes.

The following rules apply to melodic writing in each species for each part:
1. The final note must be reached by moving up or down one note at a time. In minor keys (like Dorian or Aeolian), the note before the final one must be raised (e.g., C♯ in the Dorian mode on D).
2. Allowed intervals include perfect unison, fourth, fifth, and octave, as well as major and minor second, third, and ascending minor sixth. The ascending minor sixth must be followed by downward motion.
3. If two skips (jumps) occur in the same direction, the second must be smaller than the first. The three notes should come from the same triad, or not outline more than one octave. Avoid more than two skips in the same direction.
4. After a skip, move step-by-step in the opposite direction.
5. Avoid the tritone (a specific interval) in three notes, such as F–A–B♮, and avoid the seventh interval in three notes.
6. Each melody must have a high point, usually in the middle of the exercise, on a strong beat.
7. Avoid outlining a seventh in a single line moving the same direction.

Rules for combining parts in all species:
1. Counterpoint must begin and end with a perfect consonance (e.g., unison, octave, or fifth).
2. Contrary motion (moving in opposite directions) should be common.
3. Perfect consonances must be approached by oblique or contrary motion.
4. Imperfect consonances can be approached by any motion.
5. Avoid intervals larger than a tenth between adjacent parts unless necessary.
6. Build melodies starting from the bass (lowest note) and move upward.

In first species counterpoint, each note in the added parts matches one note in the cantus firmus. All notes sound at the same time, and movement is simultaneous. Since all notes are whole notes, there is no rhythmic independence.

A "step" is a half or whole note movement. A "skip" is a third or fourth. A "leap" is a fifth or larger.

Additional rules from Fux and later teachers include:
1. Begin and end on unison, octave, or fifth, unless the added part is below, in which case use only unison or octave.
2. Avoid unisons except at the start or end.
3. Avoid parallel fifths or octaves between parts, or "hidden" parallel fifths/octaves unless one part moves step-by-step.
4. Avoid parallel fourths, though some composers used them.
5. Do not use the same interval more than three times in a row.
6. Use up to three parallel thirds or sixths in a row.
7. Keep parts within a tenth of each other unless a special melody is created.
8. Avoid two parts moving in the same direction by skip.
9. Use as much contrary motion as possible.
10. Avoid dissonant intervals (e.g., seconds, sevenths, augmented or diminished intervals, and perfect fourths in some cases).

In second species counterpoint, two notes in each added part match one longer note in the cantus firmus. An example is shown below. Additional rules for second species include:
1. It is okay to start with an upbeat and leave a half-rest in the added part.
2. Accented beats (strong beats) can have consonances, while unaccented beats can have dissonances in the form of passing notes, neighbor notes, or escape tones.
3. Dissonance on an accented beat must resolve to a consonance on the next unaccented beat.
4. Avoid unisons except at the start or end, or on unaccented beats.
5. Be careful with repeated accented perfect fifths or octaves. Avoid similar motion in both parts.

In third species counterpoint, four (or three) notes in each

Contrapuntal derivations

During the Renaissance in Europe, many composers used a style called imitative counterpoint in their contrapuntal music. In imitative counterpoint, different musical lines (called voices) start at different times. When they begin, each line repeats a similar melody. Types of music like the fantasia, ricercar, canon, and fugue (the most famous contrapuntal form) use imitative counterpoint. This style is also common in choral works such as motets and madrigals. Imitative counterpoint led to the creation of several techniques, including:

Free counterpoint

From the Baroque period onward, most compositions with multiple melodies were written using a style called free counterpoint. This means composers focused less on how added melodies related to a fixed main melody and more on how the melodies interacted with each other.

However, according to Kent Kennan, teaching free counterpoint became common only in the late 1800s. Composers like Mozart, Beethoven, and Schumann were taught the older style of "strict" counterpoint during the 1700s and 1800s. In practice, they found ways to expand on traditional ideas of musical themes.

Main features of free counterpoint:
1. Chords that were not allowed before, such as second-inversion, seventh, and ninth chords, can now be used freely if they end with a pleasant-sounding triad.
2. Chromaticism, which involves using notes outside the main scale, is permitted.
3. Rules about when dissonant notes can appear are no longer strict. Passing tones can be used on strong beats.
4. Appoggiatura is allowed: dissonant notes can be approached with large musical jumps.

Linear counterpoint

Linear counterpoint is a method where each melody stays unique and clear, even when different melodies are played together. It focuses on how melodies work individually, not on how they sound when combined in harmony. This technique was used by musicians who wanted to move away from Romantic harmony and instead highlight the importance of individual melodies. The parts in the music move freely, and their combined movement is not controlled by harmonic rules. This means either the focus is on the horizontal movement of melodies or the control of melodies by harmony is ignored.

This technique is linked to neoclassicism and was first used in Igor Stravinsky's Octet (1923). It was inspired by the music of J. S. Bach and Giovanni Palestrina. However, Knud Jeppesen noted that Bach and Palestrina had different starting points. Palestrina began with melodies and created chords from them, while Bach started with a harmonic background and let the melodies develop independently.

According to Cunningham, linear harmony was a common approach in the 20th century. In this method, melodies are combined in a way that may seem random, hoping to create new "chords" and "progressions." This can be done with any type of melody, whether it follows traditional scales or more complex systems like the duodecuple scale.

Dissonant counterpoint

Dissonant counterpoint was first developed as a theory by Charles Seeger. He described it as a method of teaching music, based on traditional counterpoint rules but with all the rules reversed. In this approach, the first type of counterpoint, called first species, uses only dissonant sounds instead of harmonious ones. This makes dissonance the main rule, and harmonious sounds are fixed by moving to a different note instead of a stepwise motion. Seeger believed this method helped musicians understand music more clearly. Other elements of music, like rhythm, could also be changed using the same principle.

Seeger was not the first person to use dissonant counterpoint, but he was the first to explain and share the idea widely. Other composers who used dissonant counterpoint, though not always in the exact way Seeger described, include Johanna Beyer, John Cage, Ruth Crawford-Seeger, Vivian Fine, Carl Ruggles, Henry Cowell, Carlos Chávez, John J. Becker, Henry Brant, Lou Harrison, Wallingford Riegger, and Frank Wigglesworth.