# Music Theory/Counterpoint
## Fugue

Fugue is sometimes considered to be the goal of contrapuntal study. Its definition may vary, depending upon the person asked but this book will consider three definitions.

### Fugue

    (1) A formal compositional structure, consisting of a fugal exposition of a subject, a development with one or more middle entries, and a recapitulation
    (2) A compositional device for the elaboration and development of a single (or multiple) musical idea(s)
    (3) A texture within a larger compositional structure or form which utilizes fugal devices or parts of a formal fugue
Traditional part writing is often based on counterpoint (adjective contrapuntal), in which a second melody is written to use harmonic intervals that create primarily consonant (good sounding) relationships between notes. First, a leading melodic line must be written, then contrapuntal methods may be used to add additional melodic lines. Consonants are further divided into perfect and imperfect consonants and contrasted by dissonance (which sounds bad). Traditionally the unison, the octave and the fifth were considered perfect consonances while the third and sixth were considered imperfect consonances; the fourth was considered a dissonance if it was above the leading line but a consonance if below, though modern theorists have questioned the need for this distinction. Other notes are considered dissonant. Additional melodic lines beyond the second may be added by writing them in counterpoint to all the lines already written.
In addition, the types of movement made a difference. In early theory there were three types of movement: direct, in which both melodic line move in the same direct, oblique in which one line moves and the other does not, and contrary in which both lines move in opposite directions. Some modern theorists further divide direct motion into similar motion, in which both lines move different distances in the same direction, and parallel motion in which both lines maintain the same interval between them. Early counterpoint texts (such as Johann Joseph Fux's Gradus ad Parnasum) listed several rules for moving lines, however these can be boiled down to one: Don't move from perfect consonance to perfect consonance by direct motion. The addition of dissonance may be added on an off beat, to be resolved on the next on beat. Large jumps are discouraged, and it is recommended that they be followed by stepwise motion in the opposite direction. The octave should be used vary sparingly and the unison avoided, however, as these tend to make the melodic lines blur together.
Various rhythms could be used in the various melodic lines, and this was encouraged with the goal being "florid counterpoint" in which various rhythms are mixed in an elaborate way. If a smaller note is used in the following line then dissonance must be on the second note; if the following note starts after the leader, then it must start on the dissonance. Of course, neither have to be dissonant, and a consonance may be used in both places. A series of stepwise movements may also be used to move over several dissonances between consonances. However, two notes one semitone apart (sometimes described as "fa on mi") should always be avoided.
Double Counterpoint is the practice of writing a line so that both the intervals and their inversion (usually octave inversions, but tenth and twelfth inversions may sometimes be used) follow an acceptable pattern of consonance and dissonance. This is done so that a transposed version of the resulting melodic line may be used as a separate part. This is especially common in classical and baroque fugues.
Note that a modern composition that follows a chord progression may just write all parts to fit the same chord, however, use of contrapuntal principles may still be useful in order to make to melodic lines work better together.