Complex melodic motion is the quality of movement of a melody

Melody

A melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...

, including nearness or farness of successive pitches

Pitch (music)

Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

 or notes in a melody

Melody

A melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...

. This may be described as conjunct or disjunct, stepwise or skipwise

Steps and skips

In music, a step, or conjunct motion, is a linear or successive interval between two pitches which are consecutive scale degrees. Any larger interval is called a skip , or disjunct motion...

, respectively and involves the use of the complex number, i, in its calculation.

Bruno Nettl

Bruno Nettl

Bruno Nettl is an active ethnomusicologist and musicologist.Bruno Nettl was born in Czechoslovakia in 1930, moved to United States in 1939, studied at Indiana University and the University of Michigan, and has taught since 1964 at the University of Illinois, where he is Professor Emeritus of...

 (1956, p.51-53) describes various types of melodic movement or contour:

• Ascending
• Descending
• Undulating: equal movement in both of the above directions
• Pendulum: extreme undulation which uses a large range and large intervals
  Interval (music)
  In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
  
  
• Tile, terrace, or cascading: a number of descending phrases
  Phrase (music)
  In music and music theory, phrase and phrasing are concepts and practices related to grouping consecutive melodic notes, both in their composition and performance...
  
   in which each phrase begins on a higher pitch than the last ended
• Arc
• Rise: may be considered a musical form
  Musical form
  The term musical form refers to the overall structure or plan of a piece of music, and it describes the layout of a composition as divided into sections...
  
  , a contrasting section of higher pitch, a "musical plateau"

Other examples include:

• Double tonic
  Double tonic
  A double tonic is a chord progression, melodic motion, or shift of level consisting of a, "regular back-and-forth motion," in melody similar to Bruno Nettl's pendulum type though it uses small intervals, most often a whole tone though may be almost a semitone to a minor third .It is extremely...
  
  : smaller pendular motion in one direction

These all may be modal frame

Modal frame

In music a melodic mode or modal frame is one of, "a number of types permeating and unifying African, European, and American song" and melody. "Mode" and "frame" are used in this context interchangeably. Melodic modes allow melodies which are not chord-based or determined by the harmony but...

s or parts of modal frames.