Progression: ordered change in mathematics, music, and other fields

Mathematical progressions

In mathematics a progression (often called a sequence) is a list of numbers generated by a fixed relation. Two classical types are:

• Arithmetic progression: each term increases by a constant difference. If a1 is the first term and d the difference, the nth term is a1 + (n-1)d.
• Geometric progression: each term is multiplied by a fixed ratio r. The nth term is a1 · r^(n-1).

These patterns are fundamental in algebra, series summation, financial calculations (simple versus compound growth), and in modelling regular change.

Musical progressions

In music, a progression usually refers to a sequence of chords that provides harmony and direction during a piece. Common patterns, such as the I–IV–V sequence or the ii–V–I cadence in jazz, create expectations and resolution. For background on harmonic movement see chord progression.

Specialized types include the backdoor progression (an alternative dominant route) and the omnibus progression (chromatic voice-leading connecting distant harmonies). Ragtime harmony follows characteristic patterns known as the ragtime progression, which helped define an early American popular style.

Other uses and examples

• Age progression — visual or forensic methods for estimating appearance as a person ages.
• Color progression — ordered palettes and gradients used in design, visualization, and art to indicate change or scale.
• Semantic progression — linguistic change in meaning over time, including processes like broadening, narrowing, or grammaticalization.

Although the term is simple, its applications vary: mathematical progressions emphasize exact rules, musical progressions emphasize perceived tension and resolution, and applied progressions model or communicate gradual transformation. Recognizing the governing rule — additive, multiplicative, tonal, visual, or semantic — is the key to understanding any progression.