progressioni

Progressioni is the Italian plural of progressione, a term used across disciplines to denote a sequence, a developing order, or a progression of events. The root is Latin progressio, meaning advancement or stepping forward. In everyday language, progressioni can refer to mathematical sequences, musical chord structures, or stages in a process.

In mathematics, a progressione refers to a sequence governed by a rule. The arithmetic progression (progressione aritmetica) has a constant difference d between consecutive terms; if the first term is a, the nth term is a_n = a + (n-1)d. The sum of the first n terms is S_n = n/2 [2a + (n-1)d]. The geometric progression (progressione geometrica) has a constant ratio r; terms satisfy a_n = a r^{n-1}, and the sum is S_n = a (1 − r^n)/(1 − r) for r ≠ 1.

In music, progressione armonica or simply progressione refers to a sequence of chords that establishes a sense of direction and tonal function. Common progressions include I–IV–V–I and ii–V–I, and the circle of fifths often governs voice-leading and modulation. Cadences such as authentic (V–I) or plagal (IV–I) endings are typical features.

In other contexts, progressioni describe the evolution of a process or condition. In medicine, for example, disease progression notes changes in symptoms or prognosis over time. In project planning or narrative analysis, progressioni designate planned stages or the sequence of events that unfold.