Origins
The word 'cadenza' is about falling — delayed falling, virtuosic falling, falling that has been turned into art. It derives from Latin 'cadere' (to fall), and the concept of a musical 'fall' — the resolution of tension, the descent to a final chord — is at its etymological and musical core.
Italian 'cadenza' means 'cadence' — a falling. In music, a cadence is the chord progression that brings a phrase, a section, or an entire movement to a close. The most fundamental cadence in Western music is the 'authentic cadence,' a progression from the dominant chord (V) to the tonic (I), which creates a strong sense of resolution — a feeling of 'falling' into place. The word 'cadence' captures this sensation: Latin 'cadere' means 'to fall,' and a cadence is the musical moment when the harmony falls to rest.
A 'cadenza' began as an elaboration of this falling moment. In the Baroque and Classical concerto, the soloist was expected to display virtuosic skill near the end of a movement by improvising an extended passage over the sustained dominant harmony — delaying the final cadence while dazzling the audience with runs, trills, arpeggios, and thematic transformations. The orchestra would stop, the soloist would play alone, and the audience would hear a private display of inventiveness and technique. When the cadenza was over, the soloist would signal the orchestra's re-entry (traditionally with a long trill on the dominant), and the orchestra would play the final cadence — the long-delayed 'fall' to the tonic.
Latin Roots
This tradition of improvised cadenzas meant that no two performances were alike, and some of the greatest cadenzas were never written down. Mozart's cadenzas for his own piano concertos survive because he wrote some of them out — possibly for his students — but many performers throughout the Classical period improvised freely. Beethoven changed this practice definitively: in his Piano Concerto No. 5 (the 'Emperor'), he wrote the cadenza directly into the score, with the instruction 'Non si fa una Cadenza' ('Do not make a cadenza' — i.e., do not improvise your own). From Beethoven onward, composers increasingly wrote out their cadenzas, and the improvised cadenza became rare.
The Latin root 'cadere' is among the most fertile in the English vocabulary. 'Case' (a situation, a thing that has befallen — from Latin 'cāsus,' a fall, from 'cadere'). 'Accident' (from Latin 'accidēns,' falling upon — something that happens to you). 'Occasion' (from Latin 'occāsiō,' a falling toward — an opportunity). 'Cascade' (from Italian 'cascata,' a falling — a waterfall). 'Decay' (from Old French 'decair,' to fall away). 'Deciduous' (from Latin 'deciduus,' falling down — of trees that shed their leaves). 'Cadaver' (from Latin 'cadāver,' one who has fallen — a corpse). All of these English words, and many more, trace to the same Proto-Indo-European root *ḱad- (to fall).
The journey from 'falling' to 'virtuosic solo display' is one of the more dramatic semantic expansions in musical terminology. A cadenza is a fall that has been suspended, elaborated, and transformed into a moment of individual brilliance — gravity defied, temporarily, by art.