The year 1721 in music involved some significant events.

Events

• December 3 – Johann Sebastian Bach marries his second wife, Anna Magdalena.
• Antonio Maria Bononcini becomes maestro di cappella in his native city of Modena.
• Georg Philipp Telemann becomes director of music in Hamburg.

Classical music

• Johann Sebastian Bach
  • Brandenburg Concertos, a collection of five concerti grossi and one ripieno concerto presented to Christian Ludwig, Margrave of Brandenburg-Schwedt.
  • Capriccio in E major, BWV 993
• Giovanni Bononcini – Cantate e Duetti
• Evaristo Felice Dall'Abaco – 6 Concerti à più istrumenti, Op. 5
• Christoph Graupner – Sonata in G minor, GWV 724
• George Frideric Handel
  • Crudel tiranno amor, HWV 97
  • Keyboard Sonatina in G minor, HWV 583
• Pietro Locatelli – 12 Concerti grossi à 4 e à 5, Op. 1 (including concerto in F minor "Christmas Concerto")
• Jean-Philippe Rameau – Orphée, RCT 27
• Valentin Rathgeber – Octava musica clavium octo musicarum, R 295
• Georg Philipp Telemann
  • Ich halte aber dafür, TWV 1:840
  • Ich hoffe darauf daß du so gnädig bist, TWV 1:847
• Francesco Maria Veracini – 12 Violin Sonatas, Op. 1
• Silvius Leopold Weiss – Tombeau sur la mort de M. Comte de Logy arrivée, WSW 20

Opera

• Filippo Amadei, Giovanni Bononcini & George Frideric Handel – Muzio Scevola
• Giovanni Bononcini – Crispo
• George Frideric Handel – Floridante, HWV 14
• Michel Richard de Lalande – Les élémens, S.153 (composed with André Cardinal Destouches)
• Giuseppe Maria Orlandini – Nerone
• Nicola Porpora
  • Eumene
  • Gli orti esperidi
• Alessandro Scarlatti – La Griselda
• Georg Philipp Telemann – Der Geduldige Socrates
• Antonio Vivaldi – La Silvia, RV 734
• Johann Adolph Hasse – Antioco

Theoretical Writings

• Alexander Malcolm – A Treatise of Musick
• Johann Mattheson – Das forschende Orchestre
• Franz Xaver Murschhauser – Academia Musico-Poetica Bipartita
• Thomas Walter – The Grounds and Rules of Musick Explained

Births

• April 1 – Pieter Hellendaal, organist, violinist and composer (died 1799)
• April 7 – Matthias Vanden Gheyn, composer (died 1785)
• December 9 – Peter Pelham, organist, harpsichordist and composer (d. 1805)
• December 27 (bapt.) – John Garth, composer (died 1810)
• date unknown
  • Barbara Campanini, dancer
  • Quirino Gasparini, composer (died 1778)
  • Giovanni Battista Lorenzi, Italian librettist (died 1807)

Deaths

• February 22 – Johann Christoph Bach, organist, elder brother of Johann Sebastian (born 1642)
• September 3 – Jan Antonín Losy, lute player and composer (born c. 1643)
• date unknown
  • Jacques Paisible, recorder virtuoso and composer (born c. 1656)
  • Jerónimo de Carrión, composer (b. 1660)