You may think that the word "improvisation" does not belong in the world of classical music, but in fact, composers and performers often made music off the tops of their heads. Here's Ether Game's list of instances of improvisation in different historical works:

• Wolfgang Amadeus Mozart, Variations on 'Ah! Vous Dirai-Je Maman' K. 265: Fans of the movie Amadeus will remember a scene in the movie at the court of Emperor Joseph where Mozart rewrites Salieri's sonata on the spot, and proceeds to "improve" the piece through an elaborate set of improvised variations. Mozart was renowned for his ability to improvise, and one spectator wrote that "One had only to give him the first subject which came to mind for a fugue or an invention: he would develop it with strange variations and constantly changing passages as long as one wished." The form that Mozart improvised in became known in the Classical era as the theme and variations form. A short musical idea, often of eight or sixteen measures, is reworked through different styles and textures. It is likely that many of Mozart's written-out theme and variations are basically transcriptions of his improvisations. For instance, this set of variations is based on an old French folk song that you may recognize as "Twinkle Twinkle Little Star."

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• Scott Joplin, "Maple Leaf Rag":Â Scott Joplin was the undisputed "king of ragtime piano." He grew up in northeast Texas, but in 1894, he settled temporarily in Sedalia, Missouri and often performed at the local Maple Leaf Club. A few years later, he published the "Maple Leaf Rag," which references his old employer. Joplin helped to make ragtime a popular genre among sheet music publishers, but the origins of his music are in improvisation. Ragtime (like jazz) is a distinctly American music, developing from the unwritten style improvised by African American street bands in Saint Louis. "Rag" was used to identify the syncopated swing-rhythms played in the bass lines of the dance tunes. Joplin however, produced all of his work by manuscript, and preferred to align himself with the great composers of Western art music.

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• Franz Liszt, Concert Paraphrase on Verdi's "Rigoletto": Franz Liszt is widely regarded as one of the greatest piano virtuosos of his or any age. He enjoyed an almost "rock star" level of celebrity throughout his life. His concerts would draw huge crowds of people who came from far and wide to see him perform. In addition to playing his own compositions, he would also take popular arias of the time and improvise elaborate embellishments on them. Later, he published transcriptions in the style of these improvised arrangements which he called concert paraphrases, including this one of the famous quartet from Rigoletto.

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• Ralph Vaughan Williams, Fantasia On A Theme Of Thomas Tallis: Ralph Vaughan Williams was particularly inspired by music of the English Renaissance and would often incorporate themes or styles from that period in his music. His first major success as a composer came with the premiere of his Fantasia on a Theme by Thomas Tallis in 1910. The work incorporates two English Renaissance music practices that have roots in improvisation. The first is the fantasia, a genre of instrumental music from the 16th century that seems impromptu and wandering, requiring no set musical form to compose. The second is the use of a technique called faburden. This practice began as an early form of improvised polyphony in which outer voices were added to a melody line at intervals of a sixth and a fourth, creating lines of parallel inverted major triads. The practice became so common among English musicians that it was referred to be composers throughout Europe as the "English Sound."

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• Sergei Rachmaninoff, Piano Concerto No. 3: Though it no longer plays a significant part in modern classical music, improvisation still features in concerti, in the section of the score marked cadenza. Originating in the Baroque era, the cadenza indicates an opportunity in the piece where the orchestra drops out and the soloist improvises feats of virtuosity. That being said, particularly famous or impressive cadenzas became associated with certain pieces, especially if they had also been performed by the composer. As with this concerto by Rachmaninoff, a written cadenza might be included to reproduce what the best performers of the piece had originally improvised. Rachmaninoff for example wrote two different cadenzas for the soloist to choose from in his 3rd piano concerto. The first and most often performed is chordal and quite difficult, while the second is more wandering and quiet, in the style of the Baroque toccata.

https://youtu.be/MOOfoW5_2iE?t=10m39s

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• Arcangelo Corelli, Sonata da Camera a Tre in G:Â Despite its name, Corelli's trio sonatas often included an unspecified number of continuo parts. Instruments such as harpsihord and lute that performed continuo accompaniment or figured bass as it was also called, utilized a sort of guided improvisation to create their parts. Rather than traditional note heads, basso continuo uses a bass line and a series of numbers that referred to chord voicings. Good continuo players are skilled enough in the system to create parts from the bass line and numbers as they read the score, listening carefully to the melody lines to make sure their accompaniment interlocks with the upper voices. Basso continuo reached its highest popularity during the Baroque era, but in fact survived all the way to the early Romantic era where it remained in sacred music as a formality.

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• Dietrich Buxtehude, Toccata In G Major: At the end of 1705 and through the first months of 1706, Johann Sebastian Bach took an unauthorized leave of absence from the position of Organist in Arnstadt and walked 200 miles to observe the composer Dietrich Buxtehude in Lübeck. Bach claimed he made the journey "to comprehend one thing or another about [Buxtehude's] art." The influence this composer had on Bach is undeniable and can be heard in his early organ works and the structure of the many cantatas he wrote during his life. Bach would also have heard Buxtehude perform several toccatas. A favorite among organ players, the Toccata was an improvisatory genre with no formal characteristics other than that performer should take a musical motive and develop it through a variety of styles and musical gestures. Because they were so versatile, Toccatas were often used as interstitial music in church services.