More than likely it would be a nocturne, a short character piece, usually for solo piano. The first composer to write piano music with the title nocturne was Irish composer John Field. Arguably the nocturne was brought to perfection by Chopin.
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A battle piece is a painting or musical composition which represents a battle.
the name given to a musical piece performed by one person is called a Solo -If you have maybe a violin piece and its to be performed by one person then you could call it "Violin solo" or "solo Violin" actually you could call it anything and just add to the score that "this should be played by one player only or played solo"
A numbered musical piece might be called a concerto, an opus, or a composition. It could also be called a symphony.
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It is called and Etude
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The music piece that introduces a musical is called the overture or prelude.
A musical pieces intended as practice aid is called an étude.
A musical manuscript that serves as a blueprint for a musical piece is called a "score." This written representation includes the notes, rhythms, dynamics, and instrumentation necessary for performers to interpret and execute the work. Scores can vary in complexity, encompassing everything from simple lead sheets to elaborate orchestral scores.
The "A" melody
It's called an etude - i.e. A musical composition written solely to improve technique.
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The word for a short dreamy piece on the piano is often referred to as a "nocturne." Nocturnes are typically lyrical and expressive, capturing a sense of tranquility and reflection, often inspired by the night. Composers like Frédéric Chopin are well-known for their beautiful nocturnes that evoke a serene and contemplative mood.