Tempo Marking/Metronome Marking/Time Signature

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From my understanding, tempo markings (such as vivace, largo, etc) give the overall direction while metronome markings give the precise speed the composer intended. I was also taught that time signatures are there purely for notational purposes and have no effect on the tempo of the piece.

However, I’m getting lost at some inconsistencies between these 3 concepts. Take, for example, Chopin’s Etude Op. 10 No. 5 (Black Keys). It’s written in 2/4 time with the tempo marking Vivace, and Chopin specifies a M.M. of 116. However, when I listen to pianists perform the piece, they seem to be performing it significantly faster than what is written. Does the piece being in 2/4 mean that the M.M. should be doubled as it’s no longer 4 beats per measure?

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Time signature does not affect the stated tempo except in which note receives a beat as in cut time 2/2. The example you indicated is deceptively fast due to the quick 16th triplets.

The short answer to your question is no.

The long answer is it’s complicated. Keyboard pieces by composers like Chopin or Liszt in particular are often treated as show pieces. Musicians will play them much faster than written to show off their technical skill, tempo markings be damned.

2/4 is two beats per measure, where a beat is written in quarter notes. The piece is declared as “Vivace 1/4 = 116”, which means you put your metronome to 116 beats per minute, and one beat is one quarter note.
Since the notation is in 2/4, there are two beats per measure.

Also, interprets usually have all the freedom to perform a piece, especially regarding tempo. Some play faster, of course for “aesthetic reasons”, but mostly to show off…

And second, you don’t want to play Chopin with metronome.