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So I am going to learn Mariage d'Amour on the piano, but the sheet music constantly changes time signatures. It jumps between 4/4 and 5/4. I even saw 3/4 and 3/8!

Is there a way to learn this piece efficiently? Or I just have to deal with it, and change the metronome all the time?

This is how the sheet music looks

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    You shouldn't need to change the metronome, but I certainly wouldn't have the metronome (app) accent any beats. Just leave it in 4/4 and don't worry that the measures won't line up. Use the pulse. – The Chaz 2.0 Apr 1 '18 at 11:48
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With 3/4, 4/4 and 5/4, the pulse is constant. In other words, the crotchet beat remains the same, it's just the number of those beats per bar that changes. Probably better not to set the metronome at all. If you must use it, leave it without the 'ping' that comes on the first beat of a bar - merely clicks. But there's no advantage in having it ticking away at all.

Make sure that the first note in each bar is emphasised - that's why it's written in this way. It could have been written all in the same time sig., but various notes written with > (accent) over them, it's virtually the same thing.

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As has been said, the pulse will remain constant, so you only need to worry about how many beats you count in each measure. But in the 3/8 measure, you want to compare the length of the 8th note to the 8th notes in the other measures. Instead of thinking about the quarter note getting the beat there, think of the 8th note getting the beat. You could also think of it is half of a 3/4 measure.

5/4 is really a time signature that is subdivided into a group of three beats and a group of two beats. (It could be 3+2 or 2+3) You will see this visually in the music. The measures I took a look at were be grouped as three, then two, but I didn't check to see whether or not that was true for every 5/4 measure in the piece. This grouping will give beat one the strongest pulse and beat 4 a strong pulse, but not as strong as beat one. Hearing that second strong pulse in the measure will help to keep the feel right and help you keep track of how many beats have gone by.

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