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I would like to understand what a measure is, can someone help me understand this concept?

Wikipedia defines a measure as "a segment of time defined by a given number of beats", but this doesn't really explain how we would get the measure of a song.

If I am given a song, how do I find its time signature (i.e its beats per measure)?

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There are two different questions here. (a) what is a measure and (b) how do i find the time signature. (b) has been answered here –  Shevliaskovic Apr 2 at 19:36
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Hello KillaKem, I see you are an EE Student. Consider a measure as a segment of a time domain where time is expressed in the x axis and the y represents a pitch field. Not far from looking at an O'scope and identifing a horizontal division as one measure. –  filzilla Apr 2 at 23:38

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Let's take a simple music sheet: enter image description here

As you can see, the time signature is given, and it's 4/4. That means every measure has 4 beats of quarters. I assume you know about the duration of the notes.

Every measure/bar in the music sheet is defined by the vertical lines you can see in the image above. Between every two vertical lines, you must have notes whose duration add up to 4 quarters (1 whole, 2 halves, 4 quarters, 1 half and 2 quarters etc).

As per Wikipedia, the measures/bars are used in order to help us read music sheets:

Dividing music into bars provides regular reference points to pinpoint locations within a piece of music. It also makes written music easier to follow, since each bar of staff symbols can be read and played as a batch

In this question you can see how to find the time signature of a song by listening to it.

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When you say, "if I'm given a song", I assume you do NOT mean that you have sheet music. Is someone humming? Are you hearing a song on the radio?

You can usually find the measures and the time signature by listening carefully for the 1-beats. The 1-beats are the beginnings of musical phrases within the structure of a song.

For example, if you're listening to "Happy Birthday to You", you can hear the emphasis at the beginnings of the measures on "happy BIRTH-day to YOU" which tells you that (a) the measures begin on BIRTH-day and YOU, and that the song has three beats per measure, which typically means a 3/4 time signature.

That's a very simplified explanation that should help you without a lot of music theory knowledge.

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