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I have (as far as I know) a basic understanding in music theory. But one thing I never understood was triplets. Logically, to me, they make absolutely no sense. I know how to calculate the value: If you have a triplet of three halve notes it would equal a whole note. But that doesn't make any sense to me! 2+2+2 Does NOT equal 4. Wouldn't that mess up the time signature making it sound weird? (obviously not but I have never gotten an example) So I'm asking for an explanation of what triplets sound like(examples!), why/when I should use them, and finally how they work e.g. how somebody concluded 2+2+2=4.

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It's not a question of 2+2+2=4, though you're on the right track. The idea of a triplet is that you have three notes squeezed into the area that two notes normally take up. So if you have triplet half notes (which is kind of rare, I have to say), they'll take up the same timespan as two half notes, meaning 2+2=4, and the world is alright again.

In other words, triplet notes are not actually the length of the note value that's written. (They're the length of the triplet form of the note value that's written.)

There are also duplets, which are in some ways the other side of the same coin. It's tough for us to provide sound examples on this site, but a quick YouTube search showed me this video, which seems okay.

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    Agreed, a half note triplet in question doesn't consume two beats, and triplet notation will have a bracket and a 3 over it, or below it, to distinguish it from the regular form of the note. If you want to get technical and use math, the bracket and 3 changes the duration of the note to 2/3 of its unbracketed value. – jaybrau Aug 11 '16 at 17:43
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Triplets modify the value of the duration they are applied to. They basically tell you that 3 notes now go where 2 use to. In your example with half note triplets in 4/4 the resulting half notes do not get the value of 2 beats, but instead is 2/3 of that value. So adding them you will instead have 4/3 + 4/3 + 4/3 which is 12/3 which is 4.

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You asked for an example of what triplets & "conventional" notes sound like together. The king of this was Anton Bruckner. He used a figure consisting of a duplet and a triplet quite often in his music, so much so that it's often called a "Bruckner rhythm" to this day. For an example of this, you can't do much better than the opening of his Fourth Symphony:

The Bruckner rhythm kicks in for the first time around the 1:20 mark in the above video.

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March around the room. Sing "One, Two-and-a, One, Two-and" keeping to the beat of your step. Or try "My Cat-has-a Red Col-lar". If necessary, get a musician to show you what I mean. The "Two-and-a" or "Cat-has-a" are triplets.

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    Those words could, and probably would, be sung in a rhythm other than triplets. There must be a phrase that can only be said in triplets. How about 'Merrily, merrily' from 'Row, row, row your boat'. – Tim Aug 11 '16 at 6:54
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I don't understand exactly what your problem is. If you have two beats of whatever, that you are playing three notes in, then you have triplets. It is really not hard. If you tap your feet four times in a bar and you play three notes for every two taps of your feet then you have triplets.

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