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I am trying to help my wife work out the naming of this notation. Its from the Doctor Who theme and is accompanied after some triplets for the trumpets

Measure is 4 beats, what is the count?

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Those are simply triplets. Think of it as :

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But with the first two triplets tied:

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Which results in your notation:

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(since two eighths equal one quarter).

If you wanted a more specific name for it, you could say 'a quarter and an eighth triplet'

These two notes are to be played on the count of one quarter (one beat if your time signature is 4/4)

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This seems like triplets where the second triplet beat is skipped. Just the first and third division of that triplet is played. it goes like this when dividing each beat in triplet :

1     2
1 2 3 1 2 3
x   x x   x               (play something on each x's)

This rhythm is commonly used in blues and jazz music and is named "shuffle beat" in those musical genres contexts.

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