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I found myself telling someone to accent "the & of 3". This just seems a really awkward way of wording it, but I don't know a better way. Beat 3 and half or the 6th 8th note sound even worse.

Is there proper terminology for the positions between the beats?

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"The & of 3" is proper terminology and is very specifically tied to how you would count out the music. A very similar alternative you could say is "the offbeat of 3" which means the same exact thing as "the & of 3". You could also say the following, but they are much more wordy:

  • Halfway between 3 and 4
  • An eighth note after 3
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  • In conducting, the upbeat is the last beat of a bar. Bars have upbeats but beats themselves don't. For this reason I don't agree that "the upbeat of 3" does mean the same as "the & of 3". But I think the other explanations you gave are fine. – Brian THOMAS Sep 22 '16 at 16:25
  • @BrianTHOMAS good catch. I actually ment the offbeat of 3. Will fix now. – Dom Sep 22 '16 at 16:33
  • "3 and" or "and before 4" would be the most intuitive for me. – user1079505 Sep 8 at 21:54
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This is part of a pretty common set of terminologies used to communicate sub-beat patterns. These are, for the case of X/4 time sigs:

a) quarter notes: one, two, three,... b) eighth notes: one-and, two-and, three-and c) triplets: one-and-ah, two-and-ah d) sixteenths one-eee-and-ah, two-eee-and-ah, e) quintuplets: you're on your own :-)

For any of these patterns, the composer may have indicated emphasis on one or more of the on- and off- beat notes.

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