Counting duplets
I read about duplets in "Music Theory for Dummies" by Michael Pilhofer and it was introduced with the following graphic (not exactly the same - it should be interpreted as a rhythmic notation).
The counting for this rhythm is given as
ONE two three FOUR-and ONE-and FOUR five six
The way I understand tuplets is that they have a 2:3 ration, meaning two notes take the place of the normal three.
But I cannot makes sense of the counting, especially that there are two bars in the example. I would have expected to count twice to six - two bars with six beats each.
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Forgetting tuplets for a bit, think about how you count a measure of 4/4 with two half notes. You count "one... three...". The "two" and "four" are part of the bar, but they're not played.
In the same way, what might be confusing you is the fact that the tuplets actually span multiple beats in your example, and the "and"s happen to fall evenly between the not-played beats. The "and"s actually apply to the not-played "five" in the first bar, and the "two" in the second bar.
So, writing two half notes in 4/4 as "one (two) three (four)", the first bar would be "one two three four (five) and (six)" and the second bar "one (two) and (three) four five six".
You might find this easier to comprehend if you count 6/8 as two in a bar, not 6. Then you have a three-group followed by a two-group, both taking up one beat. ONE-and-a TWO-and, ONE-and TWO-and-a. It's the exact equivalent of using triplets in 2/4.
You can bypass this issue by counting the 6/8 as 2. That is count the beats on 1 and 4 rather than 1,2,3,4,5,6. This is a typical way of counting 6/8 especially when it's a fast tune. In this way you are reading the 2 as a proper 1-and, and the three group as a triplet. This is much easier to read.
If you think of a rhythm that splits the middle eighth note in each group of three into two sixteenths, the second note of the duplet will fall on the second of the sixteenths. For example, in the song "chim chim cheree" (by the Sherman brother, and featured in the film Mary Poppins) triplets would be counted as "CHIM CHIM-in-EE, CHIM CHIM-in-EE", while duplets would be counted as "CHIM chim-IN-ee".
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