# Is there any nesting level of tuplets above which any tuplets can be rewritten with lower-level nested tuplets?

Tuplets can be nested. I read:

In Dorico Pro, there is no limit to the number of levels you can have in nested tuplets

This made me wonder: Is there any nesting level of tuplets above which any nested tuplets of that level or higher could be rewritten with lower-level nested tuplets?

Below is an example for "rewritten with lower-level nested tuplets". These 3-level nested tuplets:

can be equivalently written as 2-level next tuplets:

As Brian Chandler commented, in the upper version the last semiquaver is 1/5 of a triplet, whereas in the lower version it is 2/9 of the triplet, which is different. But you get the idea of what I mean by "rewritten with lower-level nested tuplets".

• Your question is not about music, since in practive no-one could read more than about two levels. But it might be a mathematical puzzle question (to which the answer is almost certainly "No"). First, however, you need to show your working: in the upper version the last semiquaver is 1/5 of a triplet, whereas in the lower version it is 2/9 of the triplet, which is different. Apr 14 at 4:33
• @BrianChandler A computer could. Thanks, point taken, question edited. Apr 14 at 4:34
• Sorry: to correct mysefl, the answer is obviously "Yes, 1 level". You simply take the LCM of all the n's in the n-tuples and multiply out. So your 5- and 7-tuples would become a 35-tuple. Apr 14 at 4:40
• I question the premise that the two notations are equivalent. For example, in the first example, the 5-tuplet takes the time of 2/3 of a beat, so the sixteenth note represents 2/15 of a beat. However, in the second example, the 9-tuplet takes the time of 2/3 of a beat, so the sixteenth note represents 2/27 of a beat. Apr 14 at 4:50
• @Aaron thanks, not a premise, just an (incorrect) example to explain the meaning of "rewritten with lower-level nested tuplets". While incorrect, it nevertheless served its purpose. If no correct example exists, then the answer is negative. Apr 14 at 4:59

(Note that in the video used for the OP demonstration, the narrator says "virtually" equivalent, not actually equivalent.)

Any set of nested tuplets can be rewritten as a single tuplet whose size is found by multiplying all of levels of the original.

In the case of the first example, it would be a 2 x 3 x 5 x 7 = 210-tuplet (the 2x comes from the fact that the tuplet represents two beats).

• Each quarter note = 70/210 (total: 140/210)
• The singleton 16th note = 35/210
• Each sixteenth note from the 7-tuplet (to be rewritten as 32nd notes for clarity) = 5/210 (total: 35/210)
• I don't think you need to multiply by 2*, but you are correct that you have to multiply all of the n in the n-tuplets, not take the least common multiple as I said in my comment. (A triplet within a triplet has to become a 9-tuplet.) (* A triplet over 2 beats is a triplet, not a 6-tuplet.) Apr 14 at 6:09
• @BrianChandler It's not a 6-tuplet, but it's equivalent to one. Apr 14 at 6:25