In correct music notation, can a tuplet (like a triplet or a quintuplet) cross a barline?
I have never seen this, but if it's not allowed, why not? It looks to me like it should be possible.
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(We're all clear on the difference between irregular beam groups and tuplets? Good.)
Beamed groups across barlines (example A) are unquestionably acceptable.
Tuplet groups across barlines (example B) are 'impossible' by the rules of traditional notation, but like irrational time signatures (4/7, 5/3 etc.) are becoming more common.
I used Sibelius to make the example, and had to cheat - the bars containing triplets are really one 4/4 bar with a graphic barline. Dorico is more amenable, though this description of how to do it includes a telling remark:
"...as an editor, I rarely find barline-spanning tuplets to be notational best practice, given that usually the same material can be re-notated in much more conventional ways, thereby improving the readability of the underlying meaning without impairing it. So the thought of seeing more tuplets over barlines in the near future just because they are cool and can now easily be done is not something that fills me with joy."
(It then goes on to admit a use for cross-barline notes in some ancient music.)
I think there are several possible interpretations of "tuplets crossing a barline", but I wouldn't consider any of them to be impossible.
A few cases I can think of, from the late Romantic period (all sourced from IMSLP):
This example aligns the tuplet '3' with the final note of the beams, so the bars are six triplet-quavers each, but beamed across the barline. This choice isn't universal; some publishers align the tuplet '3' with the middle note of the beams:
Tuplets, beamed across the barline, with no clear indication of whether the notes are supposed to align with the downbeat or not (Scriabin, Prelude Op.11 No.1): and also (same piece): The first bar starts with two (quintuplet?) quavers; the final bar isn't missing any duration to compensate. Exactly how the left-hand crotchets match up with the right-hand quintuplets is unclear.
Tuplets, beamed across the barline, with notes definitely not aligning with the downbeat (Godowsky, Studies on Chopin's Etudes, No. 19): Here the lower voice in the right hand is playing triplet quavers, offset by a non-triplet quaver; the tuplet clearly crosses the bar (and the middle triplet-quaver is correspondingly split into two tied triplet-semiquavers). (The final triplet, in the last bar, is three triplet-semiquavers, to compensate for the initial non-triplet-quaver rest in the first bar, and re-align the notes with the barline again.)
Whether you'd consider the first two examples to be "tuplets crossing the barline", I don't know - I think I would, but if you wanted to argue that the tuplets were within-bar and just beamed weirdly, I think that's also valid. I think the last example is fairly unarguable, though.
It seems worth considering whether it's better to fake the bar line placement or just change the time signature...
In the first instance you would logically need to draw the dot on the bar line, or draw it before the bar line to show it starts before the bar line, but then you end up with a note value that doesn't fit the actual duration... and it's hard to read.
The second instance just removes the bar line, avoids the note value/duration problem, and is actually readable, although I imagine it will get played incorrectly by a lot of people.
This is sort of a corollary of "can anything cross a barline?" and the answer differs for semi-modern notation (starting with late Baroque) and earlier music (Medieval through Renaissance to early Baroque). Those music forms don't have the usual strong beat relations of later music and their notation will not usually use ties for splitting notes across bar lines. Modern editions often put the bar lines only between systems, and the manuscripts may not even have contained them at all.
The soprano triplets clearly stretch over multiple measures (3 notes per 2 measures) and notating them with ties would really mess up the notation and character here.