Is there any difference between these two note values?
I have never understood if these two note values are supposed to be played in the same way:
Mathematically, I can understand that they are the same, since both of them have 6 notes that are being played in 1 beat, but they are symbolized different.
Usually, the way I have seen the sixteenth triplets is like this:
(a line connecting the two groups).
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The sextuplet can also be six in the time of three, six in the time of four and Six in the time of five. It all depends on the specifics of the piece and how the rhythms work. So yes very much different than the regular three in the time of two.
You are right. They have exactly the same rhythmic values. It would make more sense to use the sextuplets unless:
the music obviously accents two groups of three notes per beat.
the music often intersperses quavers (8th) notes with groups of triplet semiquavers (16th).
In either case where you use triplet semiquavers, I would group pairs of them with a single beam, as shown in your last example. Unless, of course, you are in a time signature which has a longer beat, such as compound time or 2/2, in which case you could group 3 and 4 groups of triplet semiquavers with single beams, respectively.
The difference between a sextuplet and two triplets is that the two triplets are clearly substructured into two units. The sextuplet in contrast may either be substructured into three groups of two notes, or it may not be substructured at all.
If you have one ascending run (for example) written as sextuplets, chances are that the composer intends you to not make the fourth note stand out in any manner.
It's like the difference between 6/8 and 3/4, the first being subdivided into two units, the second in 3. However, in many cases there is only one accent on the first note of a sextuplet. Anyway, when written as two triplets the accents are clearly on the first and (possibly a bit weaker) on the fourth note. This little article describes the different rhythmic interpretations of the sextuplet.
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