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The following section (mm. 49–52) is from "Metamorphosis II" by Philip Glass with 4/4 signature.

I don't understand how beats are adding up:

There are 4x6 16th notes in each G clef measure, equivalent to 6/4 per measure.

On F clef we have 8 8th notes, equivalent to 4/4 per measure.

So clearly I'm reading the treble staff wrong.

Are those notes pointed by red arrows 16th notes or 8th notes? They have double beams, so I thought they were 16th notes.

Philip Glass, "Metamorphosis II", mm. 49–52

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    I'm sure there are several dupes where triplets have been confused with ordinary note values, just can't find them.
    – Tim
    Jul 22 at 8:02
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    @Tim Possibly music.stackexchange.com/q/61435/9862
    – shoover
    Jul 22 at 17:40
  • Here's an example with sixteenth triplets music.stackexchange.com/q/58803/9862
    – shoover
    Jul 22 at 17:42
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    That linked question is not the same. This question asks specifically about the single rather than double beam between the triplet groupings which seems ambiguous in duration to the OP. Closed, yet three answer given, including one from the person who closed it. Yet none get into the beaming. Jul 22 at 20:28
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    ... but I disagree with @Tim that this is a duplicate. Related, sure, but sufficiently different to remain open. The present question is not only about triplets but also the way they are beamed, which is specific to semiquaver triplets in comparison to quaver triplets. I've voted to reopen.
    – Aaron
    Jul 22 at 23:41

3 Answers 3

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The "3" below each group of three sixteenth notes is a triplet indicator. Rather than 4x6 16ths notes, consider that it's 8x3 16ths.

Each sixteenth-note triplet is the equivalent of one half beat, and half beats are connected with single beams. Usually, of course, the half beat in 4/4 time would be represented by eighth notes, but in this case sixteenth note triplets reside where ordinarily each eighth note would be.

One might wonder why sixteenth-note sextuplets weren't used, but short of contacting the editor, a reasonable guess is that the goal was to help visually align each triplet with its corresponding left-hand eighth note.

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    I'd say it's just a philosophy of using as little beaming as necessary to keep the beats coherent. Since the "and" of each beat is in the normal place, it's beamed like a normal eighth note "and" would be. If they were all beamed together as sixteenths, I believe, as you mentioned, it would conventionally be notated as sextuplets and not only would that be a lot of ink on the page, but possibly offputting to novices who are only used to seeing triplets. Jul 22 at 16:39
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    The beam connecting the two triplets of 16th notes could have been omitted. I think that contributes to the OP's confusion. It looks somewhat like the 3rd note of the first triplet is an 8th note.
    – Wastrel
    Jul 22 at 17:59
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Under each three notes is a little '3'. It used to be, and sometimes still is, accompanied by a bracket. It means those three notes are a little faster than normal - they take up the same time as their normal two. With that bracket shown, you'd probably have noticed, and understood. Unsurprisingly, they're called triplets, and doing the sums again will give you the correct answer!

Every single note-head is attached to a double beam or 'tail', so each is called a semiquaver (16th note), it's just that because they're in triplets, it's easier to read writing them in threes.

Looking at how they line up against the quavers underneath, it's easier to see how they all fit together timing wise.

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  • "used to be, and sometimes still is, accompanied by a bracket": early practice used a curved line; later practice preferred a straight line following the slope of the notes with vertical strokes at either end so as to be less similar to a slur. But the bracket has, I believe, always been optional when the notes are eighth notes or shorter when the beams suffice to show the grouping clearly.
    – phoog
    Jul 22 at 7:32
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    @doubleE to add on to the existing answers here: this is a type of tuplet. The system of beams is easy for splitting notes into smaller notes in multiples of 2, but this comes in handy when you want other subdivisions. Triplets are by far the most common, but you could cram in even 5, or 7, or 11 notes into the space of an 8th note, and notate it with a little 5, 7, or 11. Jul 22 at 12:47
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All of the notes have the same duration. You're overlooking the 3 by each group of three sixteenth notes. These are sixteenth-note triplets, of which there are twenty-four in a measure of 4/4 time. The broken second beam makes it clearer that there are groups of three, while the unbroken first beam fulfills the standard practice of beaming by beat.

Neither Wikipedia nor Dolmetsch online's music theory page has a good succinct explanation of triplet. The definition in Dolmetsch's dictionary is pretty good, but I don't think it's possible to link directly:

a group of three notes of equal time value performed in the time of two of them, however, (i) one or two of the notes may be rests of equivalent value, and (ii) a consecutive pair may be replaced by a note of double value.

Because three triplets take the time of two "normal" notes of the corresponding value, the duration of a triplet is 2/3 the duration of the corresponding note. For example, at quarter=50, a quarter note lasts 1200 milliseconds, a sixteenth note lasts 300 milliseconds, and a sixteenth note triplet lasts 200 milliseconds (in theory: if you are so precise in practice, the performance will be too mechanical sounding).

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  • Interesting. Never tried timing my playing notes, in milliseconds..!
    – Tim
    Jul 22 at 8:06
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    @Tim me neither, but I find that discussions of "2/3 the duration" and "3 in the time of 2" can become confusingly abstract. This is an attempt to make it a bit more concrete.
    – phoog
    Jul 22 at 8:09
  • Just being flippant - there's not enough flippancy on this site, for me! Explaining like that is a great help, but I'm sure it'll send some folk off to search for the stopwatch !
    – Tim
    Jul 22 at 8:17
  • @Tim I hope their fingers are nimble!
    – phoog
    Jul 22 at 8:27
  • I’d recommend trying to record precise timing. Good musicians are way more precise than you think, but triplets aren’t always played exactly even.
    – ojs
    Jul 22 at 11:29

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