# Is a sixteenth note quintuplet worth 4 sixteenth notes? If so, why is this wrong?

So, I'm doing a theory course, and there's this one problem where it asks you to find the time signature of this melody:

According to the theory course, the correct answer would be 6/8 time.

I completely get that for the first two measures, however NOT for the last two (and I know/see that the time signature doesn't change in the third measure). I was taught that the quintuplet is the same value/played in the same time as 4 notes of the same value. So, the quintuplet in measures 3 and 4 (m.3 and m.4 have basically the same rhythm) would be played in the same time as 4 sixteenth notes -> 2 eighth notes.

Those 2 eighth notes from the quintuplet + 3 eighth notes total from the rest of the measure = 5 eighth notes in the last two measures?

I'm sure something is wrong about the way I understand quintuplets or this problem since the time signature doesn't change in the middle of the melody, but I don't quite see where my reasoning went wrong.

Or, maybe there is an error in the problem? (Though I kind of doubt that, haha)

• Who teaches that?? Surely the most common tuplet in the world is a triplet of eighth notes, which take the same time as two of that value. Commented Jul 26 at 6:41
• @KilianFoth I don't see any mention in the question comparing the frequency of occurence of quintuplets versus triplets. Triplets being more common aside, quintuplets are also a thing, so the answer to "who teaches that quintuplets exist" should be "anyone who gets deep enough in the topic". Triplets are not being denied their spot in the sun, they're simply not what the question is about. Commented Jul 26 at 8:26
• @Divizna My mistake, I overlooked that the rule is supposed to be specifically about 5-tuplets. Commented Jul 26 at 9:57
• The tuplet rules you were taught, although officially correct now, were standardised rather late in music history and there are very famous composers that don’t follow them (IIRC even Debussy doesn’t, for example); because of this, you should simply always look at the context instead of relying too much on these rules. Although you could argue that the problem is “bad” or faulty because it doesn’t follow today’s rules, it reflects the notational reality in a lot of music older than, say, 70 years very well. Commented Jul 27 at 8:59

This music is from Chopin's Etude Op. 10, No. 9 in F minor.

Chopin correctly uses eighth note quintuplets to fit 5 notes into the space of 3 eighth notes. Additionally, later in the piece, he uses sixteenth note quintuplets in their proper context, for fitting 5 notes into the space of 2 eighth notes:

So it would seem that as the other answers suggest your teacher made a notational mistake while copying Chopin's music.

• Thank you! It’s been bugging me where the excerpt came from. I knew it was Chopin, but after not finding it in the Nocturnes I got stuck. Now I can sleep! Commented Jul 26 at 23:56
• oh, I see!! I was wondering where it came from ^^ Commented Jul 27 at 4:27

It seems like the intent of this notation is 5 notes in the span of 3 eighth notes. I've never actually come across this scenario, but some brief research finds several examples of this 5:3 notated as a quintuplet, but always with one beam, not two.

Does this make the example from the question incorrect? I'm not sure. But it's definitely not the preferred notation.

In fact, I think it may be best to explicitly write 5:3 for maximum clarity.

• Using the ratio is probably the way to ensure that the writing is properly understood, even though that would logically make it possible to extend it to otherwise "invalid" tuples (eg: 5:2 with 5 eighths in a quarter). There is quite some controversy about tuplets for compound meters: it's not uncommon to find duplets or quadruplets both written as eighths in a 3/8 based meter. See the "Compound meter" example in the related Wikipedia article. Commented Jul 26 at 10:20

In Behind Bars: The Definitive Guide to Music Notation, Elaine Gould says that there are two ways to choose note-values for tuplets (p. 203–4):

Option 1 is to add extra notes to the beat until the next standard division. This can be expressed as a contracting ratio: keep the number on the left side of the ratio larger than the number on the right. When the left-hand number doubles the right hand number add another beam instead.

In other words, notes inside a tuplet are always "contracted", i.e., they take less time than the equivalent notes outside a tuplet.

In option 2, the note-values take the nearest arithmetical unit, so that sevens are regarded as closer to eights than to fours. This produces a series of ratios that both contract and expand.

Gould then provides an example that includes a five-tuplet of sixteenth notes filling a single beat in 6/8 time, as in your question.

To summarize, the two systems would subdivide a dotted quarter in 6/8 as follows:

Tuplet Option 1 Option 2
4 eighths eighths
5 eighths sixteenths
6* sixteenths sixteenths
7 sixteenths sixteenths
8 sixteenths sixteenths
9 sixteenths (ambiguous)
10 sixteenths thirty-seconds
11 sixteenths thirty-seconds
12* thirty-seconds thirty-seconds
13 thirty-seconds thirty-seconds

*Note that a 6-tuplet or a 12-tuplet would not typically be thought of as a "tuplet"; it would just be six sixteenth notes or 12 thirty-second notes instead.

Gould prefers option 1, as she notes that option 2 makes it hard to tell the duration of a tuplet at a glance, and that triplet divisions (such as the 9-tuplet above) are "closer neither to the greater nor to the lesser binary values." In other words, Option 2 is ambiguous as to whether a triplet in 4/4 time should be denoted with eighth notes or sixteenth notes.

• Considering that triplets are, to my knowledge, always written as 3:2 and never 3:4 in either convention, I'd assume analogically 9:12 wouldn't be a thing either; it'd be 9:6 and thus sixteenth notes. Commented Jul 27 at 12:32

TL;DR: The exercise is notated incorrectly and should have used eighth notes for the 5-tuplets.

The problem arises from the fact that divisions in music notation are always binary — except for the first division of the beat in compound time, which is ternary.

The general rule is that tuplets are notated with the largest note-type that would total less than double the total duration of the tuplet.

Want a 5-tuplet in the time of a quarter note (or two eighth notes)? Use sixteenth notes, because five eighth notes would be longer than two quarters (or four eighths), but five sixteenth notes would be less.

Want a 5-tuplet in the time of a dotted quarter note (or three eighth notes)? Use eighth notes, because five eighth notes is less than two dotted quarter notes (or six eighth notes).

Looked at this way, it's straightforward and consistent across time signatures.

The problem is when talking about beats. In simple meter, for any individual note, a 5-tuplet is created using the second division of that note. As above, a quarter note is divided into 5 sixteenth notes. That's also true in compound meter: a single quarter note would be divided into 5 sixteenth notes.

But if we're talking about dividing a beat into five parts, then the answers diverge. In 4/4 time, a quarter note represents a single beat, and five sixteenth notes would be the notated 5-tuplet. But in 6/8, a dotted quarter note represents the beat, and five eighth notes would be used for a one-beat 5-tuplet — eighth notes being the largest grouping of five less than double the duration of the tuplet (i.e., six eighth notes).

Where 4/4 and 6/8 come together again is when looking at 7-tuplets or greater. Seven sixteenth notes in 4/4 time are less than two beats, so a single-beat 7-tuplet would be written with sixteenth notes. Similarly seven sixteenth notes in 6/8 time are also less than two beats (i.e., two dotted quarter notes), so a single-beat 7-tuple would also be written with seven sixteenth notes.

• I'm not 100% sure we can mark the notation as completely inaccurate. It's not uncommon to see duplets using eighths in a compound meter, meaning that, for consistency, 4 notes should be noted as sixteenths, therefore using sixteenths for a quintuplet would not be wrong. Commented Jul 26 at 10:23
• @musicamante Historically there is inconsistency, which I believe comes from thinking in terms of "beats" rather than note durations. In terms of "beats", it makes perfect sense to use two eighths in both simple and compound meters, since it makes the notation consistent. Commented Jul 26 at 17:21

A tuplet typically takes some kind of clearly outlined division of the bar, which in 6/8 time is either the first 3/8 or the last 3/8, rather than any 4/8. So five notes occupying the space of not four but instead three in this case isn't a surprise at all; it's perfectly intuitive.

It seems the convention whether one should always "round to longer note" or "round to shorter note if the numbers are closer" isn't entirely unambiguous, so I guess it's possible to write it as sixteenths... but I agree that eighths would be preferable.