Towards the end of the first movement of Beethoven's Piano Sonata Op 27 No 2 (Moonlight) the right hand is playing triplet quavers while the left has a dotted quaver and semiquaver.

If we are naively mathematical then the right hand should move to the third of its quavers (2/3 of the beat) slightly ahead of the left moving to its semiquaver (3/4 beat).

This is beyond my amateur abilities so I play the left hand's semiquaver a little early in step with the right hand's last quaver.

I just experimented with it in MuseScore and it is precisely mathematical and it sounds quite weird.

Of course, I have listened to recordings (e.g. Barenboim 1984). They certainly don't sound like MuseScore but I wouldn't dare say that they sound like me either.

Do we know what Beethoven intended? Is there a commonly agreed interpretation?

  • 3
    As a mathematician, you may be interested in improving your polyrhythms by taking Adam Neely's 7:11 challenge. youtube.com/watch?v=U9CgR2Y6XO4 May 4, 2019 at 15:32
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    I'll work on Beethoven first. One case in which I have learned to ignore the mathematical meaning of a term is irrational time signatures. I don't go looking for music in pi / 4 time. en.m.wikipedia.org/wiki/Time_signature#Irrational_meters
    – badjohn
    May 4, 2019 at 15:39
  • Similar 4-vs.-3 tactics are used in Schubert's Impromptu in C Minor, Op. 90, No. 1, as well as Elgar's Pomp and Circumstance March No. 2 in A Minor. (Of course, it's a lot easier to have dotted 8th-16th against triplets when separate instruments play them a la the Elgar march.)
    – Dekkadeci
    May 5, 2019 at 13:10
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    It is strange that you ask about the left hand at the end since this problem actually appears in the right hand early on in the movement. The left hand at the end is a repetition of the motiv that was presented in the right hand in the beginning. In my opinion it is important to play the sixteenth note after the triplet without being overly concerned about the exact math. May 5, 2019 at 13:33
  • @LarsPeterSchultz One example seemed sufficient and that one was easy to describe. Also, for some reason. I find it easier the other way around.
    – badjohn
    May 5, 2019 at 13:34

4 Answers 4


The normal interpretation is to play the dotted rhythm distinctly from the triplets. I've never heard a recognized pianist merge it into the triplets.

Anecdotally, I want to say it is often shortened in performance. Some celebrated performers are closer to a double dot - I'll add some examples when I track them down.

There are two issues at play here. One is the combination of dotted notation with triplets. In former times this was used conventionally rather than putting a quarter note and an eighth note under a triplet. In my opinion, it doesn't apply here. The other issue is how accurately rhythms in general should be observed. The literalist view is obvious. Another view, reading between the lines, is that Beethoven was expressing a contrast of rhythm and the degree is up to the player. A comment by Claudio Arrau springs to mind, from Dean Elder's interview book "Pianists at Play":

Do you think of rhythmic figures that often are not played precisely enough?

Sometimes figures are played too precisely. Rhythm should be very elastic, the notation of rhythm being only approximate. Otherwise, rhythm becomes motoric, which I hate. In such a rhythm as the fourth Variation of the Schumann Symphonic Etudes, a variation built mostly on rhythm - you should of course be very precise. But, on the other hand, there are cases in romantic music where the slight distortion of the rhythm will help the expression.

I wouldn't expect it to sound acceptable in MuseScore without tinkering, as the software won't be phrasing and balancing the parts as a human would.


People do strange things to the 'Moonlight'. My feeling is that the unifying element is the constant triplets which, while a degree of flexibility is always allowable, should not have their flow BROKEN. Many performers disagree! This version, the first that Google threw up for me, hesitates after each of the dotted quaver-semiquaver pairs.

This keeps the triplets rather more steady

This one chooses a brighter tempo and lets the triplets flow almost un-interrupted

And don't forget Benjamin Zander's opinion that we generally play it half-speed. View this from 23'00"

The one thing they all agree on is that the semiquaver falls after the triplet. No, you can't cop out of this! (It really isn't that hard to play.)

  • One-and-two-and-a-three-and-a-four-and-ardi-
    – Laurence
    May 4, 2019 at 14:28
  • As I mentioned in a comment above, I find that I cope better in the earlier section when the dotted quaver / semiquaver is in the right hand. So, I just need my left hand to catch up with my right. It is a little disappointing since I have worked hard at ambidexterity in life in general (e.g. I can use chopsticks in either hand) but I have some way to go with my left hand when playing the piano. I won't be performing Ravel's piano concerto for the left hand any time soon.
    – badjohn
    May 7, 2019 at 15:26

Don't count too much. Make music! let it flow:

The quietest it would sound if you give the 16th note half the value of a triplet eighth. But of course that's not mathematically correct. The exact note value would be 4/12 resp. 3/12t:

The triplets each with 4/12

The dotted eighths with 9/12

The 16th with 3/12

So mathematically correct would be, if the 16th note is not played halfway between the last triplet eighth and the new bar, but very soon after the triplet quaver, said to be one quarter of the triplet value.

Try to see the sixteenth note in a different way: as a "sound movement to the next note"!

Don't play it like a computer program, even some of these softwares have a humanizer that makes the "mathematical correct" played music groove.

  • Thanks. Despite being more of a mathematician than a musician, I resist imposing too much maths into music. The precise way that MuseScore played it just made me curious. An example of my resistance to bringing maths into music is saying "quaver" rather than "eighth note".
    – badjohn
    May 4, 2019 at 12:03
  • @badjohn to my American ear, your use of "quaver" rather than "eighth note" sounds more like the normal thing that anyone would say who also says "maths" rather than "math."
    – phoog
    May 6, 2019 at 17:07
  • @badjohn Oh dear, now I feel particularly slow and boring. I do note that the US use of mathematical note names seems to be derived from German practice, and Albrecht seems to be German speaker. Albrecht, does German have a short name for "mathematics," and, if so, is it plural or singular in form?
    – phoog
    May 6, 2019 at 20:02
  • Yes, teachers and students say “Mathe”: die Mathe is sing., fem. and stands for Mathematik. May 6, 2019 at 20:13

Other answers have made a good case for the common interpretation. I have always thought that the ideal should approximate two independent voices played by different people. That can be particularly difficult for a player of "amateur abilities," but I encourage you not to give up. I despaired of ever being able to achieve that, but with some practice (less than I had thought), I was able to do it, at least to my own satisfaction.

But the reason for this answer is to address this part of your question, about Beethoven's intentions:

Do we know what Beethoven intended?

If you look at the first surviving page of the manuscript, at the end of the third line of music, in the right hand, you will see the triplet written directly below the dotted rhythm, and the head of the second note of the dotted figure is placed distinctly after that of the last note of the triplet. This seems like very strong evidence that the dotted rhythm should not be assimilated to the triplet.

There's a case to be made that this layout is not significant, especially if you see how quarter notes and half notes are not generally aligned with the first note of the corresponding triplet figure, but in the other place where the dotted figure appears in the upper staff with a written-out triplet (as opposed to the slash abbreviation), on page 5, the spacing is the same.

On the other hand, when the figure appears in the bass, at the end of the movement, the 16th notes do appear to be aligned with the third triplet notes. In two of the four cases, though, the triplet is on the other staff, with a bit of crossing out and correcting/editing in one of those spots. In the two cases where the triplet is on the lower staff, the stems and beams are colliding with one another, so everything is a bit crowded, and the alignment might have suffered because of that. I would be inclined to conclude that the alignment on the last page is less likely to reflect Beethoven's intentions.

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