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First of all, I apologize if I do not get it right with the terminology.

In this sonata of Beethoven in bar 6, there is the 3/8 being played and then at the 4rth 3/8 part, the melody enters above - with G#. I could not figure out from listening if this is a polyrhythm - being notated as 4/16 (a dotted quarter note +1/16) or is it like 6/16 (each 8th is like 2/16 so I play a dotted 8th = 5/16 and the last note enters in the 6/16? and if this is so don't they have the same notation? I am confused...

i hope you can understand what i am asking. :)

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  • It's a kind of 4:3 polyrythm. You can look at this question: music.stackexchange.com/questions/6986/… ; it considers exactly the same place in the sonata, so the answers could be helpful to you.
    – Ramillies
    Commented Oct 31, 2019 at 10:27
  • This is weird. I don't understand the question yet others seem to have no problem. Bar 6 appears to have the same rhythm as the all the rest so why the talk of polyrythms? What am I missing?
    – PeterJ
    Commented Nov 1, 2019 at 14:01
  • The dotted 8th and the 16th - basically the melody that is introduced. I also find the timing challenging (although I’m terrible).
    – WillyC
    Commented Apr 30, 2020 at 2:08

5 Answers 5

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The root of the question comes from the incorrect assumption that in Beethoven's time (and earlier) the notation for dotted rhythms was performed strictly according to the math. The math was certainly "strict" in the sense of showing the mathematically correct number of beats in the bar, but that was not necessarily how they were played.

A single-dotted note could represent anything from modern "light swing" (i.e. less than 2/3 of a beat followed by more than 1/3, but not an equal division of 1/2 + 1/2) up to even 15/16 followed by 1/16 in a slow tempo (i.e. a triple-dotted note, in modern notation)

There are two practical options for playing these particular dotted notes: either 2/3 + 1/3, or 5/6 + 1/6. The choice really depends on the tempo for the whole movement - if it is too low, 2/3 + 1/3 sounds a bit lame and "dragging", and if it is too fast, 5/6 + 1/6 sounds like a little "click" at the end of the beat rather than something musically significant.

Of course if you are playing using rubato, the beat division doesn't have to be "mathematical" anyway - just play it the however you want it to sound.

Composers and music copyists didn't like writing "triplets" consisting of a quarter note plus a half note, because it meant that some sort of bracket or slur was essential to show that the notation was a triplet. when some of the notes in the triplet did not have beams. A dotted eight plus a 16th, beamed together, was simpler to write, and everybody at the time understood what it meant.

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  • 4
    I don't think that Beethoven meant a 2/3+1/3 division here, or else he would have written it that way and put the whole thing in 12/8. Beethoven was surely familiar with the convention in the baroque that would have allowed rhythms to be notated in this fashion, but it was no longer in fashion by the time of this sonata. Plus, we have for example Czerny's testimony that this was to be performed as written in a cross-rhythm, and he actually studied with Beethoven... though today I'll admit the standard interpretation tends to be more like 5/6+1/6.
    – Athanasius
    Commented Oct 31, 2019 at 23:19
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    If Czerny says Beethoven wrote it as 4 against 3, then it's 4 against 3.
    – Guy Spiers
    Commented Aug 15, 2022 at 17:59
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    "There are two practical options ... either 2/3 + 1/3, or 5/6 + 1/6." No. A very good practical option is to play what it says, a dotted rhythm against triplets. Listen to some recordings. That's what you'll hear, give or take a bit of rubato.
    – Laurence
    Commented Aug 15, 2022 at 18:28
  • @Athanasius did Czerny have anything to say about the tempo? (I'm thinking here of Benjamin Zander.)
    – phoog
    Commented Aug 15, 2022 at 19:44
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This is a notorious question, and has been asked many times in the last 2 centuries - you are not alone! You're right in suggesting a polyrhythm - 4 against 3.

The difficulty is to play it musically. If you play it exactly, I (personally) find the two notes are a bit too close for comfort, so I tend to overdot the top line a bit. It's also important to distinguish the accompaniment triplets from the repeated G sharp theme. Not easy.

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    One of my high school students played this at our fall recital last week. Not "easy" but far from impossible or "difficult" in the grand scheme of piano repertoire. The crucial element is to play the melody voice louder than the accompaniment voices. If done correctly they will not sound "too close." Dynamic contrast is hugely important, always. Commented Nov 7, 2019 at 2:31
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    Not easy-easy. But not hard. Being able to play a fluent 2:3 or 4:3 is a pretty basic skill.
    – Laurence
    Commented Aug 15, 2022 at 18:32
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Every rendition I've ever heard, as I remember it in my head, sounds like it's based over a simple subdivision of that beat into six:

                1 2 3 4 5 6

upper taa---ta: * - - - - * 

lower ta-ta-ta: * - * - * -

This is what the notation says. The upper beam of notes ends with a sixteenth, which is half of the value of the corresponding eight in the triplet below; that is reflected by our 5 6 columns above.

This ain't exactly Brazilian Jazz; don't overthink it. :)

What I'm saying is that the dot you see on the eighth in this context shouldn't be taken literally as "time and a half", like it does when that pattern occurs in 4/4 time and has the time of a quaver. That doesn't even make sense, because the note values wouldn't add up to 3 eighths. The important thing is that the note is held over until some time close to before the timing of the ending 16th.

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  • All the recordings I'm familiar with have the rhythm as written, with the sixteenth note falling one quarter of the way between the last eighth note triplet and the downbeat of the next measure.
    – phoog
    Commented Aug 15, 2022 at 19:34
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Yes it’s weird that people usually just play this rhythm oversimplifying it like 5/6+1/6, despite the mentioned recommendation of Czerny. In my opinion, the polyrhythm option gives a great room for playing around with rubato, and uncovering musical tensions which arise between the triplets and sixteen notes, turning this piece into a study on psychological un-ease and deep personal drama, instead of sentimental soap opera, which happens often when simplifying the rhythm.

Here is an attempt to use polyrhythms in order to convey a more focused musical meaning:

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As noted in a comment, it doesn't make sense for Beethoven to have used more ink and care to write a dotted rhythm if he'd wanted the notes to line up with the triplets. We can conclude that he wanted the rhythms interpreted strictly as written.

Others seem to assume that this is impractically difficult, and suggest that the sixteenth should be played as a sixteenth triplet, falling midway between the last eighth triplet and the downbeat of the next measure. But it's really quite simple to put it in the right place, which is one quarter of the way between those two points.

To see that this is so, divide each quarter note into the least common multiple of three and four, which is twelve. Each triplet eighth comprises four of these subdivisions, and each sixteenth note three of them (the dotted eighth, therefore, has nine). To borrow Kaz's notation for ease of comparison:

                                  1 1 1
                1 2 3 4 5 6 7 8 9 0 1 2

upper taa---ta: * - - - - - - - - * - -

lower ta-ta-ta: * - - - * - - - * - - -

In other words, this...

X: 1
T: Sonata quasi una fantasia
M: C
L: 1/4
K: C#min
%%staves {(RH1 RH2) LH}
V:RH1 treble up
z2 z G/>G/|G3 G/>G/|G2
V:RH2 treble down 
L:1/8
(3E,G,C (3G,CE (3G,CE (3 G,CE|(3G,DF (3G,DF (3G,DF (3G,DF|(3G,CE
V:LH bass
[C,,8G,,8C,8]|[^B,,,8G,,8^B,,8]|[C,,4C,4]

...is mathematically equivalent to this:

X:2
T: Sonata quasi una fantasia
M: 12/8
L: 1/8
K: C#min
%%staves {(RH1 RH2) LH}
V:RH1 treble up
z6 z3 G2-G//G3/4|G9 G2-G//G3/4|G6
V:RH2 treble down
E,G,C G,CE G,CE G,CE|G,DF G,DF G,DF G,DF|G,CE
V:LH bass
[C,,12G,,12C,12]|[^B,,,12G,,12^B,,12]|[C,,6C,6]
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