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I have noticed something with the scherzo from Beethoven's Ninth and particularly the beat. If I just listen to the scherzo and try to feel the beat, I feel an Allegro, say around 130 BPM at the triple meter Scherzo and around 140 BPM at the duple meter Trio. But, if I actually look at the score, I realize that the Allegro I am feeling is really the measures moving by extremely fast. The actual beats within the measures are going 3 times as fast, at 390 BPM in the Scherzo and twice as fast, at 280 BPM in the Trio. The Trio is actually slower. But it feels faster than the preceding Scherzo.

I think I know what is going on here. The beats are so fast that to my ears, they sound like mere eighth notes or triplets instead of beats and thus instead of the note feeling like the beat, the measure feels like the beat. Several pieces I have listened to have this Measure - Beat equivalence going on. So this is what I feel in the different sections:

  • Section of the Movement - Tempo I feel - Actual Tempo - Meter I feel - Actual Meter
  • Scherzo - 130 BPM, an Allegro, Tempo jump as we transition to the Trio - 390 BPM, faster than even your typical Prestissimo - 2/4, I feel a S w S w accent pattern to the measures - 3/4, the beats are just so fast that they feel like triplets instead of beats
  • Trio - 140 BPM, faster than the Scherzo, Ritardando towards the end - 280 BPM, Very fast, but slower than the Scherzo - 2/4 still - Actually in 2/4
  • Coda - 180 BPM, slow end of Presto - 360 BPM, in between the Scherzo and Trio in terms of tempo, but almost as fast as the Scherzo - 2/4 still - Actually in 2/4

Here is a video of it being played by an orchestra so that you can hear what I mean by the measures feeling like the beats:

And here is a link to the score so that you can see that what feels like the beats are actually the measures:

http://imslp.simssa.ca/files/imglnks/usimg/f/f1/IMSLP61851-PMLP01607-Beethoven_Breitkopf_Serie_1_Band_3_B_9_2.pdf

But, there are a lot of fast pieces that I listen to and don't hear this Measure - Beat equivalence in.

For example, here is a similarly fast part of a piece by Mozart:

And another similarly fast section of a piece by Chopin(which actually might be closer to the Beethoven scherzo than the Mozart example in how fast it is):

I don't hear a Measure - Beat equivalence going on here in either of these pieces and they both feel like they are at a similar tempo to the Beethoven's Ninth scherzo(both are at a Presto tempo). I hear and feel 2 beats per measure in both the Chopin and Mozart examples(Duple meter seems to be the most common meter for Presto pieces in general). Not so with the Beethoven's Ninth Scherzo where I feel that the measure is the beat.

So obviously with the Mozart and Chopin counterexamples, it can't just be a Presto tempo causing the Measure - Beat equivalence I hear in the Beethoven's Ninth Scherzo. Did Beethoven intend for the measure to be the beat in this very fast Scherzo from his Ninth Symphony? Did he intend it to feel like it is in 2/4 the entire time, even when it is in 3/4? Why am I feeling this Measure - Beat equivalence in the Beethoven's Ninth Scherzo, but not in similarly fast pieces by Chopin or Mozart?

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There are a few things going on here.

First is hypermeter: composers very frequently organize bars into two-bar groups and those two-bar groupings into four-bar groupings, and sometimes four-bar groupings into 8-bar or even 16-bar groupings. Now, you often don't tap your foot at those long intervals, but if you tried conducting them with a sort of "strong-weak" pattern at 2-bar or 4-bar or sometimes longer intervals, you can feel that there's often an "ebb and flow" at longer intervals. Beethoven clearly organized the bars in this movement early on into two-bar groupings, but then (as pointed out in comments) explicitly breaks this pattern.

This is not unique to this piece -- longer metric grouping patterns beyond the barline occur in lots of pieces of music. You just don't tend to listen to them as the primary "beats" unless you pay attention to that level of rhythm. The general name for this, again, is hypermeter or hypermetric organization.

The second issue is why you "feel a beat" at the level you do, which has to do with the psychoacoustic concept of preferred tactus. The tactus is essentially the level where you feel the primary beat. That may or may not correspond with the notated beat level according to a time signature. (It frequently does not.)

Many psychoacoustic studies have shown that people prefer tactus levels around 100 bpm (beats per minute), and will almost always prefer a beat level that falls around the range of 40-160 bpm. That is, if you have a piece actually notated with a beat slower than 40 bpm, chances are you'll naturally try to subdivide that beat as you listen to it, in order to feel a faster tactus.

And if you hear pieces with notated beat faster than around 160 bpm, you'll often tend to group beats into larger structures when you listen and feel every two beats or three beats or feat beats as "the beat" (i.e., tactus), rather than the notated "beat" according to the time signature.

Which is why in the scherzo you feel tempos around 130-140 bpm as the tactus level. There's no way you'd feel 390 bpm or 280 bpm as a primary tactus -- it goes against your whole rhythm processing functions in your brain (though one can force oneself to count at that speed if you want). The end of the Presto may go a bit faster than the 160 bpm upper limit, but you have been entrained to hear a tactus at a certain level from the first part of the movement, so you continue to hear the beat at a similar level, rather than dropping down to a slower tactus.

The Chopin example has a notated tempo of 132 bpm at the half note, and you feel the beat at the half note level. That is precisely the same tempo range as the Beethoven, so I don't know why you'd consider that a counterexample.

The Mozart example is a bit more complicated. The tempo of the recording you link varies a lot due to rubato, but probably is around 175-180 bpm on average. (I didn't check it in detail.) If I had to conduct this piece, I'd definitely conduct it in one, rather than two beats per bar. If I had to tap my foot to it, I'd tend to feel it in one as well.

But I can understand why you still want to feel the faster beat level as the beat -- it has to do with faster chord changes and harmonic rhythm around cadences, which usually correlate with preferred tactus level. So, you're using stylistic cues to likely override the normal upper-bound of tactus level and feeling a faster beat. But also the 160 bpm is not a hard limit, and it can vary from individual to individual, definitely taking into account stylistic factors as with the Mozart. (I'd also note entrainment effects can happen here too -- if you've just come from listening to a bunch of pieces with tactus levels ranging from 130-180 bpm, and then you listen to another piece with a tactus close to that 180 bpm range, you will probably be a bit more likely to hear it similar to the pieces you've just been listening to, rather than dropping down to a 90 bpm level or less.)

Bottom line: it is sort of the presto tempi here, though it depends on quite how prestissimo they tend to be. It doesn't depend on whether something is labeled "Presto" but rather the actual timing of the tempo and duration of the beats. Once you get over around 160 bpm, chances are that you will begin to feel the tactus at a longer metric level.

  • The thing that the Chopin example was a counterexample to is not the 130-140 BPM tempo but rather the 1 beat per bar. I feel 2 beats to a bar in the Chopin Piano Sonata no. 1 in C minor Presto, whereas around the same tempo range, I feel 1 beat to a bar in the Beethoven's Ninth Symphony Scherzo. – Caters Jan 3 at 2:28
  • @Caters - I don't understand your connection here. In the question, you're talking about your perception of the primary beat, which has nothing to do with where barlines necessarily lie. I could write a tempo marking for a 6/8 bar at dotted half=100, and you might feel it "in one," but I could write a tempo marking for a 6/8 bar with eighth note=100, and you definitely wouldn't feel each measure as a single beat. Your question isn't about tempo markings -- it's about where you feel the beat, which is determined by the duration of the tactus. Barlines have nothing to do with perception. – Athanasius Jan 3 at 3:34
  • Or, to put it another way, you can potentially feel beats at many different metric levels for the same piece of music. In a 4/4 bar, you could feel a "beat" at the quarter note level, or the half, or the whole bar, or on the eighth note level even. One of these levels corresponds with the barline, but why does that matter? If you're feeling the tactus at the half or eighth level, you have the same phenomenon as you discuss in your question, i.e., you aren't feeling the beat at the notated quarter level. Whereas in the Chopin example, you do feel the beat at the notated (half) beat level. – Athanasius Jan 3 at 3:39
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You can compare the meter in the Beethoven Scherzo with compound meters like 6/4 and 6/8 which have two beats when the tempo is fast and six beats when the tempo is slow. In a similar way Beethoven's Scherzo in 3/4 has one beat instead of three each bar because of the fast tempo. The conductor will only conduct one beat per bar.

The Trio is actually slower. But it feels faster than the preceding Scherzo.

Well, the beats in the trio are actually faster than the one beat per bar in the Scherzo. You wrote earlier on in your post:

If I just listen to the scherzo and try to feel the beat, I feel an Allegro, say around 130 BPM at the triple meter Scherzo and around 140 BPM at the duple meter Trio.

Thus the trio has in fact a faster beat.

Besides the tempo the beat is also emphasized in the way the music is written. In the Scherzo the first beat in each bar is accentuated with chords, and the melodic line also fits in with one beat per bar.

The Mozart and Chopin pieces are not in triple meter so it is difficult to compare. Anyway the Mozart piece, 2/4, feels like one beat per bar when I hear it although you hear two beats per bar. The Chopin piece, cut time, is clearly two beats per bar just as Chopin has indicated.

Addition: Here is a YouTube video with the Finale movement from Chopin's Piano Sonata No.3. The meter is 6/8, but with two beats per bar:

  • So treating the very fast 3/4 of the Beethoven scherzo as being a 6/4 explains both why I feel 1 beat per bar in the scherzo and why despite the triple meter, I still feel the S w S w accent pattern so intrinsic to duple meter and thus why the scherzo feels to me as though it is in 2/4, even when the notation clearly shows that it is in 3/4. – Caters Jan 2 at 23:49
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    @Caters yes, but beware that the S w S w feeling (I suppose you mean Strong Weak Strong Weak for four bars) is broken at some point later in the movement, still one beat per bar but with sequences of 3 bars, look at bar number 177 where the headline says "Ritmo di tre battute" which means that the rhythm goes in groups of three bars. Later on in bar 234 there is a headline "Ritmo di quattro battute" meaning the rhythm goes in groups of four bars. These changes are really nice and delicate in my opinion, it is kind of like spice or sparkles. – Lars Peter Schultz Jan 3 at 0:10
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As noted in a Lars Peter Schultz comment, there are "Ritmo di quattro battute" and "Ritmo di tre battute" indications in the score of the scherzo of Beethoven's Ninth Symphony, so yes, Beethoven did intend for the measure to be the beat. "Ritmo di quattro battute" signifies that the measures are grouped in 4-measure phrases, while "Ritmo di tre battute" signifies that the measures are grouped in 3-measure phrases, and the "Ritmo di tre battute" indications are found in sections in 3/4 time...so no, Beethoven did not intend it to feel like it is in 2/4 the entire time, even when it is in 3/4.

We have a fairly strong tendency to gather beats in groups of 2 and 3 (duple, triple, and compound meters being common, additive meters inevitably being given in terms of sums of 2 and 3 (e.g. "2 + 3 + 2/4")), so I believe that is why you hear the duple-meter Mozart and Chopin pieces with 2 beats to the measure, while you hear a measure-beat equivalence with the Beethoven scherzo.

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