Hypermeter refers to the use of strong/weak metrical accent at levels beyond the notated meter.

Western classical music typically does this grouping by 2s (and so then 4s, 8s, etc.), just like it does with metrical grouping within the notated meter.

Grouping in 3s is rarer, partly because ...

Grouping in 3s at multiple metrical levels (within or beyond the meter) is rare

We first met the idea of Hypermeter briefly in the Other Rhythmic Essentials chapter. As discussed there, the strong-weak style of patterns we have seen within measures do not necessarily only operate within the measure and stop at the measure-length pulse. Especially when we have short measures and fast music (like a presto in 3/4), there can be a strong sense of such metrical relations at "higher", multi-measure levels. In Western tonal music, just as the most common division/grouping within measures is by 2s (and thus 4s, 8s etc), so it is also with hypermeter. That said, division/grouping by 3s is also eminently possible, as are 5s and 7s, and when you look more closely, you notice that interesting composers frequently rove between these options fluidly in a way that would be extremely unusual at the counting, tactus, beat level.

Hypermeter in 3s: "ritmo de tre battute"

One of the most famous, explicit notations of hypermeter is the "ritmo de tre battute" section of Beethoven’s 9th symphony. This marking explicitly points out that the measures group in 3s. In doing so, the instruction also makes explicit the assumption that the grouping had been in 2s and 4s up to that point. Example 1 reproduces the 4-measure grouping seen in the Other Rhythmic Essentials chapter:

Example 1. 4-measure hypermeter at the start of Beethoven 9/iv from the Other Rhythmic Essentials chapter.

Now Example 2 shows the later "ritmo de tre battute" section (from m.152) with comparable "hypermeter counts" as well as (editorial) double barlines to show each group of 3.

Symphony No.9, Op.125 by openmusictheory

Example 2. 3-measure hypermeter from measure 152: "ritmo de tre battute".

Note that we're talking about the "period" of the grouping here: that is, grouping in 3s or 4s. There's plenty more to say about the phase: that is, which measure in the cycle should be counted as "1". There's a lot to argue for the phase shown in this movement, although the dominant-to-tonic movement in the first to second measure of the figure sure might make you wonder about starting the phase a measure later.

Finally, for the sake of comparison, let's see that same "ritmo de tre battute" section re-notated explicitly in 9/4 to place both 3-grouping metrical levels within the notated meter.

Symphony No.9, Op.125 – Ludwig van Beethoven by openmusictheory

Example 3. The 3-measure hypermeter from Example 2 re-notated in 9/4.

Additional levels of 3s at the Fast End?

Like the common practice usage of 9-unit time signatures ("9/4", "9/8", ... ),I specify "common practice" here because of divisions like 2+2+2+3 in later styles. "ritmo de tre battute" explicitly sets out two levels of 3-grouping. Arguably, it's the two adjacent levels of 3-grouping that's notable here – the fact that Beethoven notates it with one of those level in the hypermeter is less significant.

So, can we have more than 2 levels of 3-grouping? Well, "27/X" time signatures are not seen (at least in this style), and neither is anything like "ritmo de 9 battute", but Beethoven does occasionally use 3 sets of 3 levels in his late style, i.e., adding another layer of 3-time at the "short" or "fast" end of the spectrum. This is always notated with triplets: another way to get 3-grouping into the metrical structure without having to include it in the time signature. Examples include:

In movement 4 of the string quartet no.14 (Op.131), there is an "Adagio ma non troppo e semplice" section in 9/4 time (from m.186), so we already have two levels of 3s right away (3x3 quarters). Later, it briefly uses triplet eight notes long enough to perhaps be called a metrical level (mm.223–225, or 5–7 measures before the final Allegretto reprise).

the second movement of last piano sonata (No.32, Op.111) is an “Arietta” in compound time signatures: 9/16 at first but also 6/16, and 12/16 for some sections. Additionally, some of those sections use triplet sixteenth notes to give another 3-level. When this happens in 9/16, we have 3 levels of 3 grouping that could have been written as 27/32. See Example 4 for an excerpt. That said, the hypermeter is regular, so 3x3x3 seems to be the limit for now. And the triplets in 6/16 and 12/16 add another 3-grouping, but with 2-grouping in the metrical levels.

Piano Sonata No.32, Op.111 – Ludwig van Beethoven by openmusictheory

Example 4. 3 sets of 3-grouping (effectively 27) in Beethoven's Op.111.

Sensible Limits

(Where) do we stop looking for metrical levels above and below? Some accounts of hypermeter have extended the idea to entry movements, describing them, for example, as a huge upbeat-downbeat pair. The psychology literature suggests a more modest scope, with meter giving way to form not necessarily at the limit of the notated measure, but also not far beyond. The idea is that we struggle to perceive metrical cycles and patterns when the pulse is too short (about 1/10th of a second) or too long (a few seconds, depending on the internal content). So in practice, it makes sense to consider hypermeter when the measure is very short (like the