There’s a fairly common belief that until Stravinsky came along, everything was either in 4/4 or 3/4. When I was at school in the 70s, I remember one music lesson in which we had to listen to the second movement of Tchaikovsky’s 6th symphony (which is in 5/4), and marvel at how avant garde he was to have written it in 5/4. This kind of narrative still persists today – as this quote from a site about the symphony illustrates
The second [movement] is a “limping waltz,” boasting the near-miracle of a melody so smooth you’re hardly aware it’s in 5/4 time and missing a beat. The 5/4 signature occasionally surfaces in jazz (Dave Brubeck’s “Take Five”) and rarely in rock (Ginger Baker’s “Do What You Like”), but was unheard in classical music, until this. Typical of Tchaikovsky, it pulsates with doubt – brimming with grace yet constantly off-balance enough to cast a pall over the otherwise elegant mood. (Source: Classical Notes)
But this simply isn’t true. The valse à cinq temps was developed in Paris in the 1840s, and was danced by the eponymous heroine in Catarina by Perrot to music by Cesare Pugni in 1846. It may not have been that popular in the ballroom, but it still appeared in American dance manuals in the 1880s. In ballet, perhaps the most famous one of all is the dance of the Golden Idol from La Baydère, an interpolation of a piece titled ‘Persian March’ from an 1874 ballet by Minkus called Le Papillon (follow the link to see the source for this information). There are about a dozen ‘five step’ waltzes listed among the music in the American Memory collection of the Library of Congress, and Tchaikovksy wrote another valse à cinq temps in his 18 pieces Op. 17 (it’s No. 16), which is used by Cranko in his ballet Onegin.
Not only that: 5/4 is common in Russian folk music (and English, if it comes to that), and there are examples in Tchaikovsky’s folk song collections. The promenade from Moussorgsky’s Pictures at an Exhibition from 1874 is also in 5/4. The ‘Fée Sapphir’ from The Sleeping Beauty is in 5/4, but this is a different kind of 5: at this speed, what’s really happening here is a variation in the length of the underlying pulse. The same thing happens in the 3rd of Alkan’s Air à cinq temps from Deuxième recueil d’impromptus. Published in 1849, this set also included a piece in 7.
In several of these cases, you can see a dotted line between the ballroom, dancing masters, ballet, folk music, and the concert repertoire. Whether or not you think the 2nd movement of the 6th symphony is a valse à cinq temps or an evocation of Russian folk song doesn’t really matter – far from being the first time such a thing had occurred in Western music of the 19th century, the concept was already almost half a century old. And as for the idea that 5/4 has a ‘missing beat’, or ‘limps’, this seems like just one of many possible readings. Why not an extra beat? Or why not just the right number of beats because you decided to write in 5? Look at it this way, for example: by writing in 5, Tchaikovsky allows himself to start a scale passage on the 3rd degree of the scale, and end up on the tonic on a strong beat. What do you get? A perfect arc of a 6th as your first statement. How appropriate for a 6th symphony.