In a short answer, yes.

When you look at early Soviet art (and by "early," I assume you mean the 20s and 30s), there are really two contrasting periods you can look at- the 1920s, or the NEP era, and the 1930s, which saw Stalin's Five-Year Plan, collectivization campaigns, and Great Purges.

When you talk about "abstract art," this was most common in the 1920s. The New Economic Policy, which Lenin instilled after the events of the Russian Civil War, provided a sort of compromise between communism and capitalism, where most of the market was State-owned, but there was also a private sector as well. As a result, the economy grew, and the art scene began to develop. So, you have advancements in the arts, as you said, with film being one of the strongest areas due to its propagandistic potential. Anatoly Lunacharsky, the head of the Commissariat of Enlightenment, was largely responsible for promoting cinema in the Soviet Union, saying that "cinema... constitutes a visual clarion for the dissemination of ideas, and... elements of the refined, poetic, pathetic, etc. [make it] capable of touching the emotions and thus becomes an apparatus of agitation." 

So, we have a lot of new art movements popping up in the 1920s due to the optimism that ensued from an economic surge as caused by the NEP. This optimism also led to a rise in "futurism," a movement started in Italy around 1910 that later spread to Russia, which consisted of a focus on industrialization and progress. As you mentioned, constructivism, along with structuralism, appeared in the art scene. Structuralism is an offshoot of formalism, a term that is extremely important when it comes to the Soviet art scene. Its definition changed throughout time, but in the 20s, formalism essentially meant putting a work of art's structure, or "form," above any meaning the art could hold, simply defined as "art for art's sake." As a result, you'll see a lot of abstract art, literature, theatre, music, and cinema. In the 1930s, as Socialist Realism was on the rise, the term "formalism" soon became used to define a broad set of elements that were not believed to comply with the tenets of Socialist Realism- a style that emphasized simplicity, pro-government themes, and order as opposed to abstraction.

So, now to music. Like other such mediums of art, Soviet music in the 1920s took on many "formalist" elements. Igor Stravinsky (1882-1971), whose works were known for elements such as dissonance and irregular time signatures, had a strong influence on Dmitri Shostakovich (1906-1975). I'll be focusing on Shostakovich quite a bit here. Although there are many other Soviet composers of note, including Sergei Prokofiev, Aram Khachaturian, Galina Ustvolskaya, and the infamous Tikhon Khrennikov, I often use Shostakovich's music as an example, as it provides a very comprehensive account of how Soviet art changed. Given the fact that he was composing throughout much of the Soviet Union's existence, it is a fantastic primary source in showing how Soviet music evolved as a response to political, economic, and militaristic events. His music can be divided into three periods:

• The "early" period, which lasted from around the early 1920s to 1936, which is where I'll focus on here, as his music was most strongly influenced by the 1920s art scene, including the music of Stravinsky, the films of avant-garde director Vsevelod Meyerhold, and the futurist poet and playwright Vladimir Mayakovsky,
• The "middle" period, beginning in 1936, when his opera, "Lady Macbeth of the Mtsensk District," was denounced in Pravda (one of the main State newspapers) for "formalism," signifying the change in the word's definition, along with the purges that followed against many of his family members and associates, leading to a more conservative approach to music,
• The "late" period, beginning in 1953, when Stalin died, and lasting until the year of Shostakovich's death in 1975. During this period, policies loosened and he was able to compose more avant-garde music. The "late" pieces often deal with themes such as death, and employ many elements borrowed, again, from Stravinsky, along with Gustav Mahler, whose works Shostakovich had been influenced by since around his Fourth Symphony in 1936, although this piece was withdrawn and would not be performed until 1961.

The significance of this early period shows that such abstract and "formalist" elements seen in art easily found their way into music, with Shostakovich's early work providing a very clear example of such a connection between music and art. The piece that demonstrates this best is probably his Second Symphony, subtitled "To October," commissioned in 1927. This piece involves many elements common in Soviet music of the time, including atonality, a choral section praising the Revolution (choral pieces became increasingly popular in "agit-prop," or "agitation and propaganda" music, due to the idea of having a large number of voices singing together, creating a sense of unity), and most importantly, an emphasis on industrialization. You'll see such themes of industrialization in other spheres of art as well, and Shostakovich makes use of these themes in the Second Symphony through originally scoring parts for factory whistle, although this is often omitted in most performances. He also provided the music for Mayakovsky's satirical play "The Bedbug," which ridiculed bourgeois ideals and suggested that they had no place in a utopian Soviet society. And there's also his 1928 absurdist opera, "The Nose," based on the Nikolai Gogol short story. The music in this opera pulls out all the stops to subvert and satirize traditional opera, including loud, rude trombone bursts and odd xylophone or string passages, saving the most dramatic and lugubrious music for when something mundane happens, such as when the main character, who has lost his nose (which, should I add, has grown to human size and is running around St. Petersburg causing havoc), laments that his missing nose had a pimple on it.

Although folk music became very common during the NEP period, it was most often implemented in both "art music" (commonly called "classical" music, although this term can be misleading, as it can also refer to the Classical period of art music, which took place around the late 1700s to 1820) and popular music during the 1930s and onward with the rise in Socialist Realism. As Stalin rose to power and implemented his Five-Year Plans, one of his main objectives was to rapidly accelerate industrialization and reverse many NEP policies, believing these slowed the Soviet Union's advancement to becoming a global superpower. Part of reversing such policies included making large changes in the artistic and cultural spheres, which included changing the meaning of the term "formalism" (which I mentioned above) and discouraging individualism in art. Folk and military music was seen as a "safe" alternative to the avant-garde, which originally rose to popularity to contrast with styles of the Tsarist age, but soon became seen as a potentially dangerous means of expressing dissent.

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Further reading and sources:Mayakovsky and Futurism

Futurism history and concepts

Russian formalism

A previous answer I provided on this sub on Mahler's influence on Shostakovich

Robert Weinberg and Laurie Bernstein's "Revolutionary Russia: A History in Documents," pg. 100-101

Pauline Fairclough's "Critical Lives: Dmitri Shostakovich"-

Further listening:March from "The Bedbug" Suite

Shostakovich Symphony no. 2, "To October"

Interlude from "The Nose"

Stravinsky's Octet for Wind Instruments