Once I asked the great lyricist Sammy Cahn about his song “Come Dance With Me,” which he wrote at the request of Frank Sinatra. The challenge, he told me, was trying to come up with a song about music or dancing that Irving Berlin hadn’t already written. “He has written all the great songs about dancing, has he not?” Sammy said. “He covered all the bases!” Then Sammy began reeling off the titles of some of the most famous songs Berlin had written for Fred Astaire: “Cheek to Cheek,” “Let Yourself Go,” “You’re Easy To Dance With.”

The Berlin-Astaire connection was strong. They collaborated on six films together, more projects than Astaire did with any other composer or Berlin with any other star. Yet Berlin didn’t write songs about dancing because he worked with Astaire – it was the other way around. Astaire wanted to work with Berlin (who was nearing 50 when they began their collaboration) because he had captured the magic of dancing in the words and music of so many of his songs. Indeed, a majority of Berlin’s songs were about the sheer joy of music: making it, dancing to it, participating in it.

No one understood the power of music better than Berlin: It transformed him from a homeless, penniless immigrant on the streets of the Lower East Side to one of the richest, most famous, and widely respected figures in American culture. All this is wonderfully evident in the new picture book “Irving Berlin’s Show Business” (Harry N. Abrams, 250 pages, $40), compiled by David Leopold, who also curated the New York Public Library’s new exhibition “Show Business!: Irving Berlin’s Broadway,” which opens today.

From the beginning, Berlin wrote comedy songs and ethnic songs. Long before he mastered the heartfelt, romantic ballad, he practically invented a genre all his own: the party song, the clarion call to have a good time. “Dance of the Grizzly Bear” and “Everybody’s Doin’ It” were Berlin’s first notable songs about dancing. He soon was writing numerous songs about having a ball – in the literal sense – from “The Million Dollar Ball” to “Belle of the Barber’s Ball,” and even “The Old Maid’s Ball.” Thirty years later, during World War II, he paraded his support for Franklin Roosevelt with “The President’s Birthday Ball.”

Berlin made his first attempt at a lyric in 1907, when still a singing waiter in Chinatown. In the next few years, he honed his art of writing music (even though he could barely play it) and words, and his songs, notably “That Beautiful Rag” and “Ragtime Violin,” were increasingly successful. But it was “Alexander’s Ragtime Band,” penned in 1911, that put him on the map.

A comic strip from that year (reprinted in Mr. Leopold’s book) illustrates the song’s popularity: A hapless character wakes to find his alarm clock ringing to “Alexander’s” syncopated, bugle-call melody; both the fruit-seller on the street and the local ragman hawk their wares to its rhythm; even his parrot demands a cracker to its tempo. As singer-scholar Ian Whitcomb once noted, “Alexander’s Ragtime Band” was the “Rock Around the Clock” of its era.

For decades, purists attacked Berlin with the charge that “Alexander’s Ragtime Band” wasn’t real ragtime. It wasn’t – no more than his 1920 “Home Again Blues” was a real 12-bar blues or his 1917 “Mr. Jazz Himself” authentic New Orleans jazz. “Alexander” was a song about ragtime: Berlin was the first to capture the energy and spirit of the new music in a popular song. And, even though Scott Joplin and his ilk had been around for nearly 20 years, “Alexander” was the song that officially announced the emergence of the new American popular music born of ragtime, blues, and, soon enough, jazz.

Ragtime and “animal dances” like Berlin’s grizzly bear were, eventually, denounced by the Vatican. But even before that happened, Berlin satirized his critics by playfully embracing the Satanic side of the music in “At the Devil’s Ball” and “Pack Up Your Sins (And Go to the Devil).” In “Play a Simple Melody,” Berlin composed two distinct sets of words and music, both using the same chord changes and encouraging a ragtime band to get hot. He then superimposed the two strains on top of each other contrapuntally.

When the finished song was performed in Berlin’s first Broadway show, “Watch Your Step,” the crowds practically rioted. The words to the second strain could be a netherworld prayer from a 1970s metal band: “Musical demon […] won’t you play me some rag?” In 1913’s “The International Rag,” Berlin writes prophetically of how American pop music will conquer the world: “Italian opera singers / Have learned to snap their fingers!”

Some of Berlin’s joy was leavened in 1912 with the tragic death of his first wife. For the first time, he put his heart in a song with the classic waltz “When I Lost You”; he would later help establish the waltz as the time signature for nostalgic reminiscence. Throughout the teens and the 1920s, Berlin’s love songs grew more sentimental. By the time of his second marriage in 1926, his songs about music had taken a surprisingly melancholy turn.

Fifteen years earlier, Berlin had used the genre as a harbinger of the new, of everything tony and fashionable. But in his 30s and 40s, music became a Proustian trigger to remembrance. In 1919, Berlin established that “A Pretty Girl Is Like a Melody”; just a few years later, turned that same conceit around with the beautiful “The Song Is Ended (But the Melody Lingers On).” In the first, the memory of the girl haunts you in a pleasant way; in the second, a painful one.

Berlin would continue to find variations on the idea, such as the 1928 waltz “Where Is the Song of Songs for Me?”; the romantic “Say It With Music”; the 1946 ballad “You Keep Coming Back Like a Song”; and the lovely, autobiographical “I Poured My Heart Into a Song” of 1939. But the most compelling of Berlin’s “serious” songs about music is his breathtakingly moving “Russian Lullaby” of 1927.

This brief but brilliant 16-bar melody, with its minor-key, quasi-modal harmonies, is also highly autobiographical – Berlin looking back on his family in Siberia in a way that’s poignant and bittersweet. Clearly a song about a song, it tells of how the Russian mother sings her little bubbelas to sleep. Like all of Berlin’s song-songs, it has an air of mystery – Berlin never tells us exactly what the Russian lullaby says or means. But the family is obviously looking forward, both toward the future and the West, as they dream of “a land that’s free for you and me.”

“Show Business! Irving Berlin’s Broadway” at the Library for the Performing Arts until May 26 (Lincoln Center, 212-870-1630).