That's the usual knock against Shostakovich and I can definitely see it, but unless you're ready to ding Schubert on the same ground for Winterreise and much of his overall output it's really hard to justify leaving the sardonic Soviet out.
And you don't need to convince me of Gershwin's standing in the classical canon. I'm not one of those faux snobs who insist the Tin Pan Alley tunesmith wasn't a composer and I also agree his music will most likely outlive that of nearly all his compatriots (save possibly one - see below). But we're not just talking about American composers, and even by purely populist standards it's all but indefensible to insist on Gershwin's place in such exalted company while excluding such giants as Corelli, Bizet and Faure, especially given the latter's own historical popularity and influence (Corelli for his 12 concerti grossi, the most important group of orchestral music before Haydn; Bizet for you-know-what; and Faure for his songs and couple famous tunes).
BTW there IS an American composer whose music may well outlast Gershwin's, and he happens to be one of your heroes: Leonard Bernstein, whose West Side Story is just as big a crowd fave as Rhapsody in Blue and is in fact about to receive its second film adaptation headed by none other than Spielberg. And you know who'd win all the standard music competitions between Lenny and George. I actually should've included Bernstein in the "debatable" category along with the others. (For the record I'd rank the two about the same.)
As the only guy (with the only arguable exception of Handel, which is why I place him just below the Big 3) who can challenge Mozart for supremacy in opera Verdi does rank damn high all time, but he's clearly a notch below the Holy Trinity of Bach, Mozart and Beethoven. On a superficial level all his major works are for the voice - in fact he was one of the few great composers who weren't a virtuoso - and belong to a single genre to boot (even his famous Requiem is an opera in all but name, and not even a sacred one at that as he was a longtime agnostic), whereas J.S. and Wolfgang excelled in both vocal and instrumental music as well as in all forms and genres. (I'd argue many of Bach's cantatas and passions are in fact theatrical works. Ludwig OTOH did struggle with the human voice. Not that he didn't write some of the greatest music for our natural instrument - think Sampras and clay - but compared to Bach, Mozart, Schubert and others he was clearly more comfortable on his piano and violin.) But more importantly Verdi was a genius but not an immortal like the Big 3. When you listen to Bach's Chaconne, Mozart's Requiem or Beethoven's last piano sonata (to name a single example a pop) you're no longer dealing with mere beauty or even the best of humanity but what can only be explained as inexplicable, spiritual and/or godlike transcendence. Verdi for all his genius never transcended the utmost limits of us mortals.
He did come pretty damn close, of course. La traviata is the
world's most popular opera for good reason (though I'd still say Carmen is tops for the lay audience) and probably one of my three all-time faves (Violetta certainly is one of my top three soprano roles, along with Belli's Norma and Donizetti's Lucia), with his other famous operas not too far behind making him by far the most frequently performed opera composer (even Mozart is a distant second). And who can fail to be awestruck by his high-theater Requiem? Put 'em all together and you've got arguably the richest corpus of musical theater to have come from the pen of a single man.
Speaking of whom/which, here's the single greatest Violetta I've ever seen either in person (including
Venera Gimadieva in a
fine production at the Kennedy Center last spring) or on screen:
By then (2011) Dessay was obviously a tad too old for the role and no longer in her vocal prime, but in terms of sheer dramatic power I daresay Callas herself couldn't have cut a more moving Violetta. Simply sublime.
The full production, if you're interested (though you should get the
DVD!):
We were obviously discussing composers of the classical tradition only. Comparing musicians from all genres and backgrounds assumes that it's possible to be well acquainted with every one of them, which is humanly impossible and outright arrogant.
But since I'm arrogant enough to know my opinion is superior to nearly everyone else's I'll give it a shot. A few things:
- First off the music of Gershwin, R&H, Porter and other Tin Pan Alley/classic Broadway masters is just too sophisticated for the lay public to qualify as "popular." They deserve their own category unless you can devise a consistent set of criteria which can accommodate them as well as the actual pop/rock acts.
- Your list of jazz luminaries is far from complete. Miller is included while the likes of Armstrong, Duke, Basie, Parker, Davis, Coltrane and Monk are out? Just no.
- In the world of popular music - broadly defined to cover the heyday of Robert Johnson up to the present-day dominance of hip-hop but not inclusive enough to absorb more formally rigorous genres like classical and jazz as well as such non-Western musics as gagaku, bhangra, mawwal, benga and mbalax - the Beatles (who include George and that drummer guy as well, contrary to your framing), the Stones, Dylan and James Brown stand above all the others. Prince may have been the most talented of them all in the usual metrics that we dilettantes love to obsess over, but he wasn't a genius on par with his greater predecessors.
- Even among the second and lower tiers of pop some of your choices are truly baffling. Lynne for real? Sting is a somewhat more defensible choice, but still. Here, a couple more names I'd easily include over these two: Hank, Berry, Otis, Ray, Little Richard, Aretha, Davies/Kinks, Fogerty/CCR, Marley, Green, Bowie, Patti, Young, Cash, Stevie, Marvin, the Clash, MJ (yes, the gloved one), U2, Petty/Heartbreakers, R.E.M., Kurt (though I don't care much for Nirvana's cynical solipsism these days). And my all-time fave Reed/the Velvets, the only rock act in my desert-island canon.
Takemitsu is indeed a worthy contemporary composer and you should check out his film music as well if you haven't already.
BTW a fun tidbit: for all his avant-garde cred Toru was an avid karaoke singer, which I suppose couldn't be helped in Japanese culture. (I like to tell people the best nightlife in the world may well belong to either Tokyo or Seoul.)