The future looks bright for the 13-year-old Alex Prior, who is celebrating the launch of his personal label, and who will present his new Battle Cantata at the Cathedral of Christ the Saviour in Moscow tomorrow. Things look just as good for the 13-year-old Kit Armstrong, who has just starred on the Late Show with David Letterman in America, talking about the mathematical basis of music and playing his sophisticated piano compositions. Both have determined mothers behind them, but there the similarities stop: one may be of Russian extraction, and the other of Taiwanese, but they might as well inhabit different planets.

I first encountered Alex Prior at a composition class at Dartington summer school, where the composer Stephen Montague was extolling the beauties of uncomposed ambient sound. To Montague's affronted astonishment, Prior roundly ticked him off for this avant-garde dismissal of melody and harmony, and then got up and — attended by his mother Elena — walked out.

Meeting the Priors amid the books, paintings, ceramics and velvet drapes of their St. John's Wood apartment is like walking into Tsarist Russia, and yes, that's a signed photo of Chekhov on the mantelpiece. There's a youthful photo of the great theatre director Konstantin Stanislavsky on the wall over Alex's computer and keyboards: 'That was his great-great grandfather,' explains Elena. 'Alex is the last big hope of the Stanislavsky family. And Tchaikovsky and Tolstoy were connected to it by marriage. My son has much to live up to.'

But he doesn't seem oppressed by it: you couldn't imagine a more exuberantly normal kid than this product of University College School. He was a precocious infant — 'he never crawled, he walked straight away', says Elena — with an immediate penchant for singing. He started playing the piano at six, and began to take it seriously at nine.

My homework for this interview has been to listen to a plethora of works, including his new song CD ('Just a Boy'), his first piano concerto, his symphonic poems, and his first string quartet. While the latter (recently commended in the BBC's young composers' competition) has rigour and structure, the others deliver grand emotional gestures with fairly basic scoring. 'Alex would never have been a chamber-music type — he's an arena performer, the bigger the better. When he began to write music, I had to tell him 'not four tubas please, just one', and he wanted eight harps,' says Elena.

Alex talks about his heroes Mussorgsky and Wagner, and his own restless urge: 'Melodies come to me quicker than I can write them down. One melody will arrive while I'm still writing down the previous one, so I have to be very selective. I can never stop.' What is he writing now? 'Twenty things at once. I do it on Sibelius [the composers' computer-programme] so I can work fast. Let me show you my new oratorio on the computer.' Commemorating the 600th anniversary of a battle between Russians and Tatars, it is full of bells, voices, and tuba blasts. 'That's the Russian army in the distance — my music is very visual,' he remarks, as drumbeats give way to Russian Orthodox chant.

But what really transfixes me is the voice he produces for the aria he's written for himself: it's strong and startlingly masculine. How would he classify it? 'My range is as for a Rossini tenor, and my sound is like that of a dramatic tenor. I sing naturally from the chest.' Will it break soon? 'No, I think it will just slide down naturally, like Carreras and Caruso. I'll probably end up a bass. 'Nessun dorma' is for me the easiest song to sing — it's comfortable for me. 'Danny Boy' is harder.' Having won a National Eisteddfod award, he's currently putting this voice to good use in charity concerts in Moscow and in London, on behalf of Russia's street kids.

Finally, I am shown two videos of him: conducting with boyish ardour, and singing in an American arena. And I understand why agents are queueing to sign him. The white suit, the gestures, the throbbing voice — it's all there in spades, and he's got the crowd in the palm of his hand.

Kit Armstrong and his mother, May, have just moved into a spartan flat five minutes from the Royal Academy, where Kit is studying piano and composition, and not too far from Imperial, where he is concurrently reading maths in the final year of the undergraduate course. With a wide smile and luminous gaze, he's tiny for his age. Yet when I heard him play Beethoven's First Piano Concerto at a Royal Academy concert in July, I was won over by his expressive and relaxed authority.

Like the infant Prior, Armstrong was precocious, though the talent he displayed at one year was for counting. 'If he was ever asleep and needed waking up, all I had to do was give him a maths problem,' says May, and by the time he was five had already finished what was described in California — where they were then living — as high-school maths. 'He was so smart,' says May, 'that I decided to give him music as a hobby.' She got him an electronic keyboard.

Kit's account of his first compositions is disarmingly unpretentious. 'I composed monophonic music, just one single line of notes, not even particularly melodious. But for some reason I got excited about it — I liked making strings of notes. And at one point I realised I had to make it sound good, so I had to discover what those strings sounded like. Then we got a piano, then it built up exponentially, and I started to write melodies with harmonies. Just simple ones — I began to figure out which chord would like to be followed by which other chord. I worked out my own primitive version of harmony, and theory, and voice-leading.'

At seven he became the youngest maths scholar at Chapman University in California, and at nine was a full-time undergraduate, and his musical creation kept pace.

In some ways he's still very much a child, delighting in origami and juggling. When I ask him to describe the genesis of a tonally intricate piece called Sweet Remembrance, which he composed last year, he explains in mathematical detail how he constructed it in repetitions, inversions and retrogrades, 'and to my great surprise and delight, it sounded good'.