Ok, there are actually two ways to sing Happy Birthday in Russian.
The first is something close to what you would expect:
“С днём рождения тебя” (S dnyom rozhdeniya tebya)
Literally translated, this means “With the birthday, to you” and it is to be sung to the tune of the English hit “Happy Birthday”. Here’s how it goes:
S dnyom rozhdeniya tebya! (With the birthday to you!)
S dnyom rozhdeniya tebya! (With the birthday to you!)
S dnyom rozhdeniya! S dnyom rozhdeniya! (no need to insert the name you Western self-centered indvidualist!)
S dnyom rozhdeniya tebya! (With the birthday to you!)
Now listen to a Russian sing it to test your pronunciation:
I honestly have no idea why they say “With the Birthday” or “With The New Year”. I am no etymologist, to be sure.
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Suggested post: “How to say “Hello” (and a few other simple greetings) in Russian”
The SECOND way to sing Happy Birthday is much more Russian. Also, the name of the song is something like “I Play on the Accordion“. Here are the words:
Fortunately, the Happy Birthday song that includes the word “pity” is sung to an upbeat melody:
To add further confusion to a situation in which a nation refers to this as a “sad day” and mentions that we should be mournful since we can not celebrate more than once annually, my understanding is that this song was originally performed by the critically acclaimed Crocodile Gyena. Here you go:
I sort of feel this explains why Russians don’t smile.
Of course I’m joking. Russians sometimes do smile.
And lest I leave you worrying about a magician arriving in a blue helicopter, I can say that there is a third more modern and happily upbeat option. Here it goes, together with random Russian cartoon footage. You are welcome. Now choose which one to sing, the next time you are invited to a Russian birthday party.