Just a newbie question. In musical notation, is it okay to have number of notes in a bar that doesn't conform with the Time Signature?
For example, look at this music sheet:
Can you see that the first bar only has a 1/4 note while the Time Signature is 4/4 (judging from the rest of the bars).
Yes. This is a pick-up bar, also known as an anacrusis. This melody starts on beat 4 and so this note could also be called an up-beat. That is why the first bar is incomplete. When this happens the last bar should have a complementary number of beats (in other words, the number of beats in the time signature minus the pick-up bar, 3 beats in this case).
As the music starts on beat 4, the first note feels "weaker" rhythmically than the second note, which happens on beat 1, and so has rhythmic emphasis.
Also, when you have an incomplete first bar, this is usually numbered "bar 0", with the first full bar being "bar 1".
As others pointed out, the piece you cite has a "pick up measure". Note though that it is not categorically ok for measures to not add up to the number of beats in the time signature, it can only happen at the first measure.
There is another case where you can have an apparent mismatch in the number of notes and the time signature. This happens if the measure has "grace notes". Grace notes "don't count" - they don't add to the total duration for that measure.
To elaborate on @keshlam's point about older music, there are all sorts of musics for which regular measure lengths are simply not part of the genre.
Go back far enough and you'll find non-mensural music, such as Gregorian and pre-Gregorian chant. The music of the trobadors (11th-13th centuries) was not noted with rhythm, and there's an argument made that that's how it was performed.
More recently (16th century) you'll find mensural music (i.e. music in measures) where the music changes time signature very frequently. Like, randomly. There was an artistic movement in late 16th century France called musique mesureé, which was somewhat experimental: it was a form in which the composer set poems by deriving the note durations from the rules (as they were understood at the time, and applied to then-modern French) of classical Latin poetry. Astonishingly for such a mechanical gimmick, the resulting music is very attractive and accessible, even while having pretty much completely randomly fluctuating bar lengths. In my experience, modern editions of the scores for this repertoire don't even bother notating the time signature, or handwave through it by calling the beat the measure and declaring the piece to be in some flavor of 1. (I think a crucial part, for modern musicians, of learning to play Renaissance music is getting over the anxious need to be told what time signature you're in all the time.)
I understand something similar is true for znamenny polyphony, for which basically measure marks are lies, despite being contrapunctal and even in a sense syncopated. (My director: "I, uh, will be beating 1, unless any of you guys have a better suggestion.")
And then there's modern music which, really, can do whatever it feels like. I remember from when I was a kid, but now cannot find, a really gorgeous concert band piece* which was in 4/4 except whenever it felt like it being in 5/4.
So, yes, measures can have whatever number of beats in them the composer wants to put in them, and composers can dispense with measures altogether.
But that? Your example? That's a pick-up, as explained above.
* Named, completely unhelpfully, "Passacaglia". And I have no recall of the composer.
It should also be noted that older/traditional music styles do not always fit our rigid definition of measures. As with time signatures like 13/16, this is sometimes because they were written to go with specific dances and reflect the fact that some steps really do take a bit longer than other steps. It may also simply be that the composers/performers/dancers didn't feel as strongly that everything had to occur on a perfectly regular downbeat. (I've seen a number of instances of this in music from the middle ages.)
And I've seen departure from regular measure lengths in more recent pieces, where the author/composer deliberately chose to do something unexpected... which, after all, is part of the definition of good music; set up expectations, then artfully break them in a way that seems reasonable. But in those cases the sheet music will generally have an explicit time signature change.
As others have noted, yes, this is a pick-up measure and is valid.
However, in this specific case, I believe the Song of Time is not scored with a pickup measure. This is how I would score it:
In The Legend of Zelda: Ocarina of Time, all ocarina melodies start on a full measure.
Song of Time is actually the first phrase part from Temple of Time. I don't know if that Temple of Time sheet is official or not. The only official music sheet that closely resembled to song of time's tune is Door of Time / 時の扉 and it's in 3/4. Maybe you can look at the Temple of Time's video link and tell me if it's 3/4 or 4/4.
