Standard piano music sheets contain G and F clefs, which can display comfortably (without using a lot of ledger lines) around 48 keys C2-C6.
I wonder how people write the very high/low notes, i.e., C1 or C8? Thanks.
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The preferred method depends on the instrument in question. The violin and the clarinet, for examples, are accustomed to playing a couple octaves' worth of ledger lines above the trebleclef. Cello parts may have a stack of ledger lines, or they may jump from bass to clef, or get annotated "8va" .
I once had to explain to a music major (underclass) that, unlike a piano score, woodwinds did not want to see notes progressing down from the treble to the bass clef, but rather just use ledger lines below the staff :-) .
Another way is to use ledger lines. Ledger lines and different clefs (by octave or other transposition) are all common. One (supposedly) uses whatever is easiest to read.
A related complication is that some instruments are transposing; what you read (native to that instrument) isn't the note that sounds. (A clarinet or trumpet plays what that instrument calls a "C" and out comes a Bb or perhaps and A.)