Ties and rests, and the silence that counts too
How to hold a note longer than any single note value allows, and how to write silence so that it fills the bar like any note.
Note values are each half the one before, and it is a short series: semibreve, minim, crotchet, quaver. Two things do not fit in it. A note lasting five beats, because there is no five-beat value. And silence, because a note value is a note, and silence is not. The tie solves the first; the rest, the second.
A tie adds
A tie is an arc between two notes of the same pitch. It says: play the first and hold it to the end of the second, without attacking again. The two values become a single sound, lasting as long as both together.
Minim tied to a crotchet: three beats of C, then one beat of D. There is no three-beat value without a dot, and the dot is no help when the note crosses the bar line, so the tie is what is left. Listen to the passage: it is two sounds, not three.
Semibreve tied to a crotchet: five beats. A tie has no limit; it can chain as many values as the duration asks for.
Why tie instead of writing a longer value
When a note starts in the middle of a bar and continues into the next, the tie is compulsory: a single value cannot cross the bar line. But even inside one bar, a tie is usually better than a long value, because it shows where the strong beat is.
Quaver, crotchet tied to a quaver, minim. The crotchet begins in the middle of the first beat and ends in the middle of the second; written this way, with the tie cutting exactly at the turn of the beat, the reader sees beat two in the right place. An equivalent value with no tie would hide that, and reading gets slower.
A rest has an exact duration
A rest is silence written with the value of a note. Every value has one: semibreve rest, minim rest, crotchet rest, quaver rest, each worth the same time as the corresponding note.
Crotchet, crotchet rest, crotchet, rest. Four beats, two of them silent, and the bar closes. A rest is not a failure or a hole: it counts, and a bar that adds up wrong because of a forgotten rest is as wrong as one with an extra note.
A minim rest followed by two crotchets: the music comes in on the third beat. Without the rest written, the reader would not know to wait.
Rest and tie in the same bar
The two live together, and that is how most real rhythms are written: a note that starts off the beat, held through the next beat, with a silence before it.
Quaver rest, quaver tied to a crotchet, crotchet, crotchet. The note begins in the second half of the first beat and lasts to the end of the second. Count "one-and, two-and" and the C comes in on the "and" of one. That pattern is the way into syncopation, which is the next reading.
What to take from here
A tie adds values of the same pitch into a single sound, and it is how to write a duration the series of note values does not have, or one that crosses the bar line. A rest is silence with an exact value, and it counts in the bar like a note. Together they write the rhythms that begin off the beat, and without them notation could only write what starts on the pulse.