Middle C is the term musicians give to the pitch whose frequency is around 261.63 Hz:
The note whose frequency is twice that of Middle C is also called C:
...as is the note whose frequency is twice that again:
If we have enough frets on our mandolin, we could double the frequency again; likewise, if we had a fifth string, we could half the frequency of our original note. In both cases the note would still be called C.
‘C’ is an example of a pitch-class. The C's discussed so far are obviously different notes, but they share such a strong affinity that musicians consider them to be in many respects equivalent - something like different intensities of the same colour. As we are about to discover, there are basically 12 pitch-classes.
An interval is the difference in pitch between any two notes. The interval any two adjacent C's (or any other pitch-class for that matter) is known as an octave.
The octave can be divided evenly into twelve semitones. (This is our second interval, by the way.) Here are the notes in the octave between middle C and the C one octave above it ‐ there is a semitone between adjacent notes:
C C♯ D D♯ E F F♯ G G♯ A A♯ B C
We could also express this as C D♭ D E♭ E F G♭ G A♭ A B♭ B C. ‘♯’ is pronounced sharp; ‘♭’ is pronounced flat - which version of a note's name we use depends largely on context.
Because we have identical pitch-classes in each octave, the notes form a continuum:
. . . G G♯ A A♯ B C C♯ D D♯ E F F♯ G G♯ A A♯ B C C♯ D D♯ E F F♯ G G♯ A A♯ B C . . .
For this reason, it can be convenient to view the pitch classes as a ‘clock’, with a clockwise motion representing a rise in pitch: