The interval between C and G is a perfect fifth (P5).
But if the G is in a different octave does the interval name stay the same?
One concept that I might add is the distinction between simple and compound intervals.
Simple intervals are intervals of one octave and smaller; compound intervals exceed the span of an octave.
So this C up to G is a fifth. If we move the C down an octave (or the G up an octave), this becomes a twelfth, but we can also call it a "compound fifth," meaning that it's a fifth, but one larger than an octave.
To quickly translate between simple and compound versions of intervals, just remember the "Rule of 7": subtract or add by 7 to move octaves. Thus an 11th is actually just a compound fourth, because 11-7=4.
Any interval is the space between two notes. For naming, two facts are needed. Names of notes, and number of semitones between them.
So, it stands to reason that the interval name for C>E (M3) will not be the same for C>an E an octave higher.That is M10.
In every case, half the interval name stays the same! C> any E will always be M, but the number attached will vary.
If C stays in the same place and G goes an octave higher, then yes, names changes. But if bouth of them moves to one side, then they don't change
The intervals are counted in the first 2 octaves like this:
1 - Prime
2 - Sekunde
3 - Terz
4 - Quarte
5 - Quinte
6 - Sexte
7 - Septime oder Septe
8 - Oktave
9 - None
10 - Dezime
11 - Undezime
12 - Duodezime
13 - Tredezime oder Terzdezime
14 - Quartdezime
15 - Quintdezime oder Quindezime oder Doppeloktave
Source: German wiki site