Your roman numerals mean intervals right?
Er… No, actually they do not mean intervals.
A major scale has seven notes or degrees. They can be labelled by Roman numerals or named. For instance the scale of C major:
C – I – tonic
D – II – supertonic
E – III – mediant
F – IV – subdominant
G – V – dominant
A – VI – subdominant
B – VII – leading note
The advantage of this labelling is that it will be exactly the same for every major scale, even though the notes will be different in each scale.
So the roman numerals can mean two different things (not intervals!).
In a melodic context, they will mean the notes themselves.
So a perfect cadence means that the last two note of the melody will be V (dominant) followed by I (tonic). If the melody is in C major, it means that it ends with e the sequence G – C
In a harmonic context, the numerals mean the triads built upon the notes the roman numerals refer to. For instance, in a perfect cadence, the last two chords will be: V – I, that is (in C major): V = GBD (the triad built upon the dominant) followed by I = CEG (the triad built upon the tonic).
Does this help?
Best wishes,
Bernhard.