So far, I have outlined only diatonic intervals here. These are not the only intervals that exist, there are also intervals between sharp and flat notes to consider. Here is the full list of both simple and compound intervals:
Intervals from 0 up to 12 semitones are simple intervals. Intervals above 13 semiitones are compound intervals.
After 12 semitones (the perfect 8th, or octave) we continue naming the intervals in numerical order ie. 9th's up to 15th's.
So to play a major 9th interval, we would play the root note and the major 2nd note an octave higher.
Here are some rules regarding intervals:
Intervals can be inverted, so that the higher note becomes the lower note, and vice versa. When we invert an interval, we come up with a new interval, based on the following rules:
To name the inverted interval, subtract the degree of the original interval from 9 (e.g. a 7th becomes a 2nd, a 6th becomes a 3rd, a 4th becomes a 5th, and so on).
A perfect interval, when inverted, is still perfect.
A major interval, when inverted, becomes minor, and a minor interval becomes major.
An augmented inteval, when inverted, becomes diminished, and a diminished interval becomes augmented.
Note: a perfect 4th interval is also sometimes considered dissonant when not supported by a lower 3rd or 5th