Interval names (”perfect fourth,” “augmented sixth”) can be confusing at first. The key to understanding is 1) think in letter names, 2) put C in the middle. I’ll walk you through a simple visual explanation.
Interval names have two parts: quality and number. If I say “perfect 5th” the quality is “perfect” and the number is “5th.”
Number is counted by moving through the alphabet one letter at a time. We count both the starting and ending notes, so A-B is a 2nd. We ignore sharps and flats, so B-F is a fifth, and so is Bb-F#.
Once we know the number, quality fine-tunes the size of the interval. There are five options for quality: major, minor, perfect, diminished, and augmented. We can easily find examples if we start with C.
Unisons, 4ths, 5ths, and octaves can be perfect, but not major or minor. Go up or down from C by that amount and you’ll find the perfect intervals.
All of the other intervals (2nd, 3rd, 6th, 7th) can be major or minor, but not perfect.
Starting at C, go down by that amount and you get a minor interval.
Go up by that amount and you have a major interval.
And that’s all the intervals you’ll find within a major scale.
Augmented and Diminished Intervals
To get diminished or augmented intervals, add a sharp to everything but the C in the middle.
The augmented intervals are the ones that got bigger.
The diminished ones are the ones that got smaller.
Now that you have examples based on C, you can apply the same idea to different keys, always with the tonic in the middle.
Once you have this understanding, you can start working on skills like identifying intervals that don’t start on the tonic. Counting half-steps can help with this, but doesn’t tell you everything you need to know.
Remember that the names of intervals make a lot more sense in the context of scales, and scales always follow the alphabet A-G (and repeat).