Indeed, Billy and thanks. I was contemplating going into a diatribe about intervals and what they mean/how they are analyzed... so let's do that now because it is really cool and helpful stuff when dealing with diatonic scale/mode theory (and chord construction).
I learned (almost) all this stuff from that book "Elementary Rudiments of Music" Starise mentioned upthread. It really is a fabulous book (although a bit dry and clinical but it IS a textbook so yanno... no pissin' around which I personally like).
First off it needs to be noted that the Tone/Semitone sequence of the Major (Ionian) scale is what we use as our "yardstick" for ALL interval measurements. You measure from the "I" step (Root note) up to whatever note you are creating/"solving" your interval with... no matter what. There is a specific naming convention for all the intervals in the Major scale. There is also a specific naming convention/protocol when analyzing/manipulating intervals that do not conform to the Major scale Tone/Semitone sequence.
The default terms used in our (unaltered) Major scale yardstick for each interval are as follows...
I = Perfect Unison
II = Major 2nd
III = Major 3rd
IV = Perfect 4th
V = Perfect 5th
VI = Major 6th
VII = Major 7th
VIII = Perfect Octave
In proper music theory we never use the same letter note twice in the same diatonic scale/key (in theory... once you start using scales with more than 7 notes like Mixo-Blues that goes out the window). So you would never use Gb and G in the same key... you would use F# and G instead.
So to maintain that "one letter name per step" principle while being able to cover all 12 keys we instead Sharp (+1 semitone) and Flat (-1 semitone) any notes that do not conform to our Major scale yardstick. It's essentially forcing all these keys and scales to conform to the hard ass rules of the stave. As counterintuitive as it may seem ("why not have 12 letter note names?") it actually works out very elegantly and is very interesting to explore, especially on guitar and/or writing it out by hand.
Now I'm going to show you what the procedure is to acheive this on our yardstick and the terms we use (and when we use them) when identifying the interval "qualities" (quality meaning whether the interval is Major, Minor, diminished, augmented, etc) of each step. This is where you need to actually memorize some stuff but it's not too hard.
For each step I will use -1, -2 or +1 to indicate how many semitones are being added or removed to/from our interval along with what the proper term is when we do any of those things to said interval.
I* = Perfect Unison
-1 = Diminished 1st
+1 = Augmented 1st
*Since this is the Root we don't screw with it. We make everything else wrap around it HOWEVER it is fun to contemplate such as the concept of Zero and the like. But yeah, altering this step merely puts you in another key/mode so it serves no practical purpose. If you did though that's the correct terminology (I think).
II = Major 2nd
-1 = minor 2nd
-2 = diminished 2nd
+1 = Augmented 2nd
III = Major 3rd
-1 = minor 3rd
-2 = diminished 3rd
+1 = Augmented 3rd
IV = Perfect 4th
-1 = diminished 4th
+1 = Augmented 4th
V = Perfect 5th
-1 = diminished 5th
+1 = Augmented 5th
VI = Major 6th
-1 = minor 6th
-2 = diminished 6th
+1 = Augmented 6th
VII = Major 7th
-1 = minor 7th
-2 = diminished 7th
+1 = Augmented 7th
VIII* = Perfect Octave
-1 = diminished 8ve (octave)
+1 = Augmented 8ve (octave)
*Just like the "I" (root) we don't bugger with this outside of extreme naval gazing. ;-)
First thing to notice is that all steps that start out as "Perfect" can only become diminished (-1 semitone) and Augmented (+1 semitone) while all steps that start out as "Major" have three possibilities, minor (-1 semitone), diminished (-2 semitones) and Augmented (+1 semitone).
An easy way to remember this is think of your twelve bar blues. I, IV and V are the chords you are playing and the I, IV and V of our Major scale yardstick are "Perfect". You can only add (Augment) or subtract (diminish) 1 semitone with a "Perfect" interval.
The remainder of our steps are II, III, VI and VII... all of which start out as "Major". You will notice that you can subtract up to 2 semitones from these intervals but like the "Perfect" intervals you may only add 1 semitone. So whenever you are on one these four steps of our yardstick the sequence is -1 = minor, -2 = diminished and as with all steps +1 = Augmented.
If you have to go beyond these boundaries then you are breaking the laws of diatonic theory which means one of two things... you are using a scale that has more than 7 notes (such as blues scales and the like) or you are misinterpreting the interval/scale/chord/key/etc.
Okay, so that is some pretty crazy old school music nerd stuff and probably a little heavy for folks just looking for some new noodling possibilities but it's super useful for understanding a plethora of musical concepts and language.
Cheers!