and the 1 b3 b5 7 - half dimininshed?
and the 1 b3 b5 b7 - ?
and the 1 b3 b5 bb7 - full diminished?"
1 b3 b5 is a diminished triad. I'm used to jazz, where it almost never comes up (you have some indication of the seventh.) You could call it diminished, but I don't like to because to me that assumes that you have the bb7. I suppose it doesn't technically assume that, (you have to write in the dim7 for it to mean that) so you could just call it diminished.
1 b3 b5 7 isn't a chord you see a lot. I've never come across it. Putting a major7 on top of a diminished triad is a strange sound. I suppose it would be a min/maj7b5 chord? Best name I can think of (the min/maj7 chord is the same chord with a perfect fifth, so if you flat the 5, I suppose that is what it would be called.) Don't worry about this one, you'll probably never see it written out.
1 b3 b5 b7. This is a half diminished chord, more commonly called a m7b5 chord. It is what you get when you put a minor 7th on top of a diminished triad. It is a diatonic chord; it is built from the seventh degree of any major key. Half diminished as in diminished triad, but not a diminished 7th. It comes up most often in jazz when you have a ii V i in a minor key. (i.e. Dm7b5, G7, Cm7.)
1 b3 b5 bb7. This is a diminished 7 chord. While you can think of it in terms of its degrees (as written above, maybe easier to think 1 b3 b5 6) it also makes sense to think of it as a stack of minor thirds. The chord is built by taking minor thirds and stacking them on top of each other. For this reason, A diminished is the same as C diminished is the same as Eb diminished; the notes repeat every minor third. It is diatonic to no key. This one comes up a lot, so you should be used to it. It can be a passing chord, because a diminished chord always wants to resolve upwards a half step, so you can approach any chord with a diminished chord a half step below. (instead of just playing C minor, you might play B dim7, then C minor) It is also sometimes used to connect two chords a whole step apart by placing a dim7 chord between them, even if the movement is descending (instead of Dm7, Cm7, you have Dm7 Dbdim7, Cm7.) You can use it to make your own progressions more interesting, and soloing over it is fun and wierd sounding. It's a good chord to be familiar with.
