Most of the numbers we use describing chords are odd because harmony is based on thirds. The next number in the 1 - 3 - 5 series is 7. The seventh is always minor unless indicated. A C7 has a B-flat even though the B is natural in the key of C. Cma7 would contain a B-natural.
||A diminished 7th chord uses a diminished triad plus a diminished 7th (half a step smaller than a minor 7th). There are only three such chords, plus the many enharmonic spellings.|
Sometimes the "correct" spelling of a dim 7th can be awkward. The first two chords here are identical. The other two are simple spellings of the other two dim 7th chords.
The notes are E, G, B-flat and D. The name of the chord depends on where it is going.
|The same notes may be analyzed differently depending on the usage. A diminished triad with a minor 7th might be a "half-diminished 7th" as used in the first example. This accounts for the strong dominant - tonic movement in the progression.|
The middle example is a minor 6th chord. This explains the "Hollywood Cadence" sound of the sub-dominant - tonic progression.
The third pair shows the most common usage as a ninth chord missing its root!