For advice on improvisation (intermediate-advanced) see Practical use of notes from diminished 7th chords in improvisation.
Diminished seventh chords are a bit of an oddity, sometimes ignored when practising, and causing panic when suddenly spotted in a chart. Their usual symbol is either dim7 or o7.
They differ from most other chords in being chromatic, that is to say there is no major scale that contains the 4 notes of any diminished 7th chord. They don't belong to any key. Nevertheless they are widely used in the great American songbook. I've tried to identify the common situations where diminished chords are found.
The interval between each note in a diminished 7th chord and the next is a minor 3rd. A further minor 3rd brings us back to the root. It's a very symmetrical chord. So the tones present are the root, minor 3rd, diminished 5th and diminished 7th. e.g. D, F, Ab, B. The interval of a diminished 7th is one semitone smaller than a minor 7th, so it is the same as a major 6th.
The diminished triad is not chromatic, it occurs naturally on the 7th degree of the major scale. The diminished triad symbol is usually dim or o, although those symbols in a chart can sometimes imply a diminished 7th.
Due to their symmetry, there are only three different diminished 7th chords. If you raise a diminished chord by a minor 3rd you just get an inversion of the original chord. So the three diminished chords are:
and a chord of Bo7 (for example) is just an inversion of Do7.
As well as the diminished chord, there's also the diminished scale, which alternates tones and semitones and has 8 notes per octave (it's 'octatonic'). Each diminished scale, and there are three, consists of the notes from two of the diminished 7th chords.
Take the scale
D Eb F F# G# A B C.
This is sometimes referred to as the 'D half-whole' diminished scale, named after the intervals at the start, a semitone ('half') then a tone ('whole'), starting at the root, D.
The other diminished scale that contains D is:
D E F G Ab Bb B C#,
which is the 'D whole-half' diminished scale. But nomenclature like that leads to 24 different scale names when there are really only three scales. After all, that scale is the same as the 'E half-whole' scale, the 'F whole-half', and so on.
I think (and this is a general point not related solely to diminished sevenths) there are two ways to learn this kind of thing. Either you learn 24 scales so that you automatically play the correct scale without thought when you encounter the chord; or you learn 3 scales and try to associate the correct scale with the chord when you see it in a chart. The former is much better, except for the enormous amount of practice time needed for that one activity (which might be better spent on ear training, for example). It's not worth it!
The diminished scale is often used in improvising over 7b9 chords, and this is possibly its main use. The scale to choose is the one that contains the correct diminished chord, but which also contains the root.
So over D7b9 you have F# A C Eb from the chord, plus D, and that brings in F Ab (G#) and B to give the complete
D Eb F F# G# A B C
A possible approach for associating the chord with the correct scale is to go for the 3rd, (or 5th, 7th, or b9th, whichever is easiest), and play the diminished 7th chord tones based on that starting point, and interpose the scale notes that are a tone higher.
The diminished scale is also useful over a diminished 7th chord, in which case play the notes of that chord, and again interpose the scale notes that are a tone higher.
If you're interested in theory, the diminished scale is a 'mode of limited transposition'<= back to resources index.