Diminished chords, and the symmetry that erases the root
Minor thirds stacked until the chord turns symmetrical, with no settled root; the same chord has four readings, and the spelling chooses which one counts. It is the passing chord of choro.
Stack minor thirds, and only minor thirds. Three notes give the diminished triad; four give the diminished seventh chord, and in it something happens that happens in no other chord: the fifth minor third returns to the note you started from. The chord closes on itself, and in doing so loses its root. This is the diminished chord, and its symmetry is at once the problem and the use.
Two crossed tritones
C diminished seventh has C, E flat, G flat and B double flat. Between C and G flat there is a tritone; between E flat and B double flat, another. The chord is two tritones interlocked, which is why it is the most tense chord tonal harmony uses: everything in it asks to move.
The spelling with a B double flat looks like overkill, and it is not: the seventh of C has to be some kind of B, and the diminished seventh is a B lowered twice. Nobody writes that in a chart, and on the keyboard it is the A key; on the staff, the name says where the note came from.
Four readings of the same chord
Because the four notes sit at equal distances, any of them can be the root. C, E flat, G flat and A form the same chord as E flat, G flat, A and C, and as G flat, A, C and E flat. There are only three diminished seventh chords in the whole world; the other nine are inversions of them under other names.
Play the four in sequence: it is the same sound rising third by third. The spelling chooses which of the four readings counts, and the choice is made by the next chord: the diminished chord resolves a semitone above the root the chart chose for it.
The diminished as a dominant
The commonest reading of the diminished chord is as a dominant with no root. G with a seventh and a flat ninth has G, B, D, F and A flat; take the G away and what is left is B, D, F and A flat — the B diminished seventh chord. It is the dominant of C with the bass removed, and it resolves to the same place.
That is why the diminished chord on the seventh degree, a semitone below the tonic, is a cadence: it is the dominant with a flat ninth seen from another angle, and it reaches the tonic by a semitone in the bass.
The passing diminished
The use that gives the chord its place in choro and samba is as a passing chord: between two diatonic chords a whole tone apart, the diminished chord comes in on the note between and links the two by semitones.
C, C sharp diminished, D minor: the bass rises by semitones and the diminished chord is the dominant of D minor with no root, the V7 of II with the A removed. The same gesture serves between the second and third degrees, between the fourth and fifth, rising or falling, and it is the trademark of choro accompaniment.
The half-diminished is another thing
The B half-diminished chord — B, D, F, A — has only one tritone and a minor seventh, not a diminished one. It is not symmetrical, it has a clear root, and its function is preparation: it is the second degree of the minor key, the first chord of the minor II–V–I. The chart writes m7♭5, or the symbol ø, and the difference of one note, A against A flat, is the difference between a chord that prepares and one that pushes.
What to take from here
The diminished seventh chord is minor thirds stacked until the octave closes, two crossed tritones, with no root of its own: only three exist, and each has four names, chosen by the chord that follows. It is a dominant with a flat ninth minus its bass, which is why it resolves a semitone above; as a passing chord, it links degrees a whole tone apart by semitones. The half-diminished, with a single tritone, is a different chord — the one that prepares in a minor key.