Half diminished 7th is same as the minor 7th flat five chord .
Diminished chord has the spelling : root – minor 3rd – flat fifth. It is a triad along with major, minor and augmented chords.
Diminished 7th chord
Diminished 7th chord has the spelling : root – minor 3rd – flat fifth – major6th
Between each note of the diminished 7th chord, is an interval of a minor 3rd.
That is, the notes of a diminished 7th chord divides the 12 notes between octaves, into 4 equal divisions of minor 3rd intervals.
1. the diminished 7th chord is a diminished 7th chord starting from any of its notes as root and
2. makes the diminished 7th chord useful as a preceding chord to chords of 4 different roots. (V to I motion)
Diminished 7th chord from any of its notes
G# diminished 7th chord : G# B D F
B diminised 7th chord : B D F G#
D diminished 7th chord : D F G# B
F diminished 7th chord : F G# B D
The same notes, starting from each we get diminished 7th chords. This is because the interval between any nearby notes in the chord is the same, a minor 3rd.
Diminished 7th chord as entry to 4 different roots
For example, let us take G# diminished 7th chord : G# B D F
Below we group the notes starting from each of the four notes of the G# diminished 7th chord. Then we write a 7th chord which share three notes with the dim7th chord.
1. G# B D F
E7 : G# B D E
2. B D F G#
G7 : B D F G
3. D F G# B
A#7 : D F G# A#
4. F G# B D
C#7 : F G# B C#
We see that each of the note grouping or rearranging of diminished 7th chord has 3 notes in common with a 7th chord.
Those three notes in the seventh chord gives it the tension to be used as a V7 chord before a I chord. (in fact just the third and seventh notes of a 7th chord are enough to create the tension. the 3rd and 7th note form a tritone interval which is the reason for the unsettled feeling which asks for resolving.)
Since the diminished 7th chord has the tension interval (tritone) of a 7th chord, we can use the diminished 7th chord as the V7 chord in a V7 to I motion.
From the above grouping of notes, we see that the same diminished chord, can act as 4 different V7 chords in a V7 to I motion. Each of the V7 can then resolve to 4 different 'I' chords hence acting as a door to 4 different roots.
p.s. Here is a chord playing method that works: The chord code