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.

This means

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

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