Diminished 7th is fun, because if you look at it in terms of intervals, each note is the same distance apart--interval of a minor 3rd.
Because the 3 half-steps of each interval evenly divide the 12 steps of the scale, it is "symmetric", if you will. If you move the chord up (or down) by 3 frets/half-steps... it's the same chord! Same notes, just in a different order now.
Even more fun, because of that symmetry, you can call the same chord by four different names:
Cdim7: C-Eb-Gb-A
Ebdim7: Eb-Gb-A-C
Gbdim7: Gb-A-C-Eb
Adim7: A-C-Eb-Gb
Which name you choose depends on how it's being used in the song and the chords on either side of it.
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