Diminished seventh chord

Chord notation: Cdim7, C°7, Co7, or in jazz just Cdim, , or Co. Diminished chords are always used as passing chords, not as a resolution chord. And they are often used instead of a dominant seventh chord, because the top four notes in a dominant seventh flat nine chord (7b9), is a diminished chord. So here there is an interesting parallelism operating in minor thirds.

The dim7 chord and its inversions, has the same fingering! 
 
|---|---|---|  |---|---|---|  |---|---|---|  |---|---|---| 
|---|---|-Eb|  |---|---|-Gb|  |---|---|-A-|  |---|---|-C-| 
|-A-|---|---|  |-C-|---|---|  |-Eb|---|---|  |-Gb|---|---| 
|---|---|-Gb|  |---|---|-A-|  |---|---|-C-|  |---|---|-Eb| 
|---|-C-|---|  |---|-Eb|---|  |---|-Gb|---|  |---|-A-|---| 
|---|---|---|  |---|---|---|  |---|---|---|  |---|---|---| 
      3          5       7          9              12 

This parallelism is remarkable. Everybody knows that on advantage of the guitar over other instruments like the piano, is that every chord fingering in principle is movable on the the guitar, repeating itself every twelfth fret. But with the dim7 chord has another paralellims in that it works in minor thirds. Becaus of its perfect symmetry it repeats itself every fourth fret. More on this in the article Pat Martinos dim7 insights.

A diminished chord, or diminished seventh chord, is a completely symmetric four note chord, in which each note is always separated by a minor third, weather it is an inversion or not. If you make an inversion of a diminished seventh chord, moving the lowest note one octave up, the minor third pattern is intact. This symmetry is explained by the simple fact that a stack of four minor third intervals is an octave:

 Cdim7 arpeggio      Cdim7 arpeggio inversion 
|---|---|---|---|    |---|---|---|---|---| 
|---|---|---|---|    |---|---|---|---|---| 
|---|---|---|---|    |---|---|---|---|---| 
|---|-A-|---|---|    |---|-A-|---|---|-C-| 
|-Eb|---|---|-Gb|    |-Eb|---|---|-Gb|---| 
|---|---|-C-|---|    |---|---|---|---|---| 
      7       9            7       9 

So what's special about the dim7 chord is that it is perfectly symmetrical. First, there's a symmetry in that they are made of three superimposed minor 3rds (i.e. diminished triads). The diminished seventh thus consist of two tritones (#4 = augmented fourth = 6 semitones above the root. The same number of frets as the b5).

A stack of two minor thirds is a tritone, and four minor thirds on top of each other is an octave.

  |C| | |Eb| | |Gb| | |Bbb| 
   \   /  \   /  \   / 
     m3     m3     m3 
      \   /  \   /  
        b5     b5  
          \   / 
            b5 

The b5 at the bottom of the triangle is explained by the fact that the interval from Bbb back to the C again is a minor third too.

 |C| | |Eb| | |Gb| | |Bbb| | |C|  
   \   /  \   /  \   /  \   / 
     m3     m3     m3     m3 

Second. It's symmetric because all of the inversions of them sounds harmonically similar.

Third. Because of the chord's symmetrical nature, there are only three different diminished seventh chords possible. Because the interval between each note is identical, then any of the chord tones can be the root of the chord -i.e. Gbdim7 = Ebdim7 = Cdim7 = Adim7.

Fourth on the guitar, the diminished seventh chords repeat themselves without any change in fingerings, every fourth fret - the fingerings for inversions of diminished chords are identical.

There are two practical voicings of diminished sevenths. See Pat Martinos dim7 insights.