Harmony, a long with a lot of other aspects of music, has deep-rooted foundations in maths and logic. There's fractions in rhythm, algebra in learning scales, physics in harmonics and many more. Too, there is plenty of symmetry in harmony. These tend not to appear in our Diatonic scales and chords, but more in synthetic ones. I'm going to briefly look at diminished chords, augmented chords and some other synthetic scales in which these can be derived from.
A diminished chord is made up of stacked minor 3rds [3 semitones] i.e C, Eb, Gb, Bbb (this is B double-flat, which sounds the same as 'A'. Click here to read up on enharmonic equivalents, and why they're used.)
Because the chord is built up of the same stacked interval, it is symmetric, meaning C dim is same as Eb dim, Gb dim, A dim (or Bbb dim, but we're going to stop calling it that for this article).
N.B At this stage it is important not to confuse diminished chords with 'half diminished chords' they have similar structure but half-dims have a natural minor 7 at top, e.g. C, Eb, Gb, Bb. Sometimes these would be refereed to as Cmin7(b5), however these are not symmetrical.
There are two types of diminished scale associated with the diminished chord. There's a half-whole scale built on the root, and a whole-half (reason being half-step plus whole step = 3 semi-tones, same as whole-step plus half-step.)
In C:
Fig .1 'whole-half step' = C D Eb F Gb Ab A B C
Fig .2 'half-whole step' = C Db Eb E Gb G A Bb C
You'll notice both of these scales contain the notes C, Eb, Gb and A. Which go to make up the chord. If you've been following you'll also realise that this accounts for the Eb, Gb and A diminished scales. Not only that but also the D, F, Ab, B, Db, E, G and Bb diminished scales can also be attained from this.
There is just one diminished scale missing, and that is as follows:
Fig .3 G Ab Bb B Db D E F G - this is the 'half-whole step' scale starting on G, as the 'whole-half step' scale would have been covered by Fig .2
What this all translates to, is that when using a diminished chord as a substitution, there are various directions you can go. Take this simple ii V I in C:
Dm G7 Cmaj7
The notes contained in the G7 chord us G B D F. Swapping the G for an Ab gives us Abdim (though it's more likely we'd talk about B dim as its root would have a smoother transition to the I chord), Giving us:
Dm Bdim Cmaj7
Knowing that vii dim leads nicely to I, you could in-fact change key, with these examples:
Dm Ddim Ebmaj7
Dm Fdim Gbmaj7
Dm Abdim Amaj7
Notice each time, the I chords in each example are minor thirds apart. Also these resolved chords could just as easily be minor chord, Dm Bdim Cm etc
A augmented chord is a chord built upon stacked Major 3rds. For instance, a C augmented (written C+) would read C E G#. Again, this chord is symmetrical, so C+ contains the same notes as E+ and G#+, meaning, like the above examples, there are various options when using it. Commonly, the augmented chord is used instead of a V. So using the above ii V I, let's see what we can do using G+ in-place of G7:
Dm G+ Cmaj7
Dm B+ Emaj7
Dm D#+ G#maj7
Again you'll notice the resolved I chords are major 3rds apart, and they too could be substituted for minor chords i.e Dm G+ Cm etc
The augmented chord appears naturally in some variations of scales and modes, particularly the harmonic and melodic minor modes, but too in this particular synthetic scale, called the 'whole tone scale' as it is built up on simply whole tones:
In C:
C D E F# G# A# C
This would be the same as D, E, F#, G# and A# whole tone scales, meaning there is only one other whole tone scale, beginning on C#:
C# D# E# (F) G A B C#
All in all, there are a plethora of options available using symmetrical chords, just as substitutions for the V chord, meaning chaining key is easy. I will cover a little more on the diminished chord later on.