Usually when people talk about a secret harmony they are talking about some mysterious chord like the one used by Alexander Scriabin called, variously, the Extase-chord or the "mystic chord" or even, by Scriabin himself, "the chord of the pleroma" a term from Christian Gnosticism. Here is how it looks:
That is a very unsettling chord and part of the reason is the unresolved dissonances. Where did it come from and where is it going? It is a good chord to torture the listener with, even though Scriabin seems to have used it to suggest some sort of higher reality above our phenomenal world. Sometimes Scriabin's harmonies are described as combining an augmented sixth or dominant ninth with tonic anticipation (see Taruskin, The Oxford History of Western Music, vol. iv "Music in the Early Twentieth Century" pp 214 et seq.), which is an attempt to see an exotic harmony in terms of its contrapuntal origins.
That was largely what I was trying to do in my previous post: show how harmony comes from counterpoint. A lot of exotic harmony is only exotic because of the context. For example, the augmented sixth chord sounds just like a dominant seventh chord until it resolves. Here, let me demonstrate. This is a standard cadence from V7 to I:
That first chord is a simple dominant 7th. But if we spell that E flat as a D#, we get something quite different. Spelled F A D# we have an augmented sixth chord as the interval from F to D# is an augmented sixth. This harmony actually comes from the old Phrygian cadence:
The chord resolves to the dominant--spelling the E flat, which would resolve down to D--as a D# which resolves up to E, makes all the difference. Composers use this re-spelling trick to facilitate some rather interesting modulations. In this case, for example, a V7 - I cadence in B flat becomes an Aug6 - V - i cadence in A minor (if you were to continue to the tonic)! And all this comes from a chromatic passing note.
The truth is that you can get away with a lot harmonically if you use chromatic passing notes well. Here is an example from Beethoven. It starts in measure 19 of the second movement of the String Quartet op 18 no. 2:
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The augmented sixth chord is on the third beat of the third measure. The A flat in the cello and the F# in the second violin make the interval of an augmented sixth and resolve outward to the dominant G. Here, let's have a listen. The four measures shown in notation above start right at the 1:39 mark in this clip:
That augmented sixth chord certainly has a rich and expressive sound, doesn't it? Another similar kind of harmony is that of the Neapolitan 6th, but I will save that for another post.