When I was young, I discovered this intriguing connection between major and minor chords, and other types of chords, that no textbook seem to mention. Just wondering what you guys think of this.
Take any major chord, say C major, with the tonic duplicated an octave above, consisting of the notes C, E, G, C. The interval between C and E is a major 3rd, and the interval between E and G is a minor 3rd, and between G and the upper C is a perfect 4th. So we have the sequence of intervals: major 3rd, minor 3rd, perfect 4th, and this sequence of intervals duplicates itself across octaves (if we double the chord each octave above and below). We may abbreviate this as the sequence +3, -3, 4. Note that this sequence is common to all major chords, and its cyclic order is unchanged by chord inversion: a first inversion chord has intervals -3, 4, +3, which, if you wrap it around a circle going clockwise, say, has exactly the same cyclic ordering as +3, -3, 4. Similarly, the second inversion 4, +3, -3, also has the same cyclic ordering. So we may say that the +3, -3, 4 cyclic sequence of intervals is a "signature" of a major chord.
Now something interesting happens if we hold a mirror to our piano keyboard, and finger a C major chord as shown in the mirror. What we get is, of course, not C major, but the notes E, A, C, E, which is actually an inversion of A minor. If we were to measure the intervals between consecutive notes of this chord, we find that it has the cyclic sequence 4, -3, +3, 4, .... Note that this is the exact mirror-image of the major chord cyclic sequence +3, -3, 4, .... In other words, reversing the sequence of consecutive intervals in a major chord turns it into a minor chord.
What if we apply this interval reversal to other kinds of chords? For example, the dominant 7th chord, say G7, consists of the notes G, B, D, F, G, which has intervals +3, -3, -3, 2. What happens if we reverse this to 2, -3, -3, +3? Well, if we start from G, that gives us G, A, C, Eb, G, which is the second inversion of the C minor 6th chord.
This is particularly interesting, because if you fix a tonic pitch, say C, then build a chord progression on it, say C major -> F major -> G major -> C major, then you invert the intervals in each chord relative to the tonic (or equivalently, play those chords in a mirror-image piano keyboard), you get the chord progression A minor -> E minor -> D minor 6th -> A minor. Notice that the subdominant F has become the dominant E minor (without the sharpened leading tone), and the dominant has become the *subdominant* D minor 6th chord. So in some abstract sense, the dominant 7th chord in a major key is an analogue of the subdominant minor 6th chord in a minor key.
Any chord progression in a major key can be "mirror-imaged" in this way into a chord progression in a minor key, with dominant/subdominant roles reversed. Yet, because of the mirror-image equivalence of the intervals in each chord, the progression Dm6 -> Am sounds "almost like" the traditional full cadence G7 -> C. Weird, huh?
We can apply this going the other way, too. If we sharpen the leading tone in a minor key, we get a major dominant chord, which is E major or E major 7th in the key of A minor. What happens if we invert this? In fact, we get nothing other than F minor, or respectively, F minor 6th, which is the minor subdominant of C major. In other words, sharpening the leading tone in a minor key is somehow analogous to flattening the 6th degree of in a major key. If we invert a minor key cadence E -> Am in this way, we get the analogue Fm -> C, which is a modified Amen cadence with a minor subdominant. If we use the dominant 7th chord in the minor key, E7 -> Am, we find that the inverted analogue is Fm6 -> C. Again, we see the duality between the dominant 7th chord and the minor 6th chord. This seems to suggest that the amen cadence and the closed cadence are somehow duals of each other!
One may wonder, then, whether all chords have "duals" constructed from this mirror-image process. The answer is that some chords are singular: they do not change their "species" under this mirror-imaging!
One such chord is the minor 7th chord. Take Am7, for example, consisting of the notes A, C, E, G, which gives us the interval sequence +3, -3, +3, +2. The interesting thing about this sequence is that it is cyclically palindromic: if you reverse the order of intervals, you get +2, +3, -3, +3, which is just a single inversion away from the original +3, -3, +3, +2. If you do this in the mirror, you find that you end up playing the notes G-E-C-A, which is the same chord, just in a different inversion.
Other examples of singular chords are the diminished 7th and the augmented 5th. C aug 5, for example, has the notes C, E, G#, C, which gives the interval sequence +3, +3, +3, which is clearly palindromic, so mirror-imaging it does not change it. Similarly, the diminished 7th chord has interval sequence -3, -3, -3, -3, which is also palindromic.
An even more interesting chord to consider is the major 9th chord, for instance G-B-D-F-A. Mirror-imaging this chord gives G-B-D-F-A -- again the same chord! So major 9th chords are also singular. Looking at the interval sequence +3, -3, +3, -3, +3, we see that it's palindromic, as expected.
So where does all this lead to? It suggests a new kind of harmonic device, a kind of "harmonic reversal", or more simply "mirror image" :-P, analogous to the melodic device of inversion, where major changes into minor and vice versa, dominant and subdominant exchange roles, and chords that are neither major nor minor ("singular") remain the same. Because all the relationships between intervals are preserved (except in reverse order), there is no need to worry about things like whether to sharpen the leading tone when replacing the dominant chord in a major key with the dominant chord in a minor key; under this reversal, it becomes instead the subdominant chord, perfectly reflecting the function of the original progression. Everything just fits together exactly as before, except in upside-down pitch order.
It also suggests an interesting way of discovering new chord progressions: take any existing progression and mirror-image it to find the dual progression where major/minor are swapped, subdominant/dominant are exchanged, etc.. Note that this works outside of subdominants and dominants too: a progression that changes from major to relative minor, like C -> E -> Am, becomes Am -> Fm -> C, which is just as "smooth" a progression as the original! Similarly, C -> D -> G, which smoothly modulates from a tonic major to dominant major, becomes Am -> Gm -> Dm, which smoothly modulates from tonic minor to subdominant minor. All kinds of such dual progressions can be discovered this way.
Isn't it interesting? :-)