This is an abbreviated introduction to traditional harmony that covers music roughly from the time of Mozart up to the the "Tristan chord" of Wagner. It is not intended to be a text on harmony, but only an introduction, and from a rather economical viewpoint. For a real text, see [Piston 1941]. In symbolics of chordal notation, Piston is often sloppy about subscripts and superscripts; be warned.

Emphasis here is on harmony in the context of major scales; they are not called major for nothing. See [Overtones]. Begin then in the key of C major, because it has pedigree in such beginnings and because it is easiest to visualize on a keyboard. On each note (abstractly, the degree) of the scale, build triadic chords:

            G   A   G   C   D   E   F   (G)
            E   F   G   A   B   C   D   (E)
            C   D   E   F   G   A   B   (C)

            I   II  III IV  V   VI  VII  I

            I   =  tonic            C major
            II  =  submediant       D minor
            III =  mediant          E minor
            IV  =  subdominant      F major
            V   =  dominant         G major
            VI  =  superdominant    a minor
            VII =  supertonic       G major 7th without its root
                                    (incomplete diminished 7th chord)

This is the first indication that the vertical order of intervals or pitches in a chord has a decided harmonic meaning.

Chordal inversions are obtained by cyclic permutation (more symmetries) of the vertical order of the pitches.

        first inversion           second inversion

              C                           E
        I^6   G                   I_4^6   C
              E                           G

The strongest permutation is the first given and called the root position of the chord.

From the physics of vibrations, one might explain its strength by saying that in root position the reinforcement of the overtones of each pitch is greater than it is in the first amd second inversions.

In the first inversion, and increasingly in the second inversion, a V flavor is being given to the chord by the interaction of overtones. The second inversion is frequently called upon to function as a cadential chord, that is, as an "appogiatura" that resolves as (GCE) -> (GBD). The root G with its controling overtones, makes the (CE) above it sound like unstable notes that must be resolved to accord with the root G. This is a simple tension reliever.

Harmony, of whatever sort, is a story of creating and relieving tension based on tone/pitch alone. In musical composition, there are other independent ways of doing the same thing, and that is what gives the effect of musical form its great complexity.

The same business with a minor chord and its inversions requires a refinement of the above argument, as does the situation with different instruments since in each case, the weighting of the overtones is different. The indroduction of the nonlinear processing by the cochlea of the ear makes life even more difficult in terms of explanations. The fact remains that the ear does not hear the reality of physics, but distorts and combines it in its own way. It is a general reality that the physical world is not as we perceive it; hence our ideas of what and how things are real can be sometimes (almost always) bogus.

There is a traditional but somewhat inconsistent chordal notation of harmonic function that describes triadal inversions and added tones, by using subscripts and superscripts on the Roman numeral notation. Here, a chord designated by a series of notes say I = (CEG) always mean in ascending order, read from left to right. Some examples:

        V^7         dominant seventh chord
                    In C major (GBDF)

        V_0^7       the VII chord being understood as a V substitute

        IV^6        first inversion of IV
                    In C major (ACF)

        IV_4^6      second inversion of IV
                    and called a "6-4 chord"
                    In C major (CFA)

        V_0^9       a dimished 7th chord being understood as
                    dominant 9th chord without its root.
                    In C major (BDFAb)
                    [More on dimished 7th chords and their root
                     ambiguities later]

The foundation of all traditional harmony of major mode lies in two related relationships:

             1)  I -> V (tonic - dominant) a vertical relationship
             2)  the "leading tone" cadential formula
                 7 -> 1                a horizontal relationship

                  I -> V  is equivalent to  IV -> I
   that
                  III is a weak substitute for V
                  VII is a strong substitute for V
                  II is a weak substitute for IV
                  VI is a weak substitute for I

   which accounts for all the triads built on the I scale in
   terms of the I -> V progression.  From this viewpoint
   this traditional harmony is utterly elementry.

As a matter of simple observation of found chordal progressions, the following are often found, in order, to the right, of approximately decreasing probability:

                I    ->     IV,  V,  VI,  II, III
                II   ->      V, VI,   I, III,  IV
                III  ->     VI, IV,  II,   V
                IV   ->      V,  I,  II, III,  VI
                V    ->      I, VI,  IV, III,  VI
                VI   ->     II,  V, III,  IV,   I
                VII  ->    III,  I