Harmony/Transformations/Coda 1/Uninvertibility

It seems paradoxical, and possibly humiliating, for a scientific vision of music theory to accept the fact that certain harmonic progressions refuse to invert and apparently act as exceptions. It seemed high time that the culprit be identified and its role clearly defined.

The role of the Fundamental Bass
in Uninveribility

The concept of the Fundamental Bass was first presented
by Jean-Philippe Rameau in his Traité de L'Harmonie (1722).
The very first musical example (of the G7/C progression)
is presented in 5 voices,
the top 4 for the Orbit Lines
and the bottom voice for the Fundamental Bass.
Even with considerable hindsight,
it would be difficult to evaluate which was the greater discovery,
the Fundamental Bass or the use of 5 performance voices.

The Fundamental Bass, which consists of the root of each chord, is thus by definition uninvertible, contrary to the four Orbit Lines which are invertible. In certain 4-voice chord progressions, one of the Orbit Lines being missing, the FB plays a crucial role which permits this specific progression to take place but which prevents its inversion.

As we see in the Primary Notes section of the Voice-leading Matrix,
     the FB is either Orbit line 0-1 (in flattening, descending, seventh progressions)
          or Orbit line 1-0 (in sharpening, ascending, sixth progressions).
     Orbit line 1-0 is an integral part of Voice-leading A
          but Orbit line 0-1 does not appear in any of the Orbit line possibilities.
This is where Uninvertibility is born and this is where we will find our explanations.
     The following list will constantly be enlarged and everyone is invited to send in other examples.

The Blues Chord

To our knowledge, no academic harmony text-book
has ever spoken of the Blues Chord.
The German augmented sixth chord is composed of the same notes,
but serves a completely different function.
Jazz musicians use the chord often,
but insist on labeling it a seventh rather than a sixth chord.

We first saw, and defined, the Blues Chord
     in Harmony/Transformations/Chromaticism/Diminution.
We then saw it in greater detail, with new possibilities of usage,
     as one of the Favorites in Harmony/Sonorities.
It can also be found (and heard) as A1 in the Clips.

The explanations in the Favorites are quite explicit -
     The FB is what makes the F+6/C progression possible,
               the F in the bass resolving to C, the 1-0 Orbit Line.
          Had the F resolved to G, in the 1-1 Orbit Line,
                    which is not present in the Blues chord,
               the F# would have been obligatory, to be compatible with the D#,
                    in the Diminution transformation of the chord.
     The FB in a flattening (descending) progression like Em-7/Am -
          uses the 0-1 Orbit Line,
               preventing the inversion of the Blues chord.

The 1-1(10) Cluster

This 1-1(10) cluster (Db,D,D#) is presented in Harmony/Sonorities/Favorites.
     Explanation of the Listing code can be found in Harmony/Sonorities/List.

In this specific musical example,
     the version with the flattening (descending) progression G7/C is presented first,
          with all the fifths of the chords in the tenor voice, and
     the version with the sharpening (ascending) progression Dm6/Am is presented second,
          with all the roots of the chords in the FB.

The second example, in Diatonic Minor,
          is certainly the more fundamental and successful of the two.
     This did not seem so to us at the time,
          favoring the Diatonic Mode and starting with it,
               to eventually invert the result later.
     Once again, the 1-0 Orbit Line in the FB makes this possible.

The Augmented Triad

The situation here, where discordance is avoided,
     is the opposite to that of the preceding example,
          where discordance was saught.
     The 0-1 Orbit Line in the FB will be preferable,
               to avoid discordant conflict with the chromatic substitution in the 1(3)-2 Orbit Line,
          thus giving priority, for once, to the flattening, decending, seventh progressions.

The Diad

It is surprising that something as fundamental and usual as the DOMINANT/TONIC, G7/C cadence,
     should produce something as disappointing as an incomplete TONIC chord,
          which we called a "Diad" in Triad Harmony.


In the G7/C cadence, the 0-1 FB is evidently, as usual, responsible for the problem.
     Possessing Orbit 0 in the DOMINANT chord, it eliminates the Orbit 0-Orbit 0 line (G-G),
               if the chord is to remain complete,
          and deprives the TONIC of its Orbit 0 (G).
In the Dm6/Am cadence, the 1-0 FB will preserve the TONIC triad.
     Possessing Orbit 1 in the DOMINANT chord, it maintains the Orbit 0-Orbit 0 line (A-A),
               eliminating the Orbit 1-Orbit 1 line (D-E) which is not essential,
          and the TONIC triad remains complete.

Let us know if you have other examples.
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