When aTetradis incomplete, one of itsorbitshas been replaced by another.
Only thePrimary Notes(Orbit 1andOrbit 0) are free to substitute for another.

The two most common forms of substitution are:
     1. Removal ofOrbit 3, substitution by
           Orbit 1, indicated1(3), or possibly by
           Orbit 0, indicated0(3).
     2. Removal ofOrbit 0, substitution by
           Orbit 1, indicated1(0).

In musical texts, notes will be colored according to the orbit removed,
so that its fundamental presence is always evident.
In this manner, the 4 colors will always be present in any Triad.

A substitution may last for the entire length of a chord,
or for only part of it.

The main purpose of this concept of Incompleteness is to maintain contact with theVoice-Leadingof basic Tetrads while dealing with Triads.

Beware of Triads
In each Tetrad, two Triads are "hidden"
- the real Triad, in which theMOTRIXis missing, substitution1(3)or0(3)but
     theFRAMEremains, and
- a deceptive Triad, in which theCOMMON TONEis missing, substitution1(0)
     and in which the two secondary notes can easily be mistaken for aFRAME.

Incompletenessallows us to -
     (1) clearly distinguish between the two forms of Triads,
     (2) control the Voice-leading to and from Triads, and
     (3) "double" the most appropiriate note of a Triad,
          especially when we think in terms of Orbits.

Substitution for the entire length of the chord

Incompleteness and Voice-leading
Let us first see how this concept of Incompleteness helps us to control Voice-Leading, which was originally defined by the procuration of theMOTRIXitself (Orbit 3) but, as we will now see, can also be defined by the procuration of the substitution ofOrbit 3,1(3)or0(3).

-Voice-leading Aby the procuration fromOrbit 0,0-3or0-1(3)or0-0(3),
-Voice-leading Bby the procuration fromOrbit 1,1-3or1-1(3)or1-0(3),
-Voice-leading Cby the procuration fromOrbit 2,2-3or2-1(3).

In the six following musical examples,
     the harmony will always be the same - G⁷/ C⁶, G⁷/ C
          with the DOMINANT/TONIC progressions over a bar-line ( 2 cells):
               the Dominant chords on the Off-beat and the Tonic chords on the Beat.
     The 2 TONIC chords are presented as Real Triads, withoutOrbit 3(C).
     The first DOMINANT chord is also presented as a Real Triad, withoutOrbit 3(G).
     The second DOMINANT chord is presented as a Deceptive Triad,
          withOrbit 3but withoutOrbit 0,
               substitution1(0)in the Tenor voice,
                    not an obligatory decision but one which helps the Bass line, the Rhythm, and
               the fifth (COMMON TONE) in the Bass voice, removing all danger ofFRAME inversion,
                    the root (COMMON TONE) not being present.
     Each example will be classified according to the Voice-leading
          from the first to the second chord, A, B, or C, and
          according to the Substitution of theMOTRIXon the second chord,1(3)or0(3).
     All procurations and substitutions take place in the Soprano voice.

-Voice-leading A to substitution1(3)

-Voice-leading A to substitution0(3)

-Voice-leading B to substitution1(3)

-Voice-leading B to substitution0(3)

-Voice-leading C to substitution1(3)

-Voice-leading C to substitution0(3)is not recommended
     because the natural resolution ofOrbit 2(2-1) is not respected.

Analysis
Now, how do we analyse this kind of Triad writing?
     First, we establish the substitutions ofOrbit 3in our Real Triads
          (first and second chords - indicated by the color ofOrbit 3).
     By working back fromOrbit 2of any chord,
          it is possible to establish the substitution for theMOTRIXon the previous chord
               becauseOrbit 2must be preceded by eitherOrbit 3itself or
               by one of its possible substitutions,1(3)or0(3).

Substitutions 0(3) and 1(3), Chord by chord
On the first chord, all examples have substitution0(3),
     the note G, in the Bass.
On the second chord, we find substitutions1(3)or0(3),
     the note C or the note G, all in the Soprano.

Voice-leading, Chord by Chord
The Voice-leading of the first chord (DOMINANT) can be established
     by observing which Orbit in the first chord (always in the Soprano) procures the
          substitution ofOrbit 3(in the second chord).
The Voice-leading of the second chord (TONIC) is easily established
          because the following chord possesses itsMOTRIX(Orbit 3, the note F in the Alto).
     It is Voice-leading A in all cases.

It is interesting to note how little effect substitutions have on the Voice-leading
and on the choice of notes in the other voices.

It is also interesting to note that the "doubling" ofOrbit 1orOrbit 0on the Real Triads
is dictated by the disposition of the four orbits in the Basic Tetrad disposition.

More complex examples of Incompletenesscan be seen
in theMetamorphoses
and inNon-chordal tones

The Voice-leading A Swing

With 0(3)
In the following musical example, the original example of the Voice-leading A Swing
     will be subjected to Incompleteness0(3),
          removal ofOrbit 3and substitution byOrbit 0, (inscribed C⁶and G⁷).

This will produce an orbit chain0-0(3)-2-1, which will be repeated in the 4 voices.
     We now have only Real Triads, for both theTONIC and DOMINANT chords,
          clearly chords ofIandV.

With 1(0)
In the following musical example, the original example of Voice-leading A
     will be subjected to Incompleteness1(0),
          removal ofOrbit 0and substitution byOrbit 1, (inscribed C⁶and G⁷).

This will produce an orbit chain1(0)-3-2-1, which will be repeated in the 4 voices.
     We now have only Deceptive Triads, for both theTONIC and DOMINANT chords,
          which can easily be mistaken for chords ofVIandVII
               (with seemingly doubledMEDIANs, in realityOrbit 1of the TONIC and DOMINANT),
          clearly chords ofIandV.

Same sound
We would have achieved a very similar sound by removing the Alto voice
from the original musical example of theVoice-leading B Swing.

Substitution for part of the length of the chord

Here is a simple example using the Voice-leading C swing,
     with DOMINANT / TONIC chords (G⁷/ C⁶), and
          with substitutions1(3)1(3) and0(3)as well asOrbit 3itself on the same chord,1(3)-0(3)-3.
They will all have the sameOrbit 3color to remind us of what is "underneath".

-Note that the use of the substitution1(3)at the beginning of the bar
     enables us to resolveOrbit 2properly (2-1).
This example might also strike a bell (Bell-3)
- otherwise, wait to see (and hear) itdominantized !

NOTE that partial substitution1(3)-3was used in the swings of themajor mode,
     on the DOMINANT and COUNTER chords.

Triads again

In the fundamental MAJOR mode, key of C, it is interesting to note that,
     in the four Tetrads G⁷, C⁶, F⁶, and C⁺⁷we have every Triad available.
In the G⁷Tetrad, we have
- the Real Triad ofV(G, B, D) and the Deceptive Triad ofVII(B, D, F).
In the C⁶Tetrad, we have
- the Real Triad ofI(C, E, G) and the Deceptive Triad ofVI(A, C, E).
In the F⁶Tetrad, we have
- the Real Triad ofIV(F, A, C) and the Deceptive Triad ofII(D, F, A).
In the C⁺⁷Tetrad, we have
- the Real Triad ofI(C, E, G) and the Deceptive Triad ofIII(E, G, B).

-This musical example would normally be analysed as the Triads ofI - III - IV - II ₆ - I.
The Triad ofIis a Real Triad, part of C⁺⁷, but
     the Triad ofIIIis a Deceptive Triad, also part of C⁺⁷, in which we have -
          theMOTRIXB descending to theMEDIANA, Orbits3-2,
          theMEDIANE ascending to thePROPER TONEF, Orbits2-1,
          the "doubling" of thePROPER TONEG,Orbit 1, the result of the substitution1(0).

Displacements on the same chord
From the Triad ofIto the Triad ofIII,
note the displacement in the 3 lower voices, of Orbits0,1, and2.
Don't forget substitution1(0)in the Tenor voice.

The Triad ofIVis a Real Triad, part of F⁶, but
     the Triad ofII ₆is a Deceptive Triad, also part of F⁶, in which we have -
          theMOTRIXD ascending to theMEDIANE, Orbits3-2,
          theMEDIANA descending to thePROPER TONEG, Orbits2-1,
          the "doubling" of thePROPER TONEF,Orbit 1, the result of the substitution1(0),
               including its role (as root of the chord) in the Fundamental Bass at the end.

What's underneath
We have here a C⁺⁷- F⁶- C progression expressedonly in Triads, with -
     the Real Triad (on theBeat), followed by
     the Deceptive Triad (on theOff-beat).

These triads come directly from the Basic Tetrad version
     withVoice-leadingB on the first chord.
The Tenor voice has been transported from theOrbit 0(0-0) line to theOrbit 1(1-1) line
     by the two substitutions1(0)on the Deceptive Triads.

Voice-leading possibilities

With this transformation of Incompleteness,
     we now have a complete picture of the possibilities of Voice-Leading.
          In theStructure of the Major Mode, we found the "normal resolutions" -
                Orbit 3resolves toOrbit 2, (3-2), and
                Orbit 2resolves toOrbit 1, (2-1),
                Orbit 1andOrbit 0move freely to each other -
                     0-0and1-1in the orbit lines,
                     0-1and1-0in the Fundamental Bass,
                         depending on the direction of the chord progression.
          InVoice-leading, we found that -
                Orbit 0,Orbit 1, andOrbit 2can procureOrbit 3(0-3,1-3,2-3).
          Now, in Incompleteness, we found -
               the Substitutions ofOrbit 3which gave us0(3)-2and1(3)-2.

Recap
Orbit 3can only resolve toOrbit 2.
Orbit 2can
     resolve toOrbit 1, (2-1) or
     procureOrbit 3, (2-3).
Orbit 1can go anywhere (1-3,1-2,1-1,1-0).
Orbit 0can go anywhere (0-3,0-2,0-1,0-0).