When a Tetrad
is incomplete, one of its orbits has been replaced by another.
Only the Primary Notes (Orbit 1 and Orbit 0) are free to substitute for another.
The two most common forms of substitution are:
1. Removal of Orbit 3, substitution by
Orbit 1, indicated 1(3), or possibly by
Orbit 0, indicated 0(3).
2. Removal of Orbit 0, substitution by
Orbit 1, indicated 1(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 the
Voice-Leading of basic Tetrads while dealing with Triads.
Beware of Triads
In each Tetrad, two Triads are "hidden"
- the real Triad, in which the MOTRIX is missing, substitution 1(3) or 0(3) but
the FRAME remains, and
- a deceptive Triad, in which the COMMON TONE is missing, substitution 1(0)
and in which the two secondary notes can easily be mistaken for a FRAME.
Incompleteness allows 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 the MOTRIX itself (Orbit 3) but, as we will now see, can also be defined by the procuration of the substitution of Orbit 3, 1(3) or 0(3).
- Voice-leading A by the procuration from Orbit 0, 0-3 or 0-1(3) or 0-0(3),
- Voice-leading B by the procuration from Orbit 1, 1-3 or 1-1(3) or 1-0(3),
- Voice-leading C by the procuration from Orbit 2, 2-3 or 2-1(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, without Orbit 3 (C).
The first DOMINANT chord is also presented as a Real Triad, without Orbit 3 (G).
The second DOMINANT chord is presented as a Deceptive Triad,
with Orbit 3 but without Orbit 0,
substitution 1(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 of FRAMEinversion,
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 the MOTRIX on the second chord, 1(3) or 0(3).
All procurations and substitutions take place in the Soprano voice.
-Voice-leading A to substitution 1(3)
-Voice-leading A to substitution 0(3)
-Voice-leading B to substitution 1(3)
-Voice-leading B to substitution 0(3)
-Voice-leading C to substitution 1(3)
-Voice-leading C to substitution 0(3) is not recommended
because the natural resolution of Orbit 2 (2-1) is not respected.
Now, how do we analyse this kind of Triad writing?
First, we establish the substitutions of Orbit 3 in our Real Triads
(first and second chords - indicated by the color of Orbit 3).
By working back from Orbit 2 of any chord,
it is possible to establish the substitution for the MOTRIX on the previous chord
because Orbit 2 must be preceded by either Orbit 3 itself or
by one of its possible substitutions, 1(3) or 0(3).
Substitutions 0(3) and 1(3), Chord by chord
On the first chord, all examples have substitution 0(3),
the note G, in the Bass.
On the second chord, we find substitutions 1(3) or 0(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 of Orbit 3 (in the second chord).
The Voice-leading of the second chord (TONIC) is easily established
because the following chord possesses its MOTRIX (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" of Orbit 1 or Orbit 0 on the Real Triads
is dictated by the disposition of the four orbits in the Basic Tetrad disposition.
More complex examples of Incompleteness can be seen
in the Metamorphoses
and in Non-chordal tones
The Voice-leading A Swing
In the following musical example, the original example of the Voice-leading A Swing
will be subjected to Incompleteness 0(3),
removal of Orbit 3 and substitution by Orbit 0, (inscribed C6 and G7).
This will produce an orbit chain 0-0(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 of I and V.
In the following musical example, the original example of Voice-leading A
will be subjected to Incompleteness 1(0),
removal of Orbit 0 and substitution by Orbit 1, (inscribed C6 and G7).
This will produce an orbit chain 1(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 of VI and VII
(with seemingly doubled MEDIANs, in reality Orbit 1 of the TONIC and DOMINANT),
clearly chords of I and V.
We would have achieved a very similar sound by removing the Alto voice
from the original musical example of the Voice-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 substitutions 1(3)1(3) and 0(3) as well as Orbit 3 itself on the same chord, 1(3)-0(3)-3.
They will all have the same Orbit 3 color to remind us of what is "underneath".
-Note that the use of the substitution 1(3) at the beginning of the bar
enables us to resolve Orbit 2 properly (2-1).
This example might also strike a bell (Bell-3)
- otherwise, wait to see (and hear) it dominantized !
NOTE that partial substitution 1(3)-3 was used in the swings of the major mode,
on the DOMINANT and COUNTER chords.
In the fundamental MAJOR mode, key of C, it is interesting to note that,
in the four Tetrads G7, C6, F6, and C+7 we have every Triad available.
In the G7 Tetrad, we have
- the Real Triad of V (G, B, D) and the Deceptive Triad of VII (B, D, F).
In the C6 Tetrad, we have
- the Real Triad of I (C, E, G) and the Deceptive Triad of VI (A, C, E).
In the F6 Tetrad, we have
- the Real Triad of IV (F, A, C) and the Deceptive Triad of II (D, F, A).
In the C+7 Tetrad, we have
- the Real Triad of I (C, E, G) and the Deceptive Triad of III (E, G, B).
-This musical example would normally be analysed as the Triads of I - III - IV - II6- I.
The Triad of I is a Real Triad, part of C+7, but
the Triad of III is a Deceptive Triad, also part of C+7, in which we have -
the MOTRIX B descending to the MEDIAN A, Orbits 3-2,
the MEDIAN E ascending to the PROPER TONE F, Orbits 2-1,
the "doubling" of the PROPER TONE G, Orbit 1, the result of the substitution 1(0).
Displacements on the same chord
From the Triad of I to the Triad of III,
note the displacement in the 3 lower voices, of Orbits 0, 1, and 2.
Don't forget substitution 1(0) in the Tenor voice.
The Triad of IV is a Real Triad, part of F6, but
the Triad of II6 is a Deceptive Triad, also part of F6, in which we have -
the MOTRIX D ascending to the MEDIAN E, Orbits 3-2,
the MEDIAN A descending to the PROPER TONE G, Orbits 2-1,
the "doubling" of the PROPER TONE F, Orbit 1, the result of the substitution 1(0),
including its role (as root of the chord) in the Fundamental Bass at the end.
We have here a C+7 - F6 - C progression expressed only in Triads, with -
the Real Triad (on the Beat), followed by
the Deceptive Triad (on the Off-beat).
These triads come directly from the Basic Tetrad version
with Voice-leading B on the first chord.
The Tenor voice has been transported from the Orbit 0 (0-0) line to the Orbit 1 (1-1) line
by the two substitutions 1(0) on the Deceptive Triads.
With this transformation of Incompleteness,
we now have a complete picture of the possibilities of Voice-Leading.
In the Structure of the Major Mode, we found the "normal resolutions" -
Orbit 3 resolves to Orbit 2, (3-2), and
Orbit 2 resolves to Orbit 1, (2-1),
Orbit 1 and Orbit 0 move freely to each other -
0-0 and 1-1 in the orbit lines,
0-1 and 1-0 in the Fundamental Bass,
depending on the direction of the chord progression.
In Voice-leading, we found that -
Orbit 0, Orbit 1, and Orbit 2 can procure Orbit 3 (0-3, 1-3, 2-3).
Now, in Incompleteness, we found -
the Substitutions of Orbit 3 which gave us 0(3)-2 and 1(3)-2.
Orbit 3 can only resolve to Orbit 2.
Orbit 2 can
resolve to Orbit 1, (2-1) or
procure Orbit 3, (2-3).
Orbit 1 can go anywhere (1-3, 1-2, 1-1, 1-0).
Orbit 0 can go anywhere (0-3, 0-2, 0-1, 0-0).