Preface

Possibilities
There is less choice than one might think in the possibilities of voice-leading:
- Orbit 3 always resolves to Orbit 2 (of the next chord);
- Orbit 2 normally resolves to Orbit 1 (of the next chord).

Resolution
Resolution is, above all, a question of lowering the tension
by one notch (to the next smaller orbit).

- Orbit 1 and Orbit 0 are relatively free to go to each other.

More possibilities
The principal choice in Voice-lerading lies in what we call the "procuration of Orbit 3"
     when we progress from one complete tetrad to another
          and this is exactly the kind of complete harmony in which we will indulge for a while.

There are 3 possibilities of procuring Orbit 3 (the MOTRIX):
- A. from Orbit 0,
     with the voice-leading 0-3, the most natural;
- B. from Orbit 1,
     with the voice-leading 1-3, more difficult to handle;
- C. from Orbit 2,
     with the voice-leading 2-3, the most convenient, interrupting the normal resolution 2-1.

Natural Canons
We will examine these forms of Voice-leading by starting at the Basic Level of Tetrads in the Window and developing a form of "Musical Tapestry", a kind of "Fashion Show" of musical events, in which the music actually composes itself without any human intervention.

This Musical Tapestry might seem a little unusual and in reality it is. We are here in the Window delimiting the 7 diatonic notes, but where there is no RHYTHM, no beginning, no end, no points of rest, no breaths, just continual movement with complete Tetrads; there is no Fundamental Bass, just the 4 lines of Voice-leading.

We call this form of music "Natural Canons" for reasons which will soon be obvious. For now, these tapestries will be in the TONIC-DOMINANT swing of C major but would be the same in any other key or in any other swing.

Voice-leading A - Procuration 0-3

The Procuration 0-3 will produce an orbit chain 0-3-2-1 which can repeat itself indefinitely.

-Note the Orbit colors of the members of each chord:
On the first chord (TONIC - C⁶) :
     Orbit 0 (the COMMON TONE) is G;
     Orbit 1 (the PROPER TONE) is C;
     Orbit 2 (the MEDIAN) is E;
     Orbit 3 (the MOTRIX) is A.
On the second chord (DOMINANT - G⁷) :
     Orbit 0 (the COMMON TONE) is still G;
     Orbit 1 (the PROPER TONE) is now D;
     Orbit 2 (the MEDIAN) is B;
     Orbit 3 (the MOTRIX) is F.
The third chord is like the first and the fourth chord is like the second.

Orbit Chains
If we verify the orbit colors in each line, voice by voice,
we will find 0-3-2-1 everywhere.
This is why we use the term "Natural Canons".

-If each voice starts on the same orbit (say Orbit 0),
     the canon is much more obvious, and so is the fact that -
          the Sopranos and Tenors have descending lines and
          the Altos and Basses have ascending lines.
With the two chords of the swing operating in opposite directions,
     this should not come as any surprise.
NOTE -
     1. the 2 descending (or ascending) lines should not be placed in adjacent voices
          because they cross, and
     2. a descending line will produce a FRAMEInversion
          if it is placed in the Bass Voice.
     Remaining in this "Primary Disposition" (with an ascending line in the Bass Voice)
               severely limits the possibilities of disposing the Voice-leading A Swing -
          one can only invert the Soprano and Tenor, or the Alto and Bass,
               not both, because that brings us back to the original disposition.
Naming our chord dispositions in ascending order as usual (starting with the Bass voice) -
     the disposition we have here, in the fifth chord, is 3,2,1,0
          (2,1,0,3, 1,0,3,2, and 0,3,2,1, in the following chords, would be considered the same disposition).
     the only other possible disposition is 0,1,2,3
          (3,0,1,2, 2,3,0,1, and 1,2,3,0, would be considered the same disposition).
     3 or 1 of the sharpening (ascending) chord (in this case, C⁶) must be in the Bass voice.

-This one might ring a bell - let's call it Bell-1 for future reference !

For more on this natural canon

Voice-leading B - Procuration 1-3

The procuration 1-3 will produce an orbit chain 1-3-2 which can repeat itself indefinitely.

V-shape
Notice the "V-shape" of this 1-3-2 movement
with Orbit 3 at the apex of the "V".
This bouncing back of Orbit 3 is most important
when one jumps to it (a movement of third or more).

If we wish to complete the harmony, Orbit 0,
          which does not appear in the chain,
     will be held in another voice (the Alto for the best disposition of the voices),
          because the three canonic lines cross.
     There is an unusual spread between the Tenor and Bass voices,
          but that is the best place for it.

-We do not have 2 descending and 2 ascending lines here,
     but rather 3 lines which are each successively ascending and descending.
This is the result of the conflict between the odd number of orbits in the chain
     and the even number of chords in the swing.
          There is no serious objection to the FRAMEinversion in the first chord of the Bass voice -
               the MOTRIX is present (in the Soprano voice), and
                    the COMMON TONE, held as a pedal tone throughout, does not attract attention.
NOTE -
     1. The 3 canonic lines should not, in principle, be placed in adjacent voices
          because they cross, but
               it is possible to spread them between the Tenor ans Bass voices.
     2. The Orbit 0 line should be placed in the Alto Voice
          to avoid crossing the three top voices.
     This severely limits the possibilities of disposing the Voice-leading B Swing -
          one can only invert any 2 of the 3 canonic lines.
Naming our chord dispositions in ascending order as usual (starting with the Bass voice) -
     the disposition we have here, in the first chord, is 1,2,0,3 (simplified 1,2,3)
          (3,1,2 and 2,3,1, in the following chords, would be considered the same disposition),
     the only other possible disposition is 3,2,1
          (2,1,3 and 1-3-2 would be considered the same disposition).
     The 3 canonic lines, being the same, present no problem of choice for the Bass voice.

Canons
Canons can be produced in contrary motion by entering at the distance of 1 chord,
     as we did for Voice-leading A, (starting arbitrarily on Orbit 3).

Canons can also be produced in similar motion by entering at the distance of 2 chords.

Voice-leading C - Procuration 2-3

The procuration 2-3 will produce the orbit chain 2-3 which can repeat itself indefinitely
     (Orbit 3 returning immediately to Orbit 2).
If we wish to complete the harmony, Orbit 1 and Orbit 0,
          which do not appear in the chain,
     can be maintained in the other two voices.
This swing is less interesting than the other two, with far less movement, but has considerable advantages -
     1. The four orbit lines may be present with or without the Fundamental Bass.
     2. Any of the four orbit lines can be placed in the Bass voice.
     3. None of the four orbit lines cross, permitting any choice of disposition.
          Voice-leading C was used exclusively in the examples of the major mode,
               both for the swings and for the circle.

-Who knows, it might strike a bell (Bell-2)
     if presented as a Beat-Off-beat accompaniment, especially if it is dominantized.

Moreover, with various forms of Incompleteness and Non-chordal tones
Voice-leading C has been used extensively in just about all styles of music.