Chordal and linear structures

In thegeneration of triads and scales, 3 diagrams were placed side by side -
1. in the center, the original series of fifths in the diatonic Window (7 notes)
     which acts as generator for both structures (chordal and linear);
2. on the left, where theFRAMEs of fifths are “divided” by theMEDIANs,
     we find the chordal structure of triads
          (where all the notes are heard at the same time); and
3. on the right, the double circle of fifths, where the fifths are “multiplied” by 2,
     we find the linear structure of the scale,
          (where notes are heard one after the other).

With Jordan Triangles, it will be possible
1. to combine both structures (chordal and linear) in one diagram,
2. to extend the possibilities of linear structure “multiplication” from 2 fifths to 3,
     (possibly more) and
3. to observe the “birth” (borrowing) of theMOTRIX(sixth or seventh) more clearly.

Definition of the Jordan Triangle

Orbits 0 and 1
Starting from the central note D, acting as theCOMMON TONE,Orbit 0, and generating, with the interval of the perfect fifth, the notes G and A, produces thePROPER TONEs,Orbit 1, of the 2FRAMEs of the swing D.

In the normal disposition of the Window,
     the 3 notes of this swing are placed vertically,
          in a column.

However, if the 2 extremities of the swing
          (G and A,Orbit 1),
     are placed to one side,
     they may be joined by a vertical green line,
          producing a “Jordan Triangle”.

Chordal and linear structural relationships
Both of these relationships are present in the triangle.
1 - The blue lines D-G and D-A represent
     (a) not only the chordal relationship of the 2FRAMEs of the swing D
          (the pivotalCOMMON TONEgives the swing its name),
     (b) but also the linear Voice-leadings 0-1 and 1-0 (the Fundamental Bass).
2 - The green line G-A represents
     (a) not only the chordal relationship between theCOMMON TONEand theMOTRIX,
     (b) but also the basic Voice-leading 1-1
          (between the two peripheralPROPER TONEs).

Assemblage of Jordan Triangles

Orbits 2 and 3
The 5 triangles of the Window (one for each swing) can be superimposed in a column
     by zig-zagging the notes of the series of fifths (F, C, G, D, A, E, B).
The linear structure of the scale is found in the vertical green lines on each side
     (C, D, E, on the right and F, G, A, B, on the left).
The triangles are separated from each other to clearly differentiateOrbit 0) andOrbit 1).
The sameFRAMEis indicated twice (with 2 parallel blue lines)
     specifying the direction of the chord (the swing to which it belongs)
          withOrbit 0andOrbit 1indicated accordingly.

Insertion of MEDIANS
     in the middle of each (double) blue line
They correspond to the notes of the Window
     but have their own specific tuning (+/-),
     being generated by the natural major third (5/4).

The Jordan Diamonds
Any 2 adjacent triangles form a diamond,
     composed of 2 adjacent swings,
     with the common chord between them.
We will concentrate on the diamonds at the ends.
     (The central swing D,
          which unites the major and minor sections,
     is, most of the time, only a one-way passage.)

The MOTRIX Diagonals
Let us start by examining the 2 red diagonals,
     between F and D, in the bottom of the Window.
1. The note F isOrbit 1of theFRAMECF and
     the note D isOrbit 1of theFRAMEGD.
2. These diagonals run past the note E-,
      Orbit 2of theFRAMEs GC and CG.
3. These diagonals indicate the linear structures
     F E- D and D E- F,
          common to both swing G and swing C.
4. (a) in the swing G, F-E--D represents
          the Orbit line 3-2-1, and
     (b) in the swing C, D-E--F represents
          the Orbit line 3-2-1.
5. TheOrbit 3F of theFRAMEGD
     is borrowed from theOrbit 1F of theFRAMECF.
TheOrbit 3D of theFRAMECF
     is borrowed from theOrbit 1D of theFRAMEGD.
One could even say that theOrbit 3F
          is borrowed from the substitution 1(3)
     and that theOrbit 3D
          is also borrowed from the substitution 1(3).
6. There seems to be a strong symbiosis between the twoMOTRIX)s
     which resolve to theMEDIANE- of the TONIC chords GC or CG.
7. TheMOTRIX, in these F E- D, D E- F diagonals,
          seems to have strong linear heritage,
     but one must not neglect the chordal relationship of 2 fifths (9/8)
          between F and G (on the chord of G⁷) and
          between D and C on the chord of F⁶.
Note that the chordal relationship is betweenOrbit 0andOrbit 3
     and that the linear relationship is betweenOrbit 1andOrbit 1.

Where do MOTRICES come from?
Each chord surrounding the TONIC (DOMINANT and COUNTER)
     borrows itsMOTRIXfrom the substitution of theMOTRIX1(3) of the other chord.
This becomes all the more evident when chromaticism is involved (like the D#--).
The same procedure ofMOTRIXborrowing will apply elsewhere in the Window
     not only in the D-B diagonals, for theMOTRIXof Dm⁶and Em⁷
     but also in the C-A diagonals, for theMOTRIXof C⁶and Dm⁷, and
     in the G-E diagonals, for theMOTRIXof G⁶and Am⁷.

1. Comparing the triangle and the diamond
(a) In the triangle, we have the primary notes -
     oneOrbit 0which is pivotal and
     twoOrbit 1s which are peripheral.
(b) In the diamond, we have the secondary notes -
     oneOrbit 2, in the center, which is generated and
     twoOrbit 3s, at the ends, which are borrowed.

2. Comparing the 2 sets of diagonals
In the F-C-D-G diamond there are
     the C E- G, G E- C diagonals which are chordal, and
     the F E- D, D E- F diagonals which are linear.
Orbit 2(E-) belongs to the 4 diagonals,
     with both linear and chordal structure.

It is interesting to observe Orbit 2 (E-) surrounded by a “court” of four Orbits 1 -
one of which (D or F) will be borrowed to become Orbit 3,
and another (G or C) will be chosen to be Orbit 0,
depending on the swing (C or G) involved at that moment.

The periphery of the Jordan Diamond
This is all the more evident when we examine the complete periphery of the diamond,
     with the MEDIANS added, B- in theFRAMEGD and A- in theFRAMECF
     (see the following illustration in The Four Metamorphoses).
(a) On the upper left side, delimited by the F E- DMOTRIXdiagonal,
     we have the complete DOMINANT chord, D, B-, G, F. (G⁷).
(b) On the lower right side, delimited by the D E- FMOTRIXdiagonal,
     we have the complete COUNTER chord, F, A-, C, D, (F⁶).
The orbits are inversed in perfect symmetry with
      the TONIC triad, on the short diagonals, between its 2 neighboring tetrads.

The Four Metamorphoses

This seems to be the best way we have yet found
          to illustrate Metamorphosis 3, (G⁷/ F⁶) -
      Orbit 1andOrbit 3(D, F/ F, D) exchange notes,
      Orbit 2of each chord (B- / A-) acts as a
          non-chordal tone ofOrbit 0of the other chord (C / G).

The linear resolutions of these non-chordal tones
     could be clearly perceived inMOTRIXdiagonals
          which do not appear in this diagram -
     (a) B- to C in the diagonal; A B- C, and
     (b) A- to G in the diagonal; Bb A- G.

Metamorphosis 3 is the most “compact” of the four,
     occupying only 2 Jordan triangles (1 diamond)
          because both tetrads use the same diagonal.

Metamorphosis 4 (F⁶/ Dm⁷)
     is not as compact as Metamorphosis 3,
          occupying 3 Jordan triangles.

Its chords use 2 sets ofMOTRIXdiagonals -
     F E- D / D E- F, of which we show only
          D E- F for the chord of F⁶(F, A-, C, D) and
     A B- C / C B- A , of which we show only
          C B- A for the chord of Dm⁷(A, F+, D, C).
Orbits 0 and 3 (C, D / D, C) exchange notes,
Orbits 1 and 2 (F, A- / A, F+) also exchange notes,
     but change pitch in the process (F / F+, A- / A).
It is the “spread” over 3 triangles
     that is responsible for this change of pitch,
          a comma higher in the minor section of the Window.

Metamorphosis 1 (G⁶/ G⁷) is even more spread than
     Metamorphosis 4, occupying 4 Jordan triangles.

Its chords use 2 sets ofMOTRIXdiagonals -
     G F+ E / E F+ G, of which we show only
          E F+ G for the chord of G⁶(G, B-, D, E) and
     D E- F / F E- D , of which we show only
          F E- D for the chord of G⁷(D, B-, G, F).
As we are dealing with the same triad, G (major),
     Orbits 0 and 1 (D, G / G, D)
          will evidently exchange notes of theFRAME.
Being theMEDIANof the same triad, G (major),
      Orbit 2(B-) evidently remains in the same voice.
Orbit 3remains in the same voice but changes note (E / F),
     causing a considerable spread for the same triad.

Metamorphosis 2 (G⁷/ Dm⁶)
     is even more spread than Metamorphosis 1,
          occupying 5 Jordan triangles (the entire Window).

Its chords use 2 sets ofMOTRIXdiagonals -
     D E- F / F E- D , of which we show only
          F E- D for the chord of G⁷(D, B-, G, F), and
     D C+ B / B C+ D, of which we show only
          B C+ D for the chord of Dm⁶(D, F+, A, B)
Orbit 1 (D) belongs to both chords,
     and remains in the same voice.
Orbits 2 and 3 (B-, F / F+, B)
     exchange notes but change pitch (B- / B, F / F+).
     Once again, it is this “spread” over 5 triangles
          that is responsible for this change of pitch,
               a comma higher in the minor section of the Window.
Orbit 0 remains in the same voice but changes note (G / A).
     TheOrbit 0of each chord acts as a
          non-chordal tone ofOrbit 0of the other chord.
     The linear relationship of these non-chordal tones
          could be clearly perceived in theirOrbit 1relationship (G-A)
               which does not appear in this diagram.

The Complete Swing

In 1 triangle
We could very well have inserted theMEDIANs in the first illustration and produced a diagram of the swing in one single triangle, but an important element of the swing would have been missing - theMOTRIXs.

In 3 triangles
In order to “borrow” theMOTRIXs from the appropriate diagonals (as we did in the second illustration), we must add another triangle on each side of the triangle of the swing (as we did in the fifth illustration but, this time, showing only the appropriate diagonal in each set.

We have here the complete swing G,
     which consists of the chords
          - C⁶(C, E-, G, A), and
          - G⁷(D, B-, G, F).
TheMOTRIXs are borrowed from the diagonals
     A B- C / C B- A, of which we show only
          A B- C for the chord of C⁶(C, E-, G, A), and
     F E- D / D E- F, of which we show only
          F E- D for the chord of G⁷(D, B-, G, F).
TheMOTRIXs F and A are symmetrically disposed
     around theCOMMON TONEG,
          by the chordal structure of the tone (9 / 8),
          which was originally the linear structure
               between thePROPER TONEs F, G and A, G.

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