Disposition of the 4 Strong Modes

The best disposition seems that of the Platonic regular tetrahedron,
          an equilateral triangular-based pyramid, seen from above,
     with the Flattening Diatonic Major (+++, indicatedM) centrally placed at the apex,
     and with the three other modes (+--), placed at the base around it,
          the Sharpening Diatonic Minor (indicatedw) top-left,
          the Sharpening Chromatic Major (indicatedW) top-right,
          the Flattening Chromatic Minor (indicatedm) below.

Disposition of the Parameters

TheWindow Parameteris placed on the ridges :
     between the twoDiatonic Modeswith its+line, and
     between the twoChromatic Modeswith its-line ;
TheMode Parameteris placed on the ridges :
     between the twoMajor Modeswith its+line, and
     between the twoMinor Modeswith its-line ;
TheDirection Parameteris placed on the ridges :
     between the twoFlattening Modeswith its+line, and
     between the twoSharpening Modeswith its-line .

Choice of Transformation

The only way to transform the Primary+++Mode into the three other Secondary+--Modes,
          is that of symmetrical inversion, with a choice of two possible levels of inversion.
     1. The large, global, tonal, level of the complete Window, theFull-inversion,
               along the+ridge of theWindow Parameter,
,          transforms the Flattening Diatonic Major (C) into the Sharpening Diatonic Minor (Am),
               with the note correspondents D-D, G-A, C-E, F-B.
     2. The small, nucleus, modal, level of a half-window, theHalf-inversion,
               along the+ridge of theMode Parameter,
,          transforms the Flattening Diatonic Major (C) into the Sharpening Chromatic Major (C),
               with the note correspondents E-E, D-F, C-G, B-A(b), A-B(b).
          NOTE the flats in parentheses (b),
               which only apply to the TONIC/DOMINANT Swing of the Chromatic Mode (Fm⁶/C⁷),
                    and not to the TONIC/COUNTER Swing (G⁷/C⁶).
     3. The application of both inversions (full and half in either order,+ -, or+ -)
          transforms the Flattening Diatonic Major (C) into the Flattening Chromatic Minor (Am),
               which would have been unreliable with any kind of direct parallel transformation,
                    especially if one relied on scales, instead of swings and circles.

Observations and Conclusions

Metamorphoses
There is a striking similarity between this Inversion Generation and Metamorphoses 4 and 1 -
     1.Metamorphosis 4changes a chord ofC⁶ into a chord ofAm⁷ ,
               changing both the Mode (from major to minor)
               and the direction (from sharpening to flattening);
          theFull-inversiongenerates the Diatonic Minor Mode from the original Diatonic Major Mode,
                    bringing about a TONIC chord ofAm⁷ from a TONIC chord ofC⁶ ,
               thus producing a change of Mode (from major to minor)
               and a change of Direction of the mode (from flattening to sharpening),
               but leaving the Window unchanged (still Diatonic).
     2.Metamorphosis 1changes a chord ofC⁶ into a chord ofC⁷ ,
               changing both the Window (from Diatonic to Chromatic)
               and the direction (from sharpening to flattening);
          theHalf-inversiongenerates the Chromatic Major Mode from the original Diatonic Major Mode,
                    bringing about a TONIC chord ofC⁷ from a TONIC chord ofC⁶ ,
               thus producing a change of Window (from Diatonic to Chromatic)
               and a change of Direction of the mode (from flattening to sharpening),
               but leaving the Mode unchanged (still Major).

Direction
In this process of mode generation,
     theFull-inversionchanges theMode(from Major to Minor)
     theHalf-inversionchanges theWindow(from Diatonic to Chromatic)
     both inversions change the Direction.
The parameter ofDirectionis thus a product, a result of the other parameters,
          and not a generative parameter like the two others (ModeandWindow).
               Its 2 ridges (++++and- - - -) DO NOT appear in the generation diagram.
     There seems to be no basic, generative relationship between modes of the same direction -
          The Diatonic Minor Mode and the Chromatic Major Mode are in the same direction
               merely because they are each a generation away from the original Diatonic Major.
          The Diatonic Major Mode and the Chromatic Minor Mode are in the same direction
               merely because the latter is two generations away from the original Diatonic Major.
It is high time to dismiss the myth of the Chromatic Minor being a parallel form of the Diatonic Major.

Alternative Disposition of the Four Strong Modes
1. The Diatonic Major Mode
2. The Diatonic Minor Mode
3. The Chromatic Major Mode
4. The Chromatic Minor Mode
5. Adjacent Chromatic Modes
6. A melodic Example