Harmony/Structure of Pitch/4 Forms of Harmony/Price of Diatonicism

     We very seldom stop to observe and evaluate the advantages and disadvantages of Diatonicism. It is a thing which we take so much for granted, like gravity, and that we know so little about, that this kind of meditation seems useless or at least premature. However certain elements of Diatonicism can be quite enlightening, and we will attempt to expose some of them.
     With or without Diatonicism, certain glorious elements of Music are often totally disregarded and certainly unappreciated. One of the most important is the presence of preferential distances (in this case, the octave, the perfect fifth, and the major third) which do not exist in other cosmic manifestations. The size of the electron might be the only preferential distance in the physical universe. We are also aware of other preferential sizes in Music like that of the complete chord (at four notes and not three as was long thought to be the case) and the size of a diatonic system (at seven notes, whether in the vertical form of a string of fifths in the Window, or in the horizontal form of a scale). It would be quite simple to elaborate valid and convincing proofs for all of these preferential sizes. For the present it will suffice to examine and appreciate them.
     In Chrominicism and Just Intonation, we examined the nature of individual notes, but we will now examine the nature of complete chords.

The Window

Rhythmic considerations aside, the Window is what differentiates Free Harmony from Melodic Harmony.
     In Free harmony, all chords have the same Dominant shape, while
          in Melodic Harmony, chords in the Window have several different shapes.
     In Free harmony, there is no difference between "diatonic" and "chromatic", while
          in Melodic Harmony, one is inside the Window and the other outside.
     In Free harmony, there is no Tonic preferential position of rest, while
          in Melodic Harmony, there are 2 Tonic preferential positions in the Window.
We will examine these differences more closely.

2 Levels of Direction

How far can you go in the wood ?
Only half-way, after that you're coming out.

The Window level of Direction
Very much the same way that individual notes have 2 levels of Tendency,
          individual Tetrads have 2 levels of Direction -
     1. The larger level of the entire series of fifths,
          in which the MOTRIX choice determines the direction,
               a flattening direction with a seventh as MOTRIX
                    and sharpening direction with a sixth as MOTRIX,
               placed in columns (below).
     2. The smaller level of the Window (which acts as the proverbial wood),
          with Inward chords directed toward the center, and Outward chords directed toward the edge.
               placed in rows (below).
     The BUCKLE chord (both flattening and sharpening) has been placed in the top of the table,
          being, technically, an Inward chord.
     All examples (chord symbols) are in the Natural Window.





2-3-3 Sonority
Heptatonic span

3-3-2 Sonority
Heptatonic span


3-2-3 Sonority
Pentatonic span

3-2-3 Sonority
Pentatonic span


3-3-2 Sonority
Heptatonic span

1-4-3 Sonority
Hexatonic span

2-3-3 Sonority
Heptatonic span

1-4-3 Sonority
Hexatonic span

Diversity of Chord Shape/Sound

In both flattening and sharpening directions , we find 7 chords (the BUCKLE and 6 in the Window).
     In the first row, The BUCKLE chord (both flattening and sharpening, Bm7-5 and Bm-6-5)
          has a Heptatonic span (B-F) between its Primary FRAME Notes
               which destabilizes it completely and prevents it from being a chord of rest.
     In the second row, 3 of the Window chords will resolve Inward, toward the center of the Window,
               with a Pentatonic span between the PROPER TONE and the weak MEDIAN,
          and will have a totally homogeneous 3-2-3 Sonority
               of minor seventh chords (Em7, Am7, Dm7) or sixth chords (F6, C6, G6),
                    depending on the large-level direction (flattening or sharpening).
     In the third row, 3 of the Window chords will resolve Outward, toward the edge of the Window,
               and divide in 2 very distinct categories (a really mixed bag) -
          1. A Dominant-shape chord, the only one left from Free Harmony, is an Outward chord in position only,
               without the slightest intention of ever leaving the Window,
                    with a Heptatonic span (B-F) between its strong Secondary Notes,
               producing the cadential DOMINANT / TONIC progression essential to the Strong Modes,
                    G7 / C in Diatonic Major and Dm6 / Am in Diatonic Minor.
               This powerful, climaxing, cadence brings musical phrases to an end
                    and seems to have considerable influence on the remaining chords of the Window.
               We have left the unidirectional (Circle) terrain of the Tail
                    and are now in the bidirectional (Swing) terrain of the Nucleus.
          2. We find here 2 considerably complex chords,
                    with a Hexatonic span between the COMMON TONE and the weak MOTRIX,
               major seventh chords(C+7, F+7) or minor flat sixth chords (Am-6, Em-6),
                    depending on the large-level direction (flattening or sharpening).

Degrees of Diatonicism

1. The Inward chords, with their pentatonic span, are the most diatonic of all the Window chords,
     never involved with the notes on the edge of the Window (if they are placed in the center, of course).
2. The Outward chords, with their hexatonic span, are less diatonic than the Inward chords,
     using one of the notes on the edge of the Window, either as COMMON TONE or as MOTRIX.
3. The DOMINANT chords with their heptatonic span,
     are even less diatonic than either the Inward or Outward chords,
          using both notes on the edge of the Window, as MEDIAN and as MOTRIX.
4. The BUCKLE chords with their heptatonic span,
     are also less diatonic than either the Inward or Outward chords.

Metamorphosis 4
We will define a Metamorphosis 4 association between 2 chords
     as one in which both chords are composed of the same notes
          with inversion of the Primary and Secondary functions.
This Metamorphosis 4 association exists between -
     the 2 groups of Inward chords (F6Dm7, C6Am7, G6Em7),
     the 2 groups of Outward chords (F+7Am-6, C+7Em-6),
     the DOMINANT and BUCKLE chords (G7Bm-6-5, Dm6Bm7-5).

The homogeneous Inward sound
It would be possible to use the nucleus chords (either Major or Minor) without using the DOMINANT
     and thus maintain a soft homogeneous sound with only weak Inward chords.
We have here, in C major, the chord pattern C6 / G6, Dm7F6 / C twice,
          with a cadential disposition the second time.
     There is a Metamorphosis 1 on the first 2 chords and a Metamorphosis 4 on the third chord.
We have here, in A minor, the chord pattern Am7 / Dm7, G6Em7 / Am twice,
     with a cadential disposition the second time (same Metamorphoses).
Traveling through both modes, we have here the chord pattern Dm7F6 / C6 / G6Em7 / Am7,
          which can be repeated indefinitely.
     There is a Metamorphosis 1 on each chord, followed by a Metamorphosis 4 on the first and third chords.
The same traveling through both modes, but with Diminution applied to the end of each chord
          Dm7F6, F+6+1 / C6, C+6+1 / G6Em7, Em-7-5 / Am7, Am-7-5,
     which can be repeated indefinitely (same Metamorphoses).


The concept of Tonality evidently comes from the presence of the 2 remaining Dominant shape chords.
Fortunately, each DOMINANT resolves to a TONIC which is centrally placed in its own mode,
     with its COUNTER(DOMINANT) on the outside, at the very extremity of the Window,
          and its DOMINANT on the inside, near the center of the Window,
               ready to receive its series of unidirectional ANTEs.

Traveling between modes
Each diatonic mode (Major and Minor) has its Nucleus at one end of the Window, and its Tail at the other.
     The half circle of each mode starts on the TONIC of the other mode -
          the Major half circle (Am7 / Dm7, G7 / C) starts on the Minor TONIC (Am), and
          the Minor half circle (C6 / G6, Dm6 / Am) starts on the Major TONIC (C).
     One could even include a foray into the COUNTER Swing,
               before taking off on the journey to the other mode -
          from Major to Minor (C+7 / F6, C6 / G6, Dm6 / Am), and
          from Minor to Major (Am-6 / Em7, Am7 / Dm7, G7 / C).
In all of this traveling, one could indulge in copious dominantization,
     without shaking the solidity of the Window in the least,
          enjoying the sound of Free Harmony without losing the security of Tonality,
     such as Dominantizing each Off-beat (C7 / F, Cm6 / G, Dm6 / Am, Am6 / Em, A7 / Dm, G7 / C), or
     Dominantizing every chord except the last (C7 / Fm6, Cm6 / Gm6, Dm6 / Am, Am6 / E7, A7 / D7, G7 / C).

What about the price ?
As long as we travel within the Window,
     there is no appreciable price to pay for Tonality,
          and the advantages are considerable.
However, even the innocuous (complete) diatonic circle
     leads us into perilous terrain which we will now explore.

The Buckle Chord

In the complete circle, the link from one end of the Window to the other cannot be other than difficult,
     with the considerable difference of Chrominicism between the 2 ends.
We end up with the distorted, deformed, mis-shapen, hunchbacked Buckle Chord
     whose FRAME consists of
          the sharpest note of the Window (B,) acting as root of the chord, and
          the flattest note of the Window (F) acting as fifth of the chord,
               both in Trunk Tuning.
It is impossible to use the diatonic Buckle Chord as a (temporary) Tonic of any kind,
          but it is possible to use it as a (temporary) Dominant chord (with a flat fifth).

The Neapolitan Sixth

In circles
In the Major circle,
     it will be preferable to use the chord below the Window (Bb),
          when we need a temporary Tonic chord of rest, and
     it will be preferable to use the chord above the Window (B7),
          when we need a temporary Dominant chord of motion.
     NOTE that we are using the very first notes of Primary Chromaticism (Bb and F#)
          for the FRAMEs of these "extension chords"
               which seem to be attached to the Window even if they are outside of it,
                    a little like a balcony or porch is to a house.
Linking this together will bring us to what is traditionally called the "Neapolitan Sixth",
     which is essentially no more than a simple major triad,
          whose distinction lies in the fact that it precedes a Dominant Seventh chord,
               placed a diminished fifth below it (in flattening progressions).
     It would therefore seem preferable to speak of a "Neapolitan Sixth progression"
               which includes the following chord,
          rather than of an isolated "Neapolitan Sixth chord".
A few examples of flattening circles with each Off-beat dominantized -
     1. Traveling from TONIC to TONIC, we would have C7 - F - B7 - Em - A7 - Dm - G7 - C.
     2. Traveling from DOMINANT to DOMINANT, we would have G7 - C - F7 - Bb - E7 - Am - D7 - G.
          NOTE that in both examples, from F to B7, and from Bb to E7,
               we find the Neapolitan Sixth progression, in which there is no COMMON TONE,
               and in which the Voice-leading is specific and delicate.
     3. In the Interlaced Circles, after C - E7 - Am - C7 and F - A7 - Dm - F7,
          we find Bb - D7 - G - B7,
               in which both extension chords Bb (of rest) and B7 (of movement) appear.


In the m74 Chord Pattern (I-IV-V-I)
We have here the generation of the traditional Neapolitan Sixth (chord/progression) -
     (a) first with A - A - G# - A in the Soprano, Orbit chain 0 - 03 - 2 - 1,
          with Metamorphosis 4 on the second chord, and Metamorphosis 1 on the third chord,
     (b) then, A - Bb, A - G# - A, with a temporary chromatic superior Non-chordal Tone on the second A,
     (c) and finally, A - Bb - G# - A, in which the Non-chordal Tone is permanent.
          This (c) is what the traditional Neapolitan Sixth looks/sounds like.
The tuning is a comma higher (Bb+, D, F+) than it was for the extension Bb triad (Bb, D-, F)
     being higher in the Window (in the Minor section).
This process is completely inside the Window and does not cause any kind of problem.

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