We will divide this section into 5 parts.
     1. The examination of individual chords,
               first in their basic state, then with each of the possible forms of transformation,
          evaluating and comparing the degree of movement for each possibility.
     2. The examination of the 4 Strong Modes,
          evaluating and comparing the degree of closing for each one.
     3. The examination of individual progressions within each of the 4 Strong Modes,
          evaluating and comparing the degree of closing for each one.
     4. An over-all view of the Weak Modes,
          evaluating and comparing the degree of closing with some of the Strong Modes.
     5. Finally, thePerformance Dispositionsof individual chords and complete progressions.

1. Individual Chords

Basic State

One of the most fruitful aspects of Diatonicism (within the Window)
     is the variety of chords produced, both in shape and in sound.
We will examine them here and start evaluating their role in producing more or less movement.
There are 7 chords in the Window,
          if one includes the BUCKLE chord which ties one end of the Window to the other.
     Since each chord can move in either direction,
               in a flattening direction with a seventh asMOTRIX, or
               in a sharpening direction with a sixth asMOTRIX,
          we will have 14 diatonic Tetrads in all.

Dominants

A Dominant chord can be defined in 3 different ways -
     1. As having the distance of an augmented fourth between the Secondary Notes (MOTRIXandMEDIAN),
          which is simple and accessible to beginners but not as precise as we would like,
               because that definition also applies to theBlues Chordwhich is not a Dominant chord.
     2. As having both an activeMOTRIXand an activeMEDIAN,
          an activeMOTRIXbeing either a minor seventh or a major sixth,
               at a distance of a whole tone from theCOMMON TONE, and
          an activeMEDIANbeing major on a seventh chord and minor on a sixth chord.
               This definition is more precise because it excludes the Blues Chord
                    which has an inactiveMEDIAN.
     3. As being concentrically constructed around theCOMMON TONE-
          thePROPER TONEat a distance of a perfect fifth,
          theMEDIANat a distance of a major third, and
          theMOTRIXat a distance of 2 perfect fifths (a whole tone).
               This definition is the most technical but not more precise than the second.
There are only 2 Dominant chords in the diatonic Window, G⁷and Dm⁶,
     which should not surprise us because there is only one augmented fourth (F-B)
          and it is used for the Secondary Notes in both of these chords.
Dominant chords will produce what we believe to be the highest level of movement.

Inward

An "Inward" chord can be defined as one directed toward the center of the Window.
     These chords all have an inactiveMEDIANand an activeMOTRIX.
There are 6 Inward chords in the diatonic Window, F⁶, C⁶, G⁶, and Em⁷, Am⁷, Dm⁷,
     which include the 3 ANTEs, the COUNTER, and the TONIC directed toward the DOMINANT,
          in the 2 diatonic modes, the Flattening Diatonic Major and the Sharpening Diatonic Minor.
     They all have the same sound and even possess the same notes
          ( F⁶-Dm⁷, C⁶-Am⁷, G⁶-Em⁷, seeMetamorphosis 4).
Inward chords (with their inactiveMEDIAN) will produce a lower level of movement
          than that of Dominant chords (with their activeMEDIAN),
     which can evidently be raised by chromaticizing theMEDIAN,
          and thusDominantizingthe whole chord,
               as we willsoon see.

Outward

An "Outward" chord can be defined as one directed toward the edge (exterior) of the Window.
     These chords all have an inactiveMOTRIXand an activeMEDIAN.
There are 4 Outward chords in the diatonic Window, C⁺⁷, F⁺⁷, and Am^-6, Em^-6,
     and include the 2 first chords of the Circle in the 2 diatonic modes,
          the Flattening Diatonic Major and the Sharpening Diatonic Minor.
     They all have the same sound and even possess the same notes (F⁺⁷-Am^-6, C⁺⁷-Em^-6).
These Major Seventh (and Minor Flat Sixth) chords have a veryunusual generation,
     and the C⁺⁷/ F progression is frequently Dominantized to avoid the Major Seventh chord.
Outward chords (with their inactiveMOTRIXand their unusual generation)
     might produce an even lower level of movement
          than that of Inward chords (despite their inactiveMEDIAN).
     The movement level of Outward chords can evidently be raised by chromaticizing theMOTRIX,
          and thusDominantizingthe whole chord, as was the case for Inward chords.

There are no chords with both an inactiveMOTRIXand an inactiveMEDIAN.

BUCKLE

TheBUCKLEcan be defined as the link between the Outward and Inward chords.
     ItsMEDIANis neither major nor minor,
          being inTrunk Tuningand not inShort Branch Tuning,
               as are all other self-respectingMEDIANs.
There are 2 versions of the BUCKLE, one in each direction, Bm^7-5and Bm^-6-5,
     the Bm^7-5having the same sound and the same notes as the DOMINANT Dm⁶, and
     the Bm^-6-5having the same sound and the same notes as the DOMINANT G⁷.
It is not easy to estimate the movement level of the BUCKLE -
     Even if the interval of diminished fifth (augmented fourth) is between the Primary Notes
               and not between the Secondary Notes (as it normally should)
          it is nevertheless present and confers appreciable movement to the chord.
     It is tempting to place its movement level
          between that of the DOMINANT and that of the Inward chords.
     It is almost impossible to imagine a satisfactoryOnBUCKLE because it is not sufficiently static.
Since it is neither appropriate as ultimate (TONIC) or penultimate (DOMINANT) chord,
     its veritable function seems to be that of antepenultimate (ANTE-1) chord,
          as eloquently exposed in theModulations Chapter.

Transformations

Intro
Two of the transformations, Voice-leading and Metamorphoses, do not seem particularly appropriate to the evaluation of movement, at least not for the present. We will therefore concentrate on the three others, Incompleteness, Chromaticism, and Non-chordal Tones.

Incompleteness

1. Substitutions1(3)and0(3), which remove theMOTRIXof the Tetrad, leaving the Real Triad,
     evidently reduce the movement of the chord.
2. Substitutions1(0), which remove theCOMMON TONEof the Tetrad, leaving the Deceptive Triad,
     also reduce the movement of the chord but less than in the preceding case.
3. If we wished to test the difference between Off Tetrads, Off Real Triads, and Off Deceptive Triads,
     we would probably see (and hear) that -
          all other performance conditions being equal (same Bass and same Soprano),
               the Off Tetrad would have the highest level of closing,
                    followed by the Off Deceptive Triad (with theMOTRIX),
                         and by the Off Real Triad (without theMOTRIX).

Chromaticism

Dominantizationbrings all chords to the Dominant level of movement.

It is not easy to evaluate theOrbit 4transformation on isolated chords,
          because it imposes aMOTRIXon the following chord, raising its kinetic level.
     Thus if we have an Orbit 4 transformation on an Off Dominant chord,
          theMOTRIXof the following On chord
               will considerably lower the closing level of the whole cell.

Diminution, in its complete form, applies to Inward chords,
     especially those near the edge of the Window (F⁶, C⁶, and Em⁷, Am⁷),
          altering them to F⁺⁶⁺¹, C⁺⁶⁺¹, and Em^-7-5, Am^-7-5,
               appreciably raising their movement level.

Augmentationapplies to Outward chords,
     especially those near the center of the Window (G⁽⁷⁾, C⁽⁷⁾, and Dm⁽⁶⁾, Am⁽⁶⁾),
          altering them to G⁺⁵, C⁺⁵, and Dm^-1, Am^-1,
               probably placing their movement level
                    between that of the complete Tetrad and that of the Real Triad.

Non-chordal Tones

Contrary to Chromaticism which, once applied, is there to stay and cannot be removed,
     Non-chordal Tones are there to be removed even if they are stable enough to stay.
Despite the fact that Non-chordal Tones make a chord more dissonant,
     it seems doubtful that Permanent Non-chordal Tones give a chord more movement.
However, Temporary Non-chordal Tones play an important role in highlightingrebounds,
     and are advantageously placed on the Kinetic beginning of the rebound,
          as we will see inMelo-lines.

2. The 4 Strong Modes

Traveling between the modes

When we combine the 4 strong modes inone of our Round Canons,
          the best possible balance of closing seems to be achieved by the following order -
     Diatonic Minor, Chromatic Minor, Chromatic Major, Diatonic Major.
          The sharpening (ascending) modes alternate with the flattening (descending) modes, with
               the stronger flattening DOMINANT G⁷placed incA and cC,
               the weaker sharpening DOMINANT Fm⁶placedaA and aC.
          The 2 minor modes are placed first, the 2 major modes last, with
               the simpler resonance (3-4-5) of the TONIC C major triad placed inaC, cC,
               the more complex resonance (10-12-15) of the TONIC C minor triad placed inaA, cA.
          The diatonic modes are placed first and last, the chromatic modes in the middle,
               merely the result of the 2 other more pertinent priorities.
When one hears the chord pattern of these 4 progressions, Fm⁶/ Cm, G⁷/ Cm, Fm⁶/ C, G⁷/ C,
     one does not really hear the Fm⁶as the DOMINANT of a sharpening mode,
          but rather as the COUNTER of a flattening mode, producing the chord patternsm64andM64,
               in Flattening Chromatic Minor and Flattening Diatonic Major respectively.
NOTE that the importance attached to the strength of the two Flattening Modes,
     might seem, at first sight, to be in contradiction,
          with thenon-generative nature of the Direction parameter.
     But basic generative priority and power in performance do not always coincide.
          There will certainly be more to come on this point.

3. Progressions Within The 4 Strong Modes

(a) Comparing Fundamental Progressions

The Flattening Diatonic Major mode is evidently (with its+++)in a class by itself, and we must not expect the others to be able to compete with it, either in strength or in popularity. The Flattening Chromatic Minor mode runs a respectable second and these 2 flattening modes have had exclusive control of our theory (if not of our music) for several centuries. The Sharpening Modes are not only weaker but also far less popular. However, we are starting to suspect that there is an appreciable cultural and social bias in this aversion and even disdain for sharpening progressions which should not affect our evaluation of quantification. This will be elaborated inMelo-lines.

In the Flattening modes

We have here the 2 Fundamental Progressions,G⁷ / CandE⁷ / Am-
     The Off DOMINANTs (G⁷ andE⁷ ) are the same in both progressions, but
          the On TONIC C is more static than the On TONIC Am for 2 reasons -
               the C major Triad has a simpler frequency relationship (4-5-6)
                    than the A minor Triad (10-12-15), and
               the C major Triad has a weakMEDIANin its direction back to the DOMINANT,
                    whereas the A minor Triad has a strongMEDIAN,
               both of these factors endowing the G⁷/ C progression with greater closing.
                    One must thus not be surprised by the considerable number of "Picardy thirds"
                         found at the end of pieces in the Flattening Chromatic Minor Mode.

In the Sharpening modes

We have here the 2 Fundamental Progressions,Dm⁶ / AmandFm⁶ / C-
     The Off DOMINANTs (Dm⁶ andFm⁶ ) are the same in both progressions, but
          the On TONIC Am is more static than the On TONIC C because
               the A minor Triad has a weakMEDIANin its direction back to the DOMINANT,
                    whereas the C major Triad has a strongMEDIAN, but
          the On TONIC C is more static than the On TONIC Am because
               the C major Triad has a simpler frequency relationship than the A minor Triad,
          these 2 factors seeming to neutralize each other.
                    "Picardy thirds" are not as effective or as required
                         at the end of pieces in the Sharpening Diatonic Minor Mode.

(b) Comparing Fundamental and COUNTER/TONIC Progressions

In each mode, we will now compare its DOMINANT/TONIC (Fundamental) Progression with its COUNTER/TONIC progression to see the extent to which the Fundamental Progression dominates the other in its degree of closing. We might be in for some surprises.

In The Flattening Diatonic Major Mode

As expected, its Fundamentalal Progression G⁷/ C is considerably more closing
               than its COUNTER/TONIC progression F⁶/ C -
     1. the Off dominant-shape G⁷has more movement than the Off inward-shape F⁶,
     2. the Off Flattening G⁷has more movement than the Off Sharpening F⁶,
     3. the On C is more static with a weakMEDIANback to the G⁷,
               less static with a strongMEDIANback to the F⁶,
          all factors increasing the closing level of the cell G⁷/ C over that of the cell F⁶/ C.

In The Flattening Chromatic Minor Mode

The situation is not as obvious when comparing G⁷/ Cm and Fm⁶/ Cm
               where the Off Fm⁶is also a dominant-shape chord -
     1. the Off Flattening G⁷has more movement than the Off Sharpening Fm⁶but
     2. the On C minor is more static with a weakMEDIANback to the Fm⁶,
               less static with a strongMEDIANback to the G⁷,
          one factor weighing each way but certainly in favor of the Direction of the G⁷.
               Its Sharpening Direction does not allow the Fm⁶to rival the G⁷within this Minor Mode.
          Nevertheless, there areChord Patternslikem4464, in standards likeDark EyesandLa Cumparsita,
               which are not normally found in theirM4464major state.

In The Sharpening Diatonic Minor Mode

The situation actually becomes precarious when comparing Fm⁶/ Cm and Gm⁷/ Cm -
     1. the Off dominant-shape Fm⁶has more movement than the Off inward-shape Gm⁷, and
     2. the On Cm is more static with a weakMEDIANback to the Fm⁶,
               less static with a strongMEDIANback to the Gm⁷, but
     3. the Off Flattening Gm⁷has more movement than the Off Sharpening Fm⁶,
          with Direction competing with both (1.) Off chord shape-strength and (2.) On rest.
Although we would rather rely on scientific and philosophical arguments,
     we will, for the present, refer to historical usage and preference,
          where we find that far, far more Diatonic Minor cadences
               are made with Gm⁷/ Cm (despite its Off Inward chord)
                    than with Fm⁶/ Cm (despite its Off Dominant chord),
          not to mention the preference for the Orbit Line1-1
               D-C, on the Gm⁷/ Cm cadence, rather than F-G, on the Fm⁶/ Cm cadence,
                    which we will see inMelo-lines,
                         whereTasting Pollswill be possible with both the chord progressions and the Orbit Lines.
          Once again the parameter responsible for this preference seems to be that of Direction,
               playing a more important and influential role than we had ever thought possible,
                    possibly confering onflattening weak modesmore closing than on sharpening strong modes.
          In all of this process of evaluation we must keep in mind the role which cultural prejudice might play.

In The Sharpening Chromatic Major Mode

     Precariousness is even greater as we compare Fm⁶/ C and G⁷/ C,
               where the Off G⁷is also a dominant-shape chord -
     1. the Off Flattening G⁷has more movement than the Off Sharpening Fm⁶and
     2. the On C is more static with a weakMEDIANback to the G⁷,
               less static with a strongMEDIANback to the Fm⁶,
          both factors increasing the closing of the whole cell,
               making this COUNTER/TONIC progression (G⁷/ C ) completely overpower
                    the DOMINANT/TONIC progression of the mode (Fm⁶/ C ).

          This mode must evidently be used with the greatest care,
               preferably avoiding the G⁷/ C progression completely,
                    though not necessarily the C⁶/ G progression,
               so as to allow the Fm⁶/ C progression to dominate the entire mode.

4. The Weak Modes

Weak modes are defined as those having a weak Fundamental Progression that does NOT have an Off dominant-shape chord. This Off chord is usually an inward-shape chord (though it may occasionnally be an outward-shape cord). We will, for the present, examine here only the first three Flattening Off Inward progressions.

The Em7 / Am Mode

This is the mode, with Em⁷/Am as Fundamental Progression,
          and Dm⁶/Am as Secondary COUNTER / TONIC Progression,
     which might rival the strong Sharpening Diatonic Minor Mode
          with Dm⁶/Am as Fundamental Progression,
               and Em⁷/Am as Secondary COUNTER / TONIC Progression,
           which we saw abovein the Strong Modes.
     Both of these modes are somewhat precarious and ambiguous,
          because the progressions from both sides of the TONIC are so nearly equal in strength.
     You might enjoy one of ourTasting Pollswhich deals with this very problem.

The Am7 / Dm Mode

This is an interesting minor mode, which is probably more popular than either of the preceding two.
     The problem of ambiguity, with relatively equal cadences from both sides, is solved
          not by strenghtening (dominantizing) the weak side (A⁷/Dm rather than Am⁷/Dm),
               but by weakening the strong side (G⁶/Dm rather than Gm⁶/Dm),
                    thus removing all appreciable competition
                         to the weak flattening progression (Am⁷/Dm).

The Dm7 / G Mode

Like the preceding mode, this one is also weak on each side,
          with flattening Dm⁷/G and sharpening C⁶/G,
     but, this time, with a major TONIC (G)
          instead of a minor TONIC (Dm), as was previously the case.

5. Performance Dispositions

Bass And Soprano
For the strongest possible progression,
     1. the Soprano voice should have theOrbit 1toOrbit 1line,
          possibly theOrbit 2toOrbit 1line, and
     2. the Bass voice should have theFundamentalBass,
          the root of each chord.

For theQuantification of Melody