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, the Performance Dispositions of individual chords and complete progressions.
1. Individual Chords
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 as MOTRIX, or
in a sharpening direction with a sixth as MOTRIX,
we will have 14 diatonic Tetrads in all.
A Dominant chord can be defined in 3 different ways -
1. As having the distance of an augmented fourth between the Secondary Notes (MOTRIX and MEDIAN),
which is simple and accessible to beginners but not as precise as we would like,
because that definition also applies to the Blues Chord which is not a Dominant chord.
2. As having both an active MOTRIX and an active MEDIAN,
an active MOTRIX being either a minor seventh or a major sixth,
at a distance of a whole tone from the COMMON TONE, and
an active MEDIAN being 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 inactive MEDIAN.
3. As being concentrically constructed around the COMMON TONE -
the PROPER TONE at a distance of a perfect fifth,
the MEDIAN at a distance of a major third, and
the MOTRIX at 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, G7 and 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.
An "Inward" chord can be defined as one directed toward the center of the Window.
These chords all have an inactive MEDIAN and an active MOTRIX.
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, see Metamorphosis 4).
Inward chords (with their inactive MEDIAN) will produce a lower level of movement
than that of Dominant chords (with their active MEDIAN),
which can evidently be raised by chromaticizing the MEDIAN,
and thus Dominantizing the whole chord,
as we will soon see.
An "Outward" chord can be defined as one directed toward the edge (exterior) of the Window.
These chords all have an inactive MOTRIX and an active MEDIAN.
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 very unusual generation,
and the C+7 / F progression is frequently Dominantized to avoid the Major Seventh chord.
Outward chords (with their inactive MOTRIX and their unusual generation)
might produce an even lower level of movement
than that of Inward chords (despite their inactive MEDIAN).
The movement level of Outward chords can evidently be raised by chromaticizing the MOTRIX,
and thus Dominantizing the whole chord, as was the case for Inward chords.
There are no chords with both an inactive MOTRIX and an inactive MEDIAN.
The BUCKLE can be defined as the link between the Outward and Inward chords.
Its MEDIAN is neither major nor minor,
being in Trunk Tuning and not in Short Branch Tuning,
as are all other self-respecting MEDIANs.
There are 2 versions of the BUCKLE, one in each direction, Bm7-5 and Bm-6-5,
the Bm7-5 having the same sound and the same notes as the DOMINANT Dm6, and
the Bm-6-5 having 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 satisfactory On BUCKLE 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 the Modulations Chapter.
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
1. Substitutions 1(3) and 0(3), which remove the MOTRIX of the Tetrad, leaving the Real Triad,
evidently reduce the movement of the chord.
2. Substitutions 1(0), which remove the COMMON TONE of 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 the MOTRIX),
and by the Off Real Triad (without the MOTRIX).
Dominantization brings all chords to the Dominant level of movement.
It is not easy to evaluate the Orbit 4 transformation on isolated chords,
because it imposes a MOTRIX on the following chord, raising its kinetic level.
Thus if we have an Orbit 4 transformation on an Off Dominant chord,
the MOTRIX of 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.
Augmentation applies 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.
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 highlighting rebounds,
and are advantageously placed on the Kinetic beginning of the rebound,
as we will see in Melo-lines.
2. The 4 Strong Modes
Traveling between the modes
When we combine the 4 strong modes in one 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 G7 placed in cA and cC,
the weaker sharpening DOMINANT Fm6 placed aA 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 in aC, cC,
the more complex resonance (10-12-15) of the TONIC C minor triad placed in aA, 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 Fm6 as the DOMINANT of a sharpening mode,
but rather as the COUNTER of a flattening mode, producing the chord patterns m64 and M64,
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 the non-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 in Melo-lines.
In the Flattening modes
We have here the 2 Fundamental Progressions, G7 / C and E7 / Am -
The Off DOMINANTs (G7 and E7) 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 weak MEDIAN in its direction back to the DOMINANT,
whereas the A minor Triad has a strong MEDIAN,
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 / Am and Fm6 / C -
The Off DOMINANTs (Dm6 and Fm6) 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 weak MEDIAN in its direction back to the DOMINANT,
whereas the C major Triad has a strong MEDIAN, 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 G7 has more movement than the Off inward-shape F6,
2. the Off Flattening G7 has more movement than the Off Sharpening F6,
3. the On C is more static with a weak MEDIAN back to the G7,
less static with a strong MEDIAN back 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 Fm6 is also a dominant-shape chord -
1. the Off Flattening G7 has more movement than the Off Sharpening Fm6 but
2. the On C minor is more static with a weak MEDIAN back to the Fm6,
less static with a strong MEDIAN back 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 Fm6 to rival the G7 within this Minor Mode.
Nevertheless, there are Chord Patterns like m4464, in standards like Dark Eyes and La Cumparsita,
which are not normally found in their M4464 major state.
In The Sharpening Diatonic Minor Mode
The situation actually becomes precarious when comparing Fm6 / Cm and Gm7 / Cm -
1. the Off dominant-shape Fm6 has more movement than the Off inward-shape Gm7, and
2. the On Cm is more static with a weak MEDIAN back to the Fm6,
less static with a strong MEDIAN back to the Gm7, but
3. the Off Flattening Gm7 has 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 Line 1-1
D-C, on the Gm7 / Cm cadence, rather than F-G, on the Fm6 / Cm cadence,
which we will see in Melo-lines,
where Tasting Polls will 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 on flattening weak modes more 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 G7 is also a dominant-shape chord -
1. the Off Flattening G7 has more movement than the Off Sharpening Fm6 and
2. the On C is more static with a weak MEDIAN back to the G7,
less static with a strong MEDIAN back 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 above in 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 our Tasting Polls which 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 the Orbit 1 to Orbit 1 line,
possibly the Orbit 2 to Orbit 1 line, and
2. the Bass voice should have the Fundamental Bass,
the root of each chord.
For the Quantification of Melody