Harmony/Transformations/Non-chordal Tones

Let us first define what is meant by a Chordal Tone and a Non-chordal Tone
     and how they differ from each other.
A Chordal Tone is an integral part of the chord
     and cannot be resolved independantly of the chord.
A Non-chordal Tone is not an integral part of the chord
     and can be resolved independantly (ahead) of the chord.

Try this!
As an example, let's look at G7 and G9. (See Chord Symbols)
     on the chord of G7 one cannot resolve the F
          before the resolution of the chord, but
     on the chord of G9 one can resolve the A
          before the resolution of the chord.

The ear tells us immediately!
The F is a Chordal Tone and the A is a Non-chordal Tone.

4 Chordal Tones
This seems a convincing proof that there are 4 Chordal Tones (no more, no less)
     the COMMON TONE,
     the PROPER TONE,
     the MEDIAN, and
     the MOTRIX,
          Orbits 0, 1, 2, and 3. (See THE USE OF COLOR)

All possibilities
Wanting to examine all possibilities of Non-chordal Tones,
     we start from the premise that any orbit (0, 1, 2, 3)
     may be replaced by an adjacent note (above or below)
          which is not a member of the chord.

A. This replacement may be temporary or permanent


If the replacement is temporary, the duration of the chord is divided in 2 (usually equal) parts
     and the Non-chordal Tone is placed on the first part
          (contrary to Chromaticism which is always placed on the last part).

All these examples are on the fundamental DOMINANT-TONIC progression in C major.

No Non-chordal Tones
The progression without Non-chordal Tones.

On Orbit 0
A temporary Non-chordal Tone A on Orbit 0
     (indicated "s2" in the symbol suffix) commonly called a ninth, indicated "9"
     - see Chord Symbols

On Orbit 1
A temporary Non-chordal Tone E on Orbit 1
     (s6), commonly called a thirteenth, indicated "13".

On Orbit 2
A temporary Non-chordal Tone C on Orbit 2
     (s4) commonly called an eleventh, indicated "11", or a suspended fourth.

Some Non-chordal Tones on Orbit 3, and on Orbit 0,
are not advisable. (See C. Normal-Abnormal)


If the replacement is permanent, the Orbit involved must be sufficiently stable.

On Orbit 0
- A permanent Non-chordal Tone A on Orbit 0 (s2)
- A permanent chromatic Non-chordal Tone Ab on Orbit 0 (s-2)
     giving the chord a diminished seventh look/sound.

On Orbit 1
- A permanent Non-chordal Tone E on Orbit 1 (s6)
- A permanent chromatic Non-chordal Tone Eb on Orbit 1 (s-6)
- The same, with substitution of the Orbit 3 F by Orbit 0 G (in the Alto)
     giving the chord an augmented triad look/sound.

B. This replacement may be superior or inferior

If the Non-chordal Tone is superior, it is placed above the Chordal Tone and resolves down to it.
If the Non-chordal Tone is inferior, it is placed below the Cordal Tone and resolves up to it.


If the replacement is superior (s),
     it is usually diatonic (a member of the diatonic system, inside the Window),
          although chromatic superior Non-chordal Tones are possible (See H2605, H2606)

All the examples of section A used superior diatonic Non-chordal Tones (s).

See Chord Symbols for definitions of "s" (sus or suspended) and "p" (push).


If the replacement is inferior (p),
     it is usually chromatic (outside the Window and closer to the Chordal Tone),
          although diatonic inferior Non-chordal Tones are possible.

The major diatonic mode is essentially a flattening mode,
in which the progressions and the MOTRIX are predominantly descending.
Descending (superior) Non-chordal Tones, in the direction of the mode,
resolve far more easily
than ascending (inferior) Non-chordalTones
which must "push againt the current" to rise to their resolution.
Chromaticism helps the latter considerably.
This is why the term "push" for ascending (inferior) Non-chordal Tones
seems so appropriate.

On Orbit 0
- A chromatic push7 F# on Orbit 0 (p+7)
     (See Abnormal)

On Orbit 1
- A chromatic push4 C# on Orbit 1 (p+4)

On Orbit 2
- A chromatic push2 A# on Orbit 2 (p+2)

On Orbit 3
- A diatonic push6 E on Orbit 3 (p6)
     which is only a semi-tone away from its resolution
          (as a chromatic Non-chordal Tone would be).

C. This replacement may be normal or abnormal

If the replacement is normal, it has its own space beside the Chordal Tone.
If the replacement is abnormal, it does not have its own space and must encroach on another Chordal Tone.


Most of the examples of sections A and B were normal.

Orbit 1 and Orbit 2 have space on both sides,
     sus (s) and push (p), for normal Non-chordal Tones.
Orbit 0 and Orbit 3 are, by definition, diatonic neighbors.
     Neither one has the space for a Non-chordal Tone on the side of the other.


These abnormal Non-chordal Tones of Orbit 0 and Orbit 3 are,
     when diatonic, easily mistaken for the other orbit; but
     when chromatic, more evidently identified.

On Orbit 0
- The diatonic push7 F of Orbit 0 (p7)
     can be mistaken for Orbit 3 F.
- The chromatic push7 F# of Orbit 0 (p+7)
     is more discordant (the F# againt the MOTRIX F) but clearer.
          (See Inferior)

On Orbit 3
- The diatonic sus1 G of Orbit 3 (s1)
     can be mistaken for Orbit 0 G
          and is usually considered a substitution of incompleteness 0(3).
- If we add a sus2 A of Orbit 0, (s2), especially the chromatic Ab, (s-2)
          to the previous example,
     we no longer have the impression of a substitution 0(3)
          because the A(b) gives the G a discordant "Non-chordal Tone sound".
- The chromatic sus1 Gb of Orbit 3 (s-1)
     is seldom used, but it is certainly clearer.

D. This replacement may be single or double

If the replacement is single, it will be distanced a second from the Chordal Tone.
If the replacement is double, it will be distanced a third from the Chordal Tone.


All the examples of sections A, B, and C used single Non-chordal Tones (s) or (p).


On Orbit 0
- The double diatonic sus3 B of Orbit 0 (ss3)
     can be mistaken for Orbit 2 B.
          It may be permanent or temporary, resolving to the diatonic sus2 A (s2).
- The double chromatic sus3 Bb of Orbit 0 (ss-3)
     is more discordant (the Bb againt the B) but clearer.
          It may also be permanent or temporary.
- If we apply the chromatic process of augmentation to the DOMINANT chord,
     replacing the F by the D#, (5+5)
     at the same time as the permanent double chromatic sus3 Bb of Orbit 0 (ss-3),
          we have a chord of G7 composed of 2 diminished octaves (B-Bb and D#-D).
(The really daring might wish to see H2621, H2622 and H2622b)

The possibilities of combinations are almost infinite

and we will limit ourselves to a few interesting examples on existing material.

Swing A without Orbit 2
- Let's go back to the Voice-leading A swing,
- and add a diatonic Non-chordal Tone of Orbit 2 to each chord:
     on the TONIC Chord it will be a sus4 F (s4) and
     on the DOMINANTChord, it will be a push2 A (p2).
This gives us a kind of oriental, pentatonic sound,
     with only fourth, fifths, seconds, and sevenths, no thirds.
Notice that the notes F, G, and A are present at all times
     and that the absence of MEDIAN gives us a music which is sexless
          - neither major nor minor.

Sing this in the key of G.

Playing with the Median
While we are on the subject of Non-chordal Tones of Orbit 2,
     how about this one -

On the descending TONIC triad (7), we have 2 lines:
one using the FRAME of the chord
     - Orbit 0 C, Orbit 1 G, Orbit 0 C (2 octaves higher), and Orbit 1 G, and
another using Orbit 2 with its Non-chordal Tones
     - chromatic push2 D# (p+2), Orbit 2 E,
     - (diatonic) sus4 F (s4), and Orbit 2 E.

This might ring a bell (Bell-5) for pianists with eclectic tastes (and large hands). Needless to add that it sounds great! (the original is a half-tone higher)

Dominantized A Swing
Back once again to the Voice-leading A swing,
     where we will Dominantize the TONIC Chord (Cm6),

Incompleteness, Non-chordal Tone of Orbit 0
- then apply Incompleteness 0(3) to both chords (6, 7),
- and add the (normal) chromatic Non-chordal Tone of Orbit 0 to both chords:
     - push4 F# on the dominantized TONIC Chord (p+4) and
     - sus2 Ab on the DOMINANT Chord (S-2).
This transforms the Incompleteness 0(3) into a Non-chordal Tone of Orbit 3,
     - push5 G on the TONIC Chord (p5) and
     - sus1 G on the DOMINANT Chord (s1).

Starting each voice on what seems to be the COMMON TONE G,
     (but is in reality push5 or sus1)
gives a very satisfactory natural canon.

Sing it in the key of G, starting on the note D.

More s-2, s1 Sound
Now that we have succeeded in inserting this "s-2, s1" sound in a swing,
     let's place it in a half-circle, starting on the ANTE-3.

Note the Voice-leading in these dominatized chords
- The sus1 E of Orbit 3 (s1) resolves to Orbit 2 C#.
- Orbit 2 G# resolves to Orbit 10 A (Metamorphosis 1),
     which is the sus1 A of Orbit 3 (s1),
          therefore we have Voice-leading C, Orbits 2-3.
- Orbit 1 B resolves to the sus-2 Bb of Orbit 10 (s-2), Orbits 1-1.
- The sus-2 F of Orbit 0 (s-2) resolves to Orbit 0 s1 E, .Orbits 0-0.
At first sight, it will seem complicated to follow -
     the basic Voice-leading C (Orbits 3-2, 2-3, 1-1, 0-0),
     Metamorphosis 1 imposed by the circle (Orbit exchanges 10, 01) and
     not 1, but 2 Non-chordal Tones (sus1 and sus-2),
          all at the same time!
But this seems to be the only reliable way to control each detail.

New ending
It is possible to end this more smoothly,
     by placing a discord on the TONIC Chord (another sus-2 and sus1) and
     by adding a little coda (a COUNTER/TONIC "Amen").
The chord Fmp+4p5 (the inversion of s-2s1) has a chromatic push4 B and a (diatonic) push5 C.
     You should be able to work out the Voice-leading from Cs-2s1 to  Fmp+4p5.
Some of the harshness of the sus-2, sus1 (push+4push5) sound can be reduced
     by keeping the sus-2 (and push+4) permanent
     but by making the sus1 (and push5) temporary. (Try this!)

This example is not that difficult to sing. Try it.

Other forms of 's-2, s1' sound
This "s-2s1" sound, composed of a minor third, a diminished fifth and a major seventh,
     can also be achieved by the use of other Non-chordal Tones.

On the same chord
In these three bars of DOMINANT G7, we have
- in the first bar, the double chromatic sus3 Bb of Orbit 0 (ss-3)
     which resolves to the (single) chromatic sus2 Ab of Orbit 0 (s-2), then,
- in the second bar, the sus1 G of Orbit 3 (s1)
     which resolves to Orbit 3 F, and finally,
- in the third bar, the sus6 E of Orbit 1 (s6)
     which resolves to Orbit 1 D.
The chromatic sus2 Ab of Orbit 0 (s-2) is always present.

This one might ring a bell (Bell-4) for 4-Hand aficionados.

Note that we have the same sound in the right hand for each of the 3 bars,
even if they have completely different meaning and construction.
This constant sound gives this passage great smootheness and unity.

Non-chordal Tones on each note
In the preceding example, we ended up with a descending scale in the top voice, which alternated Non-chordal Tones with their resolutions. It is also possible to have a descending scale with Non-chordal Tones on each note, by placing them in other voices when the top voice resolves.

We have here a TONIC anacrusis to facilitate the entrance into the DOMINANT G7.
Then, on the DOMINANT G7,
     with the chromatic sus2 Ab of Orbit 0 (s-2), solidily entrenched in the Alto,
- the Soprano has the sus1 G of Orbit 3 (s1).
Then, as the Soprano resolves to Orbit 3 F,
- the Alto takes the double chromatic sus3 Bb of Orbit 0 (ss-3).
Then, as the Alto resolves to the (single) chromatic sus2 Ab of Orbit 0 (s-2),
- the Soprano has the sus6 E of Orbit 1 (s6),
     (an exchange of notes D and F is made with the Tenor).
Then, as the Soprano resolves to Orbit 1 D,
- the Tenor has the sus1 G of Orbit 3 (s1).
- The Bass remains on Orbit 2 B throughout.
Finally, all resolve normally to the TONIC Chord of rest.

(Sing this in the key of G.)


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More diminished octaves
We have already seen a very curious and dissonant DOMINANT G7 in which we had 2 diminished octaves - B-Bb, between Bass and Tenor and D#-D, between Alto and Soprano.

2 on 1 chord
It is possible to place another set of 2 diminished octaves,
     (C#-C, between Soprano and Bass and F#-F between Tenor and Alto)
     at the very beginning, before the other set.
Let's examine the lines individually -
     In the Soprano, push+4 C# - Orbit 1 D / Orbit 1 C. See H2608
      In the Alto, Orbit 3 F - augmented D# / Orbit 2 E. See H2541
     In the Tenor, push+7 F# - double sus-3 Bb / Orbit 0 G. See H2611and H2613
     In the Bass, sus4 C - Orbit 2 B / Orbit 1 C. See H2604
We have already seen all this. Individually.

One can reduce some of the harshness of these diminished octaves, by placing temporary, instead of premanent, Non-chordal Tones, and by placing a little discord on the TONIC Chord to end more smoothly. But where (in which voice, on which Orbit)?

A smoother version
The Tenor is the only voice with 2 Non-chordal Tones,
     and we can improve the melodic line while we are at it,
          by resolving the push+7 F# to Orbit 0 G and
          the double sus-3 Bb to sus-2 Ab.
On the TONIC Chord, we can maintain the sus2 D in the Soprano,
     and the push+2 D# in the Alto (which produce a pleasant "Cm9 sound")
     and resolve them one at a time.
A very “Frenchy” solution!

When you sing this, start on the last TONIC Chord, third beat.

Circle with diminished octaves
What happened on the DOMINANT Chord of G7 in H2622,
     can also happen on the ANTE-3, ANTE-2, and ANTE-1 if they are dominantized.
The 3 first chords of the circle (C, F, and B) can be used to start the natural canon,
     with the diminished octaves (A#-A and F#-F) starting on the chord of B.
We are not sure if the chord of B if major or minor because it does not have a MEDIAN.
Note that the disposition has been changed from that of the preceding example.

The Last Straw
For quite a while, we were convinced that we had reached the ultimate in diminished octaves.
Then we realized that a third set of diminished octaves could be inserted before the existing two,
     with 3 sonorites (of 2 diminished octaves each) for each chordal function.
          (It is almost impossible to use the word "chord" any more.)
Each Chordal Function (bar) is composed of the following sonorities
     - s2p+2s-6p6
     - 7p+7s4p+4
     - 5+5ss-3
          We now have 27 diminished octaves in a row !
This example could also have the "smoother" version sound

This seems to be about as far as we should go
in our search for ever more complex and discordant harmonic possibilities.
We would not like to be prosecuted
for corrupting the musical purity of the youth of the nation.