Harmony/Transformations/Coda 2

Orbit Line Dispositions

At the end of Harmony/Transformations/Incompleteness, we added a "Coda 1" chapter with combinations of Voice-leading and Incompleteness. Here, at the end of the Harmony/Transformations/Metamorphoses, we are adding a "Coda 2" page in which the 15 possibilities of Voice-leading and Metamorphoses will be examined.

This might be as good a place as any to elaborate on the nature of a "generative" theory.
The essence of generation is the process and not the result.
In more practical terms this means that it is far preferable to remember
how the transformations operate rather than what they produce.
You will really appreciate this site if you scout and discover,
letting the results be your source of inspiration !

NOTE that there will be no Orbit colors in this page -
neither in the text nor in the musical examples,
because the Orbit changes are too numerous and too complex.
However, all Orbit changes will be meticulously indicated
in the text accompanying each musical example.

Metamorphosis>
Voice-leading
v

0
Swing

1
01 10

2
23 32

3
02 20 13 31

4
03 30 12 21

A
0-3

4
3-2-1-0

3 - 1
3-2-10 01

1 - 3
23 32-1-0

4
13-20-31-02

2 - 1 - 1
03-21 30 12

B
1-3

3 - 1
3-2-1 0

4
3-2-10-01

1 - 2 - 1
23 32-1 0

3 - 1
13-20-02 31

3 - 1
03-21-30 12

C
2-3

2 - 1 - 1
3-2 1 0

2 - 2
3-2 10-01

1 - 1 - 1 - 1
23 32 1 0

4
13-20-02-31

4
03-21-12-30

In the top line we find the 5 forms of Metamorphosis (0, 1, 2, 3, and 4, all hlinked within the page), and
     in the left column we find the 3 forms of Voice-leading (A, B, and C).
Each Metamorphosis is accompanied by its Orbit changes (if any) -
     Metamorphosis 0 (no Metamorphosis) has no Orbit changes and produces a Swing.
     Metamorphosis 1 exchanges Orbits 0 and 1 and produces a circle.
     Metamorphosis 2 exchanges Orbits 2 and 3 and produces a convenient half-circle.
     Metamorphosis 3 exchanges Orbits 0 and 2 as well as Orbits 1 and 3,
          and, like Metamorphosis 1, produces a circle.
     Metamorphosis 4 exchanges Orbits 0 and 3 as well as Orbits 1 and 2,
          and, like Metamorphosis 2, produces a half-circle.
Each Voice-leading is accompanied by its procuration of Orbit 3 -
     Voice-leading A with procuration 0-3
     Voice-leading B with procuration 1-3
     Voice-leading C with procuration 2-3
Each of the 15 interior squares (all hlinked within the page) indicates the disposition of the Orbit lines -
     the grouping -
          "4" when the 4 lines use the same 4-orbit chain,
          "3-1" (or "1-3") when 3 lines use the same 3-orbit chain
               and the other line uses the remaining Orbit,
          "2-2" when 2 lines use the same 2-orbit chain
               and 2 lines use the other 2-orbit chain,
          "2-1-1" (or "1-2-1") when 2 lines use the same 2-orbit chain
               and 2 lines each use the individual orbit-lines left,
          "1-1-1-1" when the 4 lines each use individual orbit-lines,
     and (below) the details of the Orbits involved.
We will see all these possibilities, one Metamorphosis at a time.

Metamorphosis 0

Since there is no Metamorphosis here,
the following examples have already been seen.

0A - with Voice-leading A

This specific Swing was already seen in Voice-leading A.
     NOTE the 4 grouping with its 3-2-1-0 Orbit chain.
          It was also presented dominantized,
               as one would find it in Free Harmony.
The Fundamental Bass is not possible here because it interferes with 1-0.

0B - with Voice-leading B

This specific Swing was already seen in Voice-leading B.
     NOTE the 3-1 grouping with its 3-2-1 Orbit chain,
                    and the "V" approach in the procuration 1-3,
               as well as the isolated line on Orbit 0.
          It could also be dominantized, even if this was not officially done on the site.
The FB is possible here.

0C - with Voice-leading C

This specific Swing was already seen in Voice-leading C.
     NOTE the 2-1-1 grouping with its 3-2 Orbit chain,
               as well as the isolated lines on Orbits 1 and 0.
          It was also presented dominantized,
               as one would find it in Free Harmony.
The FB is also possible here.

Metamorphosis 1

Since this Metamorphosis produces the "Circle",
several examples have also already been seen.
Never lose sight of its 10 01 orbit exchange.

1A - with Voice-leading A

The Voice-leading A Circle was already seen in Metamorphosis 1.
     NOTE the 3-1 grouping with its 3-2-10 Orbit chain,
               and the "V" approach in the procuration 10-3.
          The isolated line on Orbit 01 is usually left out,
               but the FB is very useful here.
     NOTE also that the 0A combination had a 4 grouping which was changed by the 10 orbit exchange.
          This Circle was also presented partly dominantized,
               in the second half, from the ANTE-3 to the end.
old1a
It would also be possible to dominantize the 2 first chords because -
     (a) the diatonic Buckle chord has an inverse dominant sonority (Bm7-5 sounding like Dm6), and
     (b) the preceding F7 ties into it perfectly,
          producing the chord pattern C7 - F7 - Bm7-5 - E7 - A7 - D7 - G7 - C.
     In the example above -
          the first line (Soprano) has the Orbit chain 2-10-3-2-10-3-2-1,
          the second line (Alto) has the Orbit chain 10-3-2-10-3-2-10-0,
          the third line (Tenor) has the Orbit chain 3-2-10-3-2-10-3-2, and
          the fourth line (Bass) has the FB.

You will find an interesting example of open Metamorphosis 1 in Free Harmony.

1B - with Voice-leading B

The Voice-leading B Circle was already seen in Metamorphosis 1.
With either (a) the jump of a sixth to produce the "V", or the 0(3) substitution to circumvent it -
     NOTE the 4 grouping with its 3-2-10-01 Orbit chain, possibly 0(3)-2-10-01.
          The FB is possible here but not absolutely necessary.
     NOTE also that the 0B combination had a 3-1 grouping which was changed by the 10 orbit exchange.

You will find an interesting example of the jump version in the Voicve-leading B of Exercise 3c.

These circles can be dominantized all the way through, as we did in 1A

1C - with Voice-leading C

The Voice-leading C Circle was already seen at the beginning of Metamorphosis 1.
     NOTE the 2-2 grouping with its 3-2 Orbit chain,
               as well as a 10-01 Orbit chain.
          The FB is possible here but not absolutely necessary.
     NOTE also that the 0C combination had a 2-1-1 grouping which was changed by the 10 orbit exchange,
               uniting both the individual 1 and 0 lines into the 10-01 Orbit chain.
          This Circle was also presented partly dominantized,
               in the second half, from the ANTE-3 to the end.

This circle can be dominantized all the way through, as we did in 1A

Metamorphosis 2

This Metamorphosis is much deeper and more subtle than the preceding Metamorphosis 1,
and we will limit ourselves to "half-circles" rather than complete ones.
Moreover, progressions from one bar to the next, will be by second, more precisely 2 fifths,
having moved 2 swings away (as in Metamorphosis 4).
One no longer has the clear harmonic effect of Metamorphosis 1,
but rather a horizontal, polyphonic effect.
Never lose sight of its 23 32 orbit exchange.

2A - with Voice-leading A

The example given in Metamorphosis 2 "cheated" a little,
     by presenting only 1 metamorphosed chord,
          and by removing its Orbit 0, thus eliminationg the change of note involved.
               There is also a dominantized version of this example.
     We will be more "honest" and straightforward here,
          by presenting several complete metamorphosed Tetrads in a row.
     NOTE the 1-3 grouping with the isolated 23 orbit line,
               as well as its 32-1-0 Orbit chain.
          The FB still consists of the root of each chord (second function),
               but now moves in seconds (2 swings away).
old2a
In the example above, we have -
     the chord pattern Cm6 - (G7)Dm6 - (A7)Em6 - (Bm-7-5)Fm6 - C,
          with the metamorphosis closed, all the chords in parentheses eliminated.
     the first line (in the Soprano voice) has the isolated 23 orbit line,
          in all, 3-23-23-23-2,
     the next 3 lines have the 32-1-0 Orbit chain -
          the Alto voice with 0-32-1-0-0,
          the Tenor 1 voice with 2-1-0-32-1,
          the Tenor 2 voice with 1-0-32-1-1, and
     the fifth line (in the Bass voice) has the FB with 1-1-1-1-0.

You might enjoy hearing it in its diatonic state without the dominantization.
     NOTE how one hears individual polyphonic lines more than actual harmonic chords.

2B - with Voice-leading B

     NOTE the 1-2-1 grouping with the isolated 23 orbit line,
          as well as its 32-1 Orbit chain,
               and another isolated 0 orbit line.
          The FB still consists of the root of each chord.
old2b
In the example above, we have -
     the same chord pattern Cm6 - G7Dm6 - A7Em6 - Bm-7-5Fm6 - C,
          with the metamorphosis now presented open, the change quite evident in the 0 line.
     the isolated 23 orbit line is in the Soprano 2 voice with 3-23-23-23-2,
     the 32-1 Orbit chain is in -
          the Soprano 1 voice with 2-1-32-1-1,
          the Alto voice with 1-32-1-32-1,
     the isolated 0 orbit line is in the Tenor voice with 0-0 0-0 0-0 0-0,
          including the note-changes, and
     the fifth line (in the Bass voice) has the FB with 1-1-1-1-0 (as in 2A).

You might enjoy hearing it in its diatonic state without the dominantization.
     We once again have a very polyphonic sound.

2C - with Voice-leading C

     NOTE the unusual 1-1-1-1 grouping with 4 isolated orbit lines, 23, 32, 1, and 0.
old2c
In the example above, we have -
     the same chord pattern C6 - G7Dm6 - Am7Em-6 - Bm7-5F6 - C,
          with the metamorphosis also presented open, the change quite evident in the 0 line.
     the isolated 23 orbit line is in the Alto voice,
          in all, 3-23-23-23-2,
     the isolated 32 orbit line is in the Soprano voice,
          in all, 2-32-32-32-1,
     the isolated 1 orbit line is in the Bass voice,
          in all, 1-1-1-1-0, acting as FB, and
     the isolated 0 orbit line is in the Tenor voice,
          in all, 0- 0 0- 0 0- 0 0- 0, with the note-changes.
This diatonic version cannot be dominantized because 2 would be placed farther from its procuration 2-3.
     We once again have a very polyphonic sound.

Metamorphosis 3

This is another deep and subtle Metamorphosis.
Since it progresses by fifths (like Metamorphosis 1),
it will enable the construction of full circles,
but in the opposite direction.
This will not produce the clear harmonic effect of Metamorphosis 1,
but rather the horizontal, polyphonic effect of Metamorphosis 2.
We have 2 orbit exchanges here, 02-20 and 13-31.

3A - with Voice-leading A

The example given in Metamorphosis 3 also "cheated" a little,
     by presenting only 1 metamorphosed chord,
          and by placing a Non-chordal Tone on each Orbit 0, to facilitate the 02-20 exchange.
               There is also a dominantized version of this example.
     We will present enough Tetrads in a row to make a complete circle,
           with each metamorphosis open.
     NOTE the 4 grouping with its 13-20-31-02 Orbit chain.
          There is no FB because it interferes with 31-02.
old3a
In the example above, we have -
     the chord pattern Cm6D7 - Gm6A7 - Dm6E7 - Am6B7 - Ebm6F7 - Bbm6C7 - Fm6G7 - C,
          with each metamorphosis open,
     the 4 voices using the same 13-20-31-02 Orbit chain -
          the Soprano voice with a 20-31-02-13-20-31-02-1 orbit line,
          the Alto voice with a 31-02-13-20-31-02-13-2 orbit line,
          the Tenor voice with a 02-13-20-31-02-13-20-0 orbit line, and
          the Bass voice with a 13-20-31-02-13-20-31-1 orbit line.

You might enjoy hearing it in its diatonic state without the dominantization.
     A little more harmonic than with Metamorphosis 2, but quite polyphonic nevertheless.
          Each bar sounds as if a Dominant-shape chord is preceded by its ANTE-1.

3B - with Voice-leading B

     NOTE the 3-1 grouping with its 13-20-02 Orbit chain,
          as well as the isolated 31 orbit line,
               with its 1-3 procuration over and over again.
          There is no FB because it interferes with 31-31.
old3b
In the example above, we have -
     the same chord pattern Cm6D7 - Gm6A7 - Dm6E7 - Am6B7 - Ebm6F7 - Bbm6C7 - Fm6G7 - C,
          with each metamorphosis open, but the parallel movements spread in time -
               02 on the second beat, and 20 on the fourth beat of each bar,
     3 lines using the same 13-20-02 Orbit chain -
          the Soprano voice with a 02-13-20-02-13-20-02-1 orbit line,
          the Alto voice with a 13-20-02-13-20-02-13-2 orbit line,
          the Tenor voice with a 20-02-13-20-02-13-20-0 orbit line, and
     the "Bass voice" using the isolated 31 orbit line,
          in all 31-31-31-31-31-31-31-1,
               starting way up high, above the Soprano, and ending below the Tenor,
                    always progressing downward to make the "V" in the 1-3 procuration.

You might enjoy hearing it in its diatonic state without the dominantization.
     Once again, a semi-harmonic, semi-polyphonic sound.

3C - with Voice-leading C

     NOTE the 4 grouping with its 13-20-02-31 Orbit chain.
          It is possible to have the FB here because there is no interference with the orbit lines.
old3c
In the example above, we have -
     the same chord pattern Cm6D7 - Gm6A7 - Dm6E7 - Am6B7 - Ebm6F7 - Bbm6C7 - Fm6G7 - C,
          with each metamorphosis open,
     4 lines using the same 13-20-02-31 Orbit chain -
          the Soprano voice with a 20-02-31-13-20-02-31-1 orbit line,
          the Alto 1 voice with a 13-20-02-31-13-20-02-1 orbit line,
          the Alto 2 voice with a 31-13-20-02-31-13-20-0 orbit line,
          the Tenor voice with a 02-31-13-20-02-31-13-2 orbit line, and
     the Bass voice has the FB with 0-0-0-0-0-0-0-1.
NOTE that while the 4 upper lines are all melodically descending,
          with flattening (descending) progressions at each bar-line,
     the FB is melodically ascending,
          with sharpening (ascending) progressions at each bar-line.

You might enjoy hearing it in its diatonic state without the dominantization.
     Still in a semi-harmonic, semi-polyphonic style.

Metamorphosis 4

Since this Metamorphosis is used in the I-IV-V-I chord progression,
(C+7-F6Dm7-G7-C)
numerous examples have also already been seen,
in all possible combinations of Voice-leading.
Never lose sight of its 2 orbit exchanges, 03 30 and 12 21.

4A - with Voice-leading A

The examples given in Metamorphosis 4 also "cheated" a little,
     by presenting only 1 metamorphosed chord in the I-IV-V-I,
          and only 2 consecutive metamorphosed chords in another example.
               There is also a dominantized version of this last example.
     We will present here 4 metamorphosed Tetrads in a row,
          producing a generous half-circle, similar to those using Metamorphosis 2,
               but in the opposite direction, with 7th chords instead of 6th chords.
     NOTE the 2-1-1 grouping with its 03-21 Orbit chain, as well as
          an isolated 31 orbit line, and
          another isolated 12 orbit line.
old4a
In the example above, we have -
     the chord pattern C7 - D7 - E7 - F7 - G7 - C,
          with each metamorphosis closed,
     2 lines using the same 03-21 Orbit chain -
          the Soprano voice with a 1-03-21-03-21-1 orbit line, and
          the Tenor voice with a 3-21-03-21-03-2 orbit line, as well as
     an isolated 2-12-12-12-12-1 orbit line in the Alto voice, and
     an isolated 0-30-30-30-30-1 orbit line in the Bass voice,
          which acts as the FB.

You might enjoy hearing it in its diatonic state without the dominantization.

4B - with Voice-leading B

     NOTE the 3-1 grouping with its 03-21-30 Orbit chain,
          and another isolated 12 orbit line.
old4b
In the example above, we have -
     the same chord pattern C7 - D7 - E7 - F7 - G7 - C,
          with each metamorphosis closed,
     3 lines using the same 03-21-30 Orbit chain,
               always jumping downward to make the "V" in the 21-30 procuration -
          the Soprano voice with a 3-21-30-03-21-1 orbit line,
          the Alto voice with a 1-30-03-21-30-0 orbit line, and
          the Tenor voice with a 0-03-21-30-03-2 orbit line, as well as
     an isolated 2-12-12-12-12-1 orbit line in the Bass voice.
          The FB could possibly have been added.
You will probably prefer to sing this a fourth higher.

You might enjoy hearing it in its diatonic state without the dominantization.

4C - with Voice-leading C

     NOTE the 4 grouping with its 03-21-12-30 Orbit chain,
          and the FB added as a fifth line.
old4c
In the example above, we have -
     the same chord pattern C7 - D7 - E7 - F7 - G7 - C,
          with each metamorphosis closed,
     4 lines using the same 03-21-12-30 Orbit chain,
          the Soprano voice with a 2-30-03-21-12-1 orbit line,
          the Alto 1 voice with a 0-03-21-12-30-0 orbit line,
          the Alto 2 voice with a 3-21-12-30-03-2 orbit line, and
          the Tenor voice with a 1-12-30-03-21-1 orbit line, as well as
     the FB in the Bass voice, 0-0-0-0-0-1.

You might enjoy hearing it in its diatonic state without the dominantization.

4CA - with Voice-leadings C and A

In the previous examples (4A, 4B, and 4C), Metamorphosis 4 was always presented closed.
     In its diatonic state, there is not much point in presenting Metamorphosis 4 open,
               unless one does it with only real triads,
          because the two tetrads are composed of exactly the same notes.
     However, in Free Harmony, with only dominant-shape chords,
          the result can be quite striking and worth-while.
     Remember that there is no Window here, no line between chromaticism and diatonicism,
          no point of reference, just travel in non-gravitational space.
old4ca
In the example above, we have -
     the chord sequence Bb7 - Ebm6C7 - Fm6D7 - Gm6E7 - Am6F#7 - BmB,
               starting anywhere and ending anywhere,
          with each metamorphosis open,
     Voice-leading A across each bar-line,
          and Voice-leading C from the beginning of a bar to the beginning of the next bar,
     in each bar, the orbit before and after the metamorphosis,
          usually (3 times out of 4) followed by another orbit of the same chord,
               both of these orbits resolving to (or procuring) the beginning of the next bar,
                    1-1 and 2-1, 2-3 and 0-3, 0-0 and 1-0, with 3-2 remaining constant,
     the 4 lines using the same 03 - 21 2 - 12 0 - 30 1 Orbit chain -
          the Soprano voice with a 1 - 03 - 21 2 - 12 0 - 30 1 - 0 2 orbit line,
          the Alto voice with a 2 - 12 0 - 30 1 - 03 - 21 2-1 orbit line,
          the Tenor voice with a 0 - 30 1 - 03 - 21 2 - 12 0 - 0 orbit line, and
          the Bass voice with a 3 - 21 2 - 12 0 - 30 1 - 03 - 2 1 orbit line.

Change of Orbit
Why do we feel the need to change orbit in the middle of a bar ?
     Let's take the first complete bar (with chords Ebm6C7) as an example -
          1. The main difficulty lies with the 21 exchange in the Bass voice -
               the Gb (2 of Ebm6) becomes 1 (more precisely a -5) of C7,
                         with an obliged, chromatic resolution to the F, 1 of the next (Fm6) chord,
                    must NOT be chromaticized to G in the C7, but followed by E (2 of the C7),
                         which ALSO resolves to the F (1 of the Fm6 chord),
                    (the resolutions 1-1 and 2-1).
          2. This obliges the Eb in the Alto voice to be followed by C (0 of the C7),
               and BOTH the Eb and the C will procure the D of the Fm6,
                    (the procurations 2-3 and 0-3).
          3. The movement from C to G (0 to 1), in the Tenor voice,
               re-establishes the equilibrium of the orbits in the chord,
                    (producing the orbit options 0-0 and 1-0).
          4. The Bb in the Soprano voice (3 of the C7 ) remains unchanged,
               the 3-2 resolution being common to both Voice-leadings.

This is a heavy piece to digest - take your time !
     You might even want to print the musical example and write in all the orbit numbers involved.

Passing Tones
It is possible to add passing (non-chordal) tones to the previous example,
     (a) in the 21 2 orbit displacement, and
     (b) in the 12 0 orbit displacement,
               a flat note in this last case, to accomodate the minor chord of the following bar,
          giving the impression of an added chord.
old4ca1
With this addition, we now have -
     the chord sequence Bb7 - Ebm6(Bbm6)C7 - Fm6(Cm6)D7 - Gm6(Dm6)E7 - Am6(Em6)F#7 - BmB,
          with each added (passing tone) chord in parentheses.
     Each voice remains very much as it was, only the passing tones have been added.

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