The Structure Of Clusters

What we commonly call “Tone Clusters” are notes which are more tightly grouped than those normally found in chords (where we find 2 thirds, with 1 second). We could thus safely define clusters as notes which are grouped entirely in seconds (with sonorities 1 and 2). Would they still be considered clusters if they were presented in more “open” position? Time and usage will tell, but we will present them here in a variety of positions, (both open and closed).

Building Clusters
They will be “built” by starting with a chord in its normal state (its basic materials) and then applying the Transformations (chromaticism and/or non-chordal tones) which produce the eventual cluster. This will enable us to retain the original chord function and voice-leading.

4-Note Diatonic Clusters

Producing a 4-note diatonic cluster is fairly simple and seems a good place to start.

Building the cluster
Let’s build it on the DOMINANT chord (in C major, the chord G7)
     and then apply the same process to all the chords of the circle.
1. If we dispose the DOMINANT chord with the notes D, F, G, B,
          in order of ascending pitch, (with sonority 3-3-2)
     we already have the “nucleus” of a cluster in the center with the notes F and G.
2. If we place a superior diatonic non-chordal tone on the note D
          (E, sus6, commonly called a 13th)
     we already have a 3-note diatonic cluster E, F, G (sonority 1-2).
3. Then, if we place an inferior diatonic non-chordal-tone on the note B
          (A, push2, commonly called a 9th)
     we complete our 4-note diatonic cluster E, F, G, A (sonority 1-2-2).

Resolving the cluster
     The safest way to resolve this cluster is to resolve the non-chordal tones first,
               and then to resolve the original chord G7 to C
          (this produces the lines E-D-C, F-E (F-D-E for one last push2), G-G, A-B-C).

If we apply the same procedure to each chord of the circle,
          (with voice-leading C) we get


Inverting the order
However, it is possible to invert the order,
     - have the chord in its normal state first (without the 2 non-chordal tones) and
     - and have it followed by the cluster (with the 2 non-chordal tones).
In this case, the 2 non-chordal tones will act as permanent non-chordal tones,
     resolving exactly as the chordal tone would have done.

Once again, if we apply this procedure to each chord of the circle, we get

The voice-leading to the resolution chord of C will thus remain the same -
     - the COMMON TONE (Orbit 0, G) will remain in place, G to G;
     - the MOTRIX (7th, Orbit 3, F) will resolve normally, F to E, the MEDIAN (3rd);
     - the superior non-chordal tone of the PROPER TONE (sus6, E)
          will resolve as the original D would have done, to the PROPER TONE, C;
     - the inferior non-chordal tone of the MEDIAN (push2, A),
          will resolve as the original B would have done, to the PROPER TONE, C.
In the circle, the MEDIAN (3rd) will procure the MOTRIX (7th) of the next chord.

Only clusters!
Here, we go all out and resolve the clusters directly to each other,
     without ever hearing the original, “normal” chords at all.

Open and close disposition
Note that, if we wish to produce a musically valid voice-leading,
only the odd-numbered chords will be in “close” disposition,
the even-numbered being more “open”.

Uniform sound (same sonority)
Note also that the sound of these clusters is not uniform -
chords 1 and 2 (C+7 and F+7) have a 2-1-2 sonority,
chords 3 and 4 (Bm7-5 and Em7) have a 2-2-1 sonority,
only chord 5 (Am7) has a 2-2-2 sonority,
chords 6 and 7 (Dm7 and G7) have a 1-2-2 sonority.

A uniform sound
The judicious addition of chromatic notes,
     produces a uniform sound of major seconds (2-2-2).

Even with the addition of chromaticism, we still consider these clusters diatonic, because the 4 notes of the cluster all bear different names and could belong to the same Window (diatonic system).

3-Note Chromatic Clusters

Chromatic clusters are produced by the Transformation of Chromaticism and not by the Transformation of Non-chordal tones as was previously the case for diatonic clusters.

Chromaticism of the Proper Tone (in the G7-C progression)
The PROPER TONE is particularly active as a substitution when a chord is incomplete.
1. It can substitute for the MOTRIX (on G7, D substituting for F and resolving to E).
     This substitution can be chromaticized (augmentation), producing the voice-leading D#-E.
2. It can also substitute for the MEDIAN (on G7, D substituting for B and resolving to C).
     This substitution can be chromaticized (diminution), producing the voice-leading Db-C.
3. During this time, the original D can remain unchanged
     and resolve to the COMMON TONE G, producing the voice-leading D-G.
We now have the 3-note chromatic cluster Db, D, D# (sonority 1-1)
          (all notes bearing the same name)
     which produce the lines Db-C, D-G, D#-E, and G-G.

Completing the orbit lines for G7-C
In order to make the substitutions clearer and enable us to tie these clusters in a circle,
          we will extend these lines backward to include the original function
     Bass - B-C-Db (original MEDIAN, inferior non-chordal tone, chromatic substitution)-C,
     Tenor - original D-G unchanged,
     Alto - F-E-D# (original MOTRIX, superior non-chordal tone, chromatic substitution)-E,
     Soprano - original G-G unchanged.

In a circle
Since this cluster only works on dominantized chords
          (from the ANTE-3 E7 to the DOMINANT G7),
     the first 3 chords of the circle must be somewhat “cheated”
          by the first entries of a natural canon (with voice-leading A).
The Tenor comes in first with the PROPER TONE line,
     then the Soprano, (eventually followed by Bass and Alto),
          has, in turn, the MOTRIX, then the MEDIAN (in the next bar),
               (each followed by the Non-chordal tone and the Chromatic Substitution),
          then the PROPER TONE resolution.

Inversion in diatonic minor
This is particularly interesting because the PROPER TONE line,
          which was formerly in the Tenor voice (it had the fifth of each chord),
     is now the Fundamental Bass (it has the root of each chord).
It has been disposed in octaves,
          the upper octave freely crossing the three other voices,
     with a 1/4 note repeated at the end of each bar,
          so as to cause as much friction as possible with the two chromatic parts.

4-Note Dia-Chrom Clusters

What we call a “dia-chrom” cluster is one in which we have both diatonic and chromatic semi-tones. We could give, as examples - F, Gb, G, A or F, G, G#, A, both of which we will see shortly.

Getting started
To the fundamental, dominantized, Voice-leading A TONIC-DOMINANT swing (Cm6 and G7),
     we will add the Orbit 4 chromaticization.
No signs of a cluster yet, only the well-known diminished 7th look/sound (sonority 3-3-3)
     on both chords Cm6-5 (Ao7) and G7+1 (G#o7).

Building the cluster
Now, let’s replace the original COMMON TONE G
               (in a fifth voice, in 3 octaves, to produce as much friction as possible)
          held throughout while we pass the Gb and the G#,
     producing the 3-note dia-chrom clusters Gb, G, A (1-2) and F, G, G# (2-1)
          with the normal tone (sonority 2) between the COMMON TONE and the MOTRIX.
Then, as a final sadistic gesture,
     we insert the sus4 F on the TONIC and the push2 A on the DOMINANT,
          removing the “sound” of the MEDIANs (3rds) completely,
     producing the 4-note clusters F, Gb, G, A (1-1-2) and F, G, G#, A (2-1-1).

Reaching the limit
It is, however, possible to go further still.
     Instead of adding only the sus4 F on the TONIC
          and the push2 A on the DOMINANT, as we did in the previous example,
     it is also possible to add the push+5 G# on the TONIC
          and the sus-1 Gb on the DOMINANT.
This produces the 4-notes clusters F, Gb, G, G# (1-1-1) and Gb, G, G#, A, (1-1-1)
     certainly as “tight” as a cluster can be, with only semi-tones.
Both non-chordal tones will be resolved, before progressing to the next chord,
     because the chromatic non-chordal tone of the MOTRIX,
          can hardly be considered permanent.

Once again, whether they will be considered bona fide clusters
     by the die-hard aficionados remains to be seen.

2 Sets Of 2-Note Chromatic Clusters

Here is another possibility which has even less chance of passing the “cluster definition” test. Instead of one 3- or 4-note cluster, it is possible to have cluster-like sounds using two 2-note chromatic clusters (either as diminished octaves or augmented unisons). We will present only the finished product here, a circle in C major, but much more detail can be found on another page to which we will frequently refer.

Building the cluster
1. Let’s start by examining the very end, the DOMINANT/TONIC cadence.
     Instead of its original basic form of B, F, D, G (reading up),
          we have 2 diminished octaves B, Bb and D#, D, (sonority 1-3-1)
          which, of course, resolve to C, E, C, G exactly as the original would have done.
2. Before this set of diminished octaves, still on the DOMINANT chord,
     we find another set of diminished octaves C#, C, F#, F, (sonority 1-4-1)
          disposed C, F, C#, F# which resolve to the original set.
3. Before this set of diminished octaves, still on the DOMINANT chord,
     we find still another set of diminished octaves E, Eb, A#, A, (sonority 1-4-1)
          disposed A#, E, Eb, A, which resolve to the second set.
     What we did to the DOMINANT chord can be applied exactly
          to the ANTE-3, ANTE-2, and ANTE-1
     There is, as usual, light ”cheating” on the first 3 chords to make the canon work.

4. We could have diminished the tension somewhat
     by immediately resolving the non-chordal tones on the COMMON TONE
          (on the DOMINANT chord, A-G, F#-G, and Bb-Ab).
5. This more moderate treatment could have been applied to the whole circle,
     rather than leaving it in the somewhat brutal form which we have here.