Spelling out some details of HEs 876
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
Checking through Nurtan’s
symmetrical nonachord table (available in full in the blog HANNAFORD – ESMEN chords
I randomly selected this set and played with the content. The following system
using transposition and rotation demonstrates the levels of consistency which I
found interesting when composing with the material. In my opinion the results
of a simple “walkthrough” presented in the Bagatelles 4 and 5 had a late
Romantic character not unlike Silvestrov’s style.
First row placed diagonally with
each opening pitch being the place holder for a transposition of the row. The
row is then rotated to fit the grid.
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
F
|
D#
|
G
|
C
|
B
|
E
|
A
|
G#
|
C#
|
F#
|
A#
|
G#
|
C
|
F
|
E
|
A
|
D
|
C#
|
C
|
F
|
A
|
G
|
B
|
E
|
D#
|
G#
|
C#
|
F#
|
F
|
A#
|
D
|
C
|
E
|
A
|
G#
|
C#
|
F#
|
B
|
A#
|
D#
|
G
|
F
|
A
|
D
|
C#
|
C
|
F
|
A#
|
A
|
D
|
F#
|
E
|
G#
|
C#
|
F#
|
F
|
A#
|
D#
|
D
|
G
|
B
|
A
|
C#
|
F
|
A#
|
A
|
D
|
G
|
F#
|
B
|
D#
|
C#
|
Pitch
content
B, D#, G#, G, C and E repeated x6,
F, A and C# x 9
Vertical content:
B,C,F,F#
|
A#,B,D#,F
|
G,G#,A,A#
|
G,A,C,D,D#
|
B,C,D,F,G
|
E,F,F#,G
|
D#,E,A,B
|
G#,A,D,D#
|
C#
|
0,1,6,7
|
0,1,5,7
|
0,1,2,3
|
0,1,3,6,8
|
0,1,3.6.8
|
0,1,2,3
|
0,1,5,7
|
0,1,6,7
|
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
F
|
D#
|
G
|
C
|
B
|
E
|
A
|
G#
|
C#
|
F#
|
A#
|
G#
|
C
|
F
|
E
|
A
|
D
|
C#
|
C
|
F
|
A
|
G
|
B
|
E
|
D#
|
G#
|
C#
|
F#
|
F
|
A#
|
D
|
C
|
E
|
A
|
G#
|
C#
|
F#
|
B
|
A#
|
D#
|
G
|
F
|
A
|
D
|
C#
|
C
|
F
|
A#
|
A
|
D
|
F#
|
E
|
G#
|
C#
|
F#
|
F
|
A#
|
D#
|
D
|
G
|
B
|
A
|
C#
|
F
|
A#
|
A
|
D
|
G
|
F#
|
B
|
D#
|
C#
|
Bottom left rising diagonally (i1 /
h1,I, 2 etc.).
F
|
F#,A#
|
C,F,A
|
F#,F,A#,D
|
F#,B,A#,D#,G
|
C,F,A#,A,D,F#
|
F#,F,A#,D#,D,G,B
|
F,A#,A,D,G,F#,B,D#
|
0,4
|
0,3,7
|
0,1,4,8
|
0,1,5,7,8
|
0,1,3,5,8,9
|
0,1,3,4,5,8,9
|
0,1,2,4,5,6,8,9
|
B,D#,G#,G,C,F,E,A,C#
|
0,1,2,4,5,6,8,9,10
|
Top right falling diagonally, (A9 / A8,B, 9 etc.).
C#
|
A,C#
|
E,G#,C#
|
F,A,D,C#
|
C,E,A,G#,C#
|
G,B,E,D#,G#,C#
|
G#,C,F,E,A#,D,C#
|
D#,G,C,B,E,A,G#,C#
|
0,4
|
0,3,7
|
0,1,4,8
|
0,1,5,7,8
|
0,1,3,5,8,9
|
0,1,3,4,5,8,9
|
0,1,2,4,5,6,8,9
|
Every set is a subset of the central group.
Now let us look at rotating the HEs formations.
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
HEs 876
Rotating HEs 876 from the central pitch A5 to A4,6 and so on
four rotations are possible before repeating. The symmetry of the nonachord permits
this to apply to each set, (if it isn’t universal I would like to know). The
following two sets are taken at random.
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
C
|
G
|
F
|
G#
|
E
|
D#
|
A
|
B
|
C#
|
E
|
G#
|
D#
|
F
|
A
|
G
|
B
|
C
|
C#
|
A
|
F
|
G
|
D#
|
B
|
G#
|
C
|
E
|
C#
|
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
The rotation breaks down the
symmetry taking B to D# as 4. D# to G# as 5 steps and so on we have
11445555
12223456
1122444523344444
Which in effect means four
distinct melodic sources, this may be of particular interest to some composers.
Let us compare at random set HEs 822
Again four rotations are possible before repeating.
C#
|
G#
|
D#
|
F
|
C
|
G
|
A
|
E
|
B
|
C
|
F
|
G
|
D#
|
A
|
G#
|
E
|
C#
|
B
|
A
|
D#
|
G#
|
G
|
E
|
F
|
C#
|
C
|
B
|
E
|
G
|
F
|
G#
|
C#
|
D#
|
C
|
A
|
B
|
C#
|
G#
|
D#
|
F
|
C
|
G
|
A
|
E
|
B
|
55255255
52461432
65131611
32352332
And finally HEs 200
G#
|
B
|
G
|
A#
|
C
|
D
|
F
|
C#
|
E
|
C
|
A#
|
D
|
G
|
F
|
B
|
C#
|
G#
|
E
|
F
|
G
|
B
|
D
|
C#
|
A#
|
G#
|
C
|
E
|
C#
|
D
|
A#
|
B
|
G#
|
G
|
C
|
F
|
E
|
G#
|
B
|
G
|
A#
|
C
|
D
|
F
|
C#
|
E
|
22333344
22244556
12233444
11113455
HEs 876, transposition
|
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
Let us now examine the transposition of the formation,
moving upwards a semitone each time:
|
B
|
D#
|
G#
|
G
|
C
|
F
|
E
|
A
|
C#
|
|
C
|
E
|
A
|
G#
|
C#
|
F#
|
F
|
A#
|
D
|
|
C#
|
F
|
A#
|
A
|
D
|
G
|
F#
|
B
|
D#
|
|
D
|
F#
|
B
|
A#
|
D#
|
G#
|
G
|
C
|
E
|
|
D#
|
G
|
C
|
B
|
E
|
A
|
G#
|
C#
|
F
|
|
E
|
G#
|
C#
|
C
|
F
|
A#
|
A
|
D
|
F#
|
|
F
|
A
|
D
|
C#
|
F#
|
B
|
A#
|
D#
|
G
|
|
F#
|
A#
|
D#
|
D
|
G
|
C
|
B
|
E
|
G#
|
|
G
|
B
|
E
|
D#
|
G#
|
C#
|
C
|
F
|
A
|
|
G#
|
C
|
F
|
E
|
A
|
D
|
C#
|
F#
|
A#
|
|
A
|
C#
|
F#
|
F
|
A#
|
D#
|
D
|
G
|
B
|
|
A#
|
D
|
G
|
F#
|
B
|
E
|
D#
|
G#
|
C
|
The coloured squares show which pitches are in common with
the top, original, row.
The colour scheme should make clear that three rows are
identical in pitch content, but not order, the remaining 9 have hexachords in
common, while the complementary pitches are D,F#,A#. The HEs 876 formation can
itself be constructed as three augmented triads:
CEG# / FAC# / GBD#.
Let us again compare the set HEs 822, the first 5 rows
should be sufficient to illustrate the similarities with the above.
|
C#
|
G#
|
D#
|
F
|
C
|
G
|
A
|
E
|
B
|
|
D
|
A
|
E
|
F#
|
C#
|
G#
|
A#
|
F
|
C
|
|
D#
|
A#
|
F
|
G
|
D
|
A
|
B
|
F#
|
C#
|
|
E
|
B
|
F#
|
G
|
D#
|
A#
|
C
|
G
|
D
|
|
F
|
C
|
G
|
G#
|
E
|
B
|
C#
|
G#
|
D#
|
We hope that our readers will examine the sets for
composition, and we would be delighted to hear your comments on any results
which interest you. Nurtan and I are spending time discussing our own findings
and I am certain we will feed back on progress as we continue.
