Hannaford - Esmen symmetrical sets walkthrough.

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.

https://www.youtube.com/watch?v=GaErmvK0Ta0

                                      

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

11224445

23344444

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.