Sunday, 25 June 2017


I don't really remember what the original question was, but it was something to do with the hexachord  B D F#  A flat  C  E flat. I think Ken asked me or told me something about it.* At that time I had plenty of idle time and thinking about it overnight I realised that it was one of the potential hexachords generated from a nine tone set symmetric about C#. With a fixed order of symmetric set generating intervals and 11 transpositions of the 9 tone set by semitone steps this will produce 33 hexachords. We exchange a few more e-mails and as usual off to the races with another project. It seems that we egg each other on to do things we ordinarily would not take on as potential waste of time. The original hexachord and its 9 tone symmetric fundamental set are given as example 1.

Clearly, one can immediately observe that there are many such sets and each set can provide a related family of hexachords. This can be most efficiently studied if we define the properties of these symmetric 9 tone sets and the hexachords generated.  Let us go through this by means of a set of Axioms and definitions as well as the mechanics of constructing the hexachords by working through an example;

AXIOM I. Each symmetric 9 tone set is symmetric about the 5th tone, conveniently chosen to be C located at a specified register conveniently chosen to accommodate all 9 tones.

AXIOM II. In each set the tones occur only once and they are fixed in their relationship to each other over the sound spectrum.

AXIOM III. These sets are not scales and the relative register of each note with respect to the others is a property of the set by 4 intervals.

The interpretation of Axiom 1 is easy. It simply suggests that the fundamental set fixes 9 tones in the sound spectrum and by Axiom III it can be transposed up or down without changing the character of the set. Axiom II specifies that any repetition or doubling is not allowed.

The defining intervals can be any number of semi-tones between 1 and 11. The first 4 columns of the table of symmetric 9 tone sets (W, X, Y, Z) are the interval basis of the set and if 1 £ W,X,Y,Z £11, then there are 14641 sets. With r as the arbitrary convenient register, these sets are generated by a simple formula:

{[Cr –z –y –x –w], [Cr –z –y –x],  [Cr –z –y], [Cr –z],  [Cr ],  [Cr +z],  [Cr +z +y],

       [Cr +z +y +x], [Cr +z +y +x +w]}

Although, a majority of these sets are not allowed by Axiom II, they can be easily identified by  the rule that Moduli 12 of the  following combinations must not be 0 (which is without mathematical jargon: None of the following combination is allowed to add up to  any multiple of 12:

2z,  z+y,  y+2z, x+y,  x+y+z,  x+y+2z, x+2y +2z, 2x +2y +2z,   w+x,  w+x+y, w+x+y+z,

w+ x+ y+ 2z,   w+ x+ 2y+ 2z,   w+2x+2y+2z, 2w+ 2x+ 2y+ 2z   


 This axiomatic restriction is easily programmed to eliminate the improper combinations. The computations show that 1752 sets are proper and can be used to generate the hexachords.

Let us look at another example: Consider the set #314. This example demonstrates that there are only 3 independent hexachords per set. Therefore with 11 semi-tone translations we have 33 hexachords per set or there are 57816 hexachords that can be generated by this simple method.

Obviously cluster chords of #1 and the most open chords of #1752 define the limits of the procedure. Within this very broad range we are convinced that many interesting harmonic structures await discovery. We have not yet started to investigate the monumental task of the investigation of harmonic functions and musicality of each chord and given the age of the two composers it is an unlikely venture. However we encourage experimentation with these chords and of course we would very much appreciate a proper attribution to Hannaford - Esmen chords in any such investigation.

Also, we would seriously and gladly consider an expanded version of this brief introduction into a more comprehensive scholarly article in a music theory journal if we are invited to do so.

*Ken’s e-mail

I was listening to a programme about the 50th anniversary of Sgt. Pepper's LHCB, its themes and construction. There was little that either of us wouldn't have worked out for ourselves, but there was a point when Howard Goodall was talking about "She's leaving home" and he explained how the song made use of the aeolian mode to create a sense of nostalgia. Something made me think back to Bridge and his piano sonata and I started playing with stacks of hexachords built on aeolian scales. 







reading these vertically we have 0,2,4,5,7,9, / 0,2,3,5,7,9 / 0,1,3,5,7,8 / 0,2,3,5,7,9 / 0,2,4,5,7,9 / 0,1,3,5,6,8 / 0,1,3,5,6,8 (then octave above). None of these form the Omega chord ),0,1,3,6,8,9 but it offers the scent of a trail.

If we look at the descending parallel chordsin the final page of the sonata we have 11 hexachords which can be mapped onto A, G, F, E, D, C, A, F', E, C', B. This scale could be stacked in thirds as


F' A B C' x E

As a collection these form a 9 pitch set of four semitones (space) central tone (space) four semitones, the sort of symmetry you enjoy.

Perhaps you can see where I am coming from?

To provide some flesh on the bones of the HEs the following provides a guide to our approach for the use of the table and this link opens up a PDF copy of the full table of ssets.

The following illustration offers some composing possibilities and how we notate for reference:

To provide some music here is a bagatelle on these sets which may be heard at

During the May – June period our e-mails were disrupted by Nurtan’s ill health, but this didn’t prevent us from discussing two issues, the relationship between internal and external activity (following a suggestion on the matter by Giorgio Sollazzi) and musical gravity. The following discussions eventually took us to the construction of the table published in the blog of the 25.06.2017.

These comments offer – should the reader wish to take some time – some of the exchanges, the humorous bits have been extracted along with the political statements which make us seem a little robotic. I promise you were are not! The posts arrived at various times between the 22nd May and the 8th June

The final frontier…let us return to this horrible question of internal and external form. To be honest if there are no useful practical outcomes to a discussion it is probably best to move onto something enjoyable, but bits of the discussion keep nagging at me. I keep coming back to space and finally it is opening up some interesting questions.

Anyway, a good definition from the astronomers, space is

Nothing as undifferentiated potential

I guess every composer knows that, and the consequences of adding something into that potential? what does that something do, add or subtract from the potential? Every note you add in should determine the final content....there we go again.

That took me to thinking about time frames, like the Feldman to start with, but time frames are everywhere in music, film scripts, dances, songs, expectations, anticipations etc. All of these can model the music before a note is played, Indian ragas, the hour of the day or night.

Then I started thinking about the third movement of Berio's Sinfonia, with all those references pasted over redesigned Mahler extracts from the Resurrection symphony. What would happen if substitute passages from other works were substituted into the music? Even if we accept that there is an underlying theme to the references substitutes could be found, after all there is a vast literature
to choose from.

Then I went a little further with the idea, let's substitute one tone row for another which shares many properties, and then make the substitution in a piece by Webern, how different would the result be? I started considering writing 3 short works with similar set material, everything in the music
remains the same apart from the pitch. This is a variant on your examination of the plainsong, or it seems so to me. Whichever method used one would have a better grasp of just how much difference the content changes. Would a change in a tone row by one pitch be as dramatic as changing a pitch in a motif by Beethoven?

There lots more jingling around in my head, that is why I played with the 4 glockenspiels idea in Space for a hundred (yet more pollution on YT), the rhythmic “string” determines everything that happens.


I am almost totally defeated by the line of discussion we were having. I still believe that there is a wealth of information and new ideas there and we have a pretty good start on it, BUT!!!! we are
missing a very crucial piece of the puzzle. Unfortunately, it leads to – or at least steers my mind to pieces, forms, techniques that are not in the realm of the rigid / flexible : rigorous - lax system or at least I cannot find the key that unlocks the method to classify and expand the internal and external forms.

I was resting all afternoon (could barely move) with a pad and pencil staring at the "every note you add in should determine the final content"... It should - at least intuitively. Well it does and it
does not. I tried different additions and subtractions in my mind with very disconcerting results. In some cases it does and in some cases it does not! Here is one of the puzzling examples. Any set of triads will do. say we have a composition fragment with chord CEG in root position suppose I
approach this with a triplet C# - F - A or A -C# _ F What would be the difference in the overall scheme? Both of them are sort of appoggiatura and serve the same function. If I like the result and use the transposition of this in sequence as a compositional format, would this be rigorous (yes
obeys a formula) or lax (yes, I have not specified the transposition and/ or back fall). What would change if I specify up resolution or down resolution strictly for each step? C# - A = down (D) and A-F = (U) suppose I create a composition: step 1: Chord = C and U C U C D C U C Step 2: transpose C (any step arbitrary to C1 repeat ups and downs then transpose C1 up or down as necessary 1 step at a time until C is reached. This gives something surprisingly musical but terrible at the same time. Why?


I am stuck at least knee deep in gravity, performance, register – all of which is complicated by dynamics. Let me explain my problem briefly. As I see it, the limitations of human ear are two -fold. The first one is the frequency cut-off. It is individually determined and likely to be strongly affected by age (Presbycusis) even keen young ears have an upper limit of hearing somewhere around 12,000 HZ. This limit is not a sharp cut off, but tapers down with increasing frequency. Let us consider a simple bowed string playing 440 A – that suggests that we should be able to hear a great deal of its upper partials – but if the fundamental is 1760 A the frequency limitation is going to interfere. In fact, there is going to be a greater limitation imposed on PPP than on fff? String and tube harmonics are relatively easy to understand and we bury it under the rubric of timbre. The mud becomes thicker in tuned plates and drums. For example the overtones of a timpani is different based on the location of impact, similar is true for xylophone marimba family etc. For bells it is even more complicated based on bell shape, e.g. orchestral tubular chimes, exactly same fundamental hand bell, cup shaped bell and traditional bells have all different partials. If I remember correctly the impact point of the clapper also changes the “sound”.  Unfortunately, I don’t remember all that much from my applied mathematics classes, a great deal of time was spent on membrane, plate and complex shape vibrations.

I wonder how one a use these complexities as a compositional tool. Also, dissonance, consonance start to take on different meanings. e.g. can the “out of tune” partials  we were told to avoid be musical?  I think, your example, C E G# C# F A hexachord takes different meanings*, f dynamics in contrast to p dynamics, no? Are these differences the same or entirely different if p is in low register and f is in the high register, vice versa? What register separation is needed to achieve this difference? Is this musically useful? If not why not?


* Sent from another server this came up when discussing the augmented triad combined with transpositions so: CEG’ + C’FA, DF’A’ etc. These produce a very limited number of possibilities 0,1,4,5,8,9 and subsets, and the whole tone-scale. The results suggested a number of composing possibilities which were worked out and shared on YT.

I'll also add the material used for "Swirling Nines" an electronic work for Reaktor and Kontakt software to add some flesh onto the bones of the table:

I was listening to a programme about the 50th anniversary of Sgt. Pepper's LHCB, its themes and construction. There was little that either of us wouldn't have worked out for ourselves, but there was a point when Howard Goodall was talking about "She's leaving home" and he explained how the song made use of the aeolian mode to create a sense of nostalgia. Something made me think of Bridge and his piano sonata and I started playing with stacks of aeolian scales, the one I looked at was built on six scales to give us the hexachords that Bridge is so consistently using. I had A, C', F, A flat, C and E flat. These vertically produce some very lovely chords, but from memory I recall there were 8 different hexachords, so if he did use a system of stacking he must have used at least two different stacks. It is rather late now and my brain aches at the thought of trawling through hundreds of possible variations. One could go back to the Bridge chord and work from there (my selection is not a million miles away from it.

Is all of this just a silly idea or a germ of an explanation that I hadn't considered before?
Hope there is something to these very interesting hexachord structures. Each produce three and with transforms we have 288 hexachords, probably there are many other symmetric structures (9 note) so we are talking about perhaps over 1,000 hexachords (retrograde and inverse included). That is not small change!