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?

Here is the opening page of the table, the full PDF is available at

I would recommend opening the file with Google sheets for viewing.

The following manuscript offers some examples, though the numbers column is not visible on the image above they are in the PDF link version.

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!

Friday, 9 June 2017

The changing character of rhythm

Quartet Op.22 Webern

“Only Webern—for all his attachment to rhythmic tradition—succeeded in breaking down the regularity of the bar by his extraordinary use of cross-rhythm, syncopation, accents on weak beats, counter accents on strong beats, and other such devices designed to make us forget the regularity of metre.”

Pierre Boulez, “Proposals,” in Stocktakings from an Apprenticeship

When I listen to the first movement of Op.22 I take away a different perception to those outlined above, it is a gentle work, and it flows. In the opening bars rests between events create a sensation of scanning a canvas at first sight, but as the music progresses a flow between events is established. Listening to the different performances available on You Tube the timings range from three minutes to three minutes 25 seconds, (Boulez conducts two of these with a wide difference between the timings). This is of course one out of many Webern compositions, but it should make us listen to and question the use of rhythm in music. Some of the discussions on Webern’s use of rhythm border on the bizarre reducing rhythm to almost insignificant on one hand and finding stacks of superimposed regular meters on the other. This blog encourages the reader to listen differently, finding larger chunks and direction in music and using our sensation of human motion to inform and guide our enjoyment of sound.

Our education system has instilled in many of us a table approach to rhythm based on the subdivision of a beat, coloured by the presence of a strong pulse. We think of the music as based on regularity and there is little doubt that for the majority of people this constancy is appealing. In “serious” music there is frequently subtle alterations within the regularities. When you engage with Schumann’s “Carnaval” as listener or performer you will encounter in the cast of characters “Paganini” and “Chopin” both of which demonstrate remarkable rhythmic ingenuity which demonstrate Boulez’s comments well, and it seems to me that the whole work is easily as rhythmically interesting as its more well-known use of melodic cells.  We should not forget either that the cross rhythms of music by Schumann and Liszt were developed by Ligeti in the 20th century, more on this later.

Teaching a module on African rhythm (and having the pleasurable experience of working with two different African dance groups) increased my awareness of the huge influence interlocking rhythms had on post 1950’s music. These go further than their obvious influence on minimalism and landmark works like Reich’s “Drumming”. For a quick overview of how to interlock rhythms I suggest this pithy but valuable article:

One cannot consider the use of rhythm without touching on regular cycles of rhythms. One of the oldest styles of music where periodic repetition features is classical Indian music and the use of Tala. A general guide to terms can be navigated from here:

while this site has a wealth of pages related to using cycles, drumming techniques, etc.

and we cannot consider cyclic composition without reference to Messiaen. This PDF provides considerable food for thought on rhythmic construction, and offer a wealth of ideas for aspiring composers:

I won't be the first to notice that notating rhythmic patterns can be a slow process particularly when using more contemporary approaches like adding very small values to each pitch (e.g. to create a rit. or subtracting values to suggest accelerando). I have found that using the grid systems on DAWs has a number of advantages over conventional notation. The example below demonstrates how layering an expanding/contracting number sequence can be overlapped with ease on a grid (number sequence 7,6,5,4,5,6,7), overlapped three times. 

While this resulting rhythm is within the performance capabilities of many ensembles, the difficulties are increased as values are taken to 32nd notes or shorter and layers increased. The desire for a precise performance may be challenging for humans but when working with computers these concerns can be put to one side, as this comment on Babbitt’s ensembles for synthesizers notes:

Babbitt's use of the synthesizer at once illustrates one of the primary advantages afforded the composer of electronic music: the ability to move with great speed and perfect accuracy among an infinite array of timbral, rhythmic, dynamic, and other gradations.

If we are to consider the use of rhythm is a more fluid manner than results from precise notation or the use of precise notation to develop a degree of complexity that breaks down regularity we would do well to consider this definition from wiki:

In the performance arts rhythm is the timing of events on a human scale; of musical sounds and silences, of the steps of a dance, or the meter of spoken language and poetry. Rhythm may also refer to visual presentation, as "timed movement through space" and a common language of pattern unites rhythm with geometry.

This broad definition widens the association of music so we can include Islamic art, the spoken word, not just poetry, the occurrence of sounds within an improvised framework and human movement. Considering “human scale” is also significant whether we work from rhythm to pitch by speeding up events or expanding rhythms designs to very long term events. The relationship between pulse and pitch is made clear on this page in an entertaining and informative collection of examples:

Recently I have been using the Native Instruments Reaktor instrument to explore the alteration of rhythm as a continual process, recording the whole event (normally lasting two to three minutes) and then treating the continuum to a variety of rhythmic treatments particularly reverb and contrapuntal layering. In one sense these are two extremes of the same process, though it is a matter of choice to make counterpoint an exact repetition. One of the fascinating outcomes of such processes is how a complex wave form can develop a several distinct identities as our perception is directed away from one aspect of scale to another.

Using the Reaktor blocks to create continuous streams of gradually evolving sounds stimulated the desire to isolate specific sections and further develop these in their own right. These fragments frequently imitated the rhythms we associate with human activity. Having recently read Steven Pinker’s “The Stuff of Thought” his verb lists of human movement constantly came back to mind. His terms of motion fall into four groups, the manner of motion, change of state, emission and extinction. The groups have been reduced to those which are more distinctly related to music.



I like the fact that bounce and swing relate (in my mind) to jazz, and as I worked on the other words noticed how some suit particular historical periods better than others, Schmann’s Chopin in “Carnaval” for the first two while revolve, slide whirl and spin are all appropriate to Paganini.

Change of state




And finally



The importance of human movement on design in music demands a different focus to the manipulation of small scale events in a grid formation of note lengths, it occupies a larger scale, gradually changing gestures which arise regularly when working with electronic music, but are also evident in the feedback between electronic and acoustic music. Let us consider the gradual transformation of a pitch or sample over a relatively long period of time and apply the terms gliding, humming and decreasing. It is possible to sustain these characteristics without writing regular rhythmic motifs or clearly defined rhythmic patterns such as gradually increasing or decreasing event lengths, as with Poppe’s “Zw√∂lf”, his discussion of the work is available here:

Using these larger scale events does not exclude the possibility of introducing regular repetition, a drone can have a pulse. Oscillating between chords can establish harmonic rhythm and represent the above characteristics. This way of thinking about rhythm is now new, it is possible to detect such characteristics in music of earlier periods, listen to Bernstein conduct the first movement of Mahler’s fourth symphony and make your focus the use of rhythm rather than melody, and it reveals a fascinating landscape of ever changing large scale paragraphs.

Before I leave this aspect of rhythm those who have an interest in computer music may find something of interest in the composition, “Flight”, which brings together many of the elements discussed above:

Rhythmic design sometimes acts as the main unifying aspect of music, dance music benefits most from the repetition of rhythmic cells. “Rosenkavalier” stands as one of the most graceful and inventive examples of a work saturated by triple time rhythms, binding the music into pastiche, quotation and nostalgia for the outdated waltz. Ravel uses the waltz as a vehicle for a different but related purpose, “La Valse” being a highly charged response to the many ways that the Great War cleaved the pre- and post-war periods of art and music in two. I cannot hear the music without hearing the waltzes as a progression from bitter-sweet innocence to a final catastrophic conclusion, a biographical account which this Rattle performance articulates so well.

Listening to the seventh minute onwards, the music increasingly distances itself from Johann Strauss and approaches post Rite of Spring thinking. This work is signals a turning point where rhythm becomes a tool to represent psychological factors.

More has been written about Stockhausen’s mathematical organisation of rhythm than his understanding of its psychological use, but both play an important part in his development.  “Gruppen” stands as a significant landmark in rhythmic construction, it uses a scale of twelve tempos (associations are made with 12 tone design, as may be seen in the link below). He develops a technique of forming relationships between smaller units and a central duration, much as overtones relate to the fundamental tone in the overtone series. The use of numeric proportions and ratios to determine the arrival of events within, and the total duration of sections to the whole composition. This link offers a useful guide to the thinking in accessible language

and a link to the music:

I will suggest that you use the table of verbs above to hear the music as larger paragraphs of events and judge for yourselves if it makes the music more accessible.

If we think of rhythm as events within a time frame their duration may vary from very small to huge according to the composing intention. Conventional uses of rhythm often articulate small scale events which are extended by exact repetition, regular phrasing and so on. Within this context accents and pauses all become highly effective in adding to the emotive content of the music. What happens when the scale changes and the music consists of a single pitch being subjected to microscopic changes is a change of gravity.  The alternative to this traditional view of  gravity, created by subdivision of a beat, is to create a different type of complexity that can be characterised by an overall characteristic, e.g. a crescendo, or have it associated with pitch to effect, e.g. a glide (back to verbs of human motion). What one is working at in these circumstances is a “packet” of activity which may or may not interact with its neighbours.

Those who enjoy Ligeti’s music may be aware of some of the above features in his music, and may have noted that his most adventurous approaches are in the use of rhythm while his harmony and structures are more conventional. Ligeti’s own comments on his Piano Etudes of 1985 are well known but deserve a quick summary, and a return to some of the issues raised earlier regarding Romantic music. He states that his music has foundations in the piano music of the Romantic period and the music of sub-Saharan regions. At first sight these seem to be at opposite ends of rhythmic design, one presenting a real or designed rubato and the other a web of precise interlocking rhythms. The feature of Romantic music that he considers is the crossing of 3x2 and 2x3 beat patterns and with this in mind the connections become more akin. It also makes us realise that the growth of minimalism is more backward looking than some might realise.

As for the complexity of organised rhythms losing their surface identity and adopting a new macro identity Ligeti notes that:

It is possible to beat both a duple or triple meter to the (African) patterns….there are no accents and consequently no hierarchy of beats, only the smoothly flowing additive pulse.

I, of course, would underline flowing.

For those who would like much greater detail on Ligeti and his use of rhythm I strongly recommend the PDF Ligeti, Africa and Polyrhythm:

Like all architects composers are always considering design, the shape of our music is as important as the contents which populates it. In some music there is little balance between shape and content, Cage’s “Atlas Eclipticalis” can be considered as a contents works, a link is offered to provide an opportunity to hear the effect of zero-gravity rhythm.

As a student I recall a performance given by a large group of performers of a text composition from Aus den sieben Tagen”, Richtige Dauern, I quote the opening lines,

Play a sound

Play it for so long

Until you feel

That you should stop.

Within the group were musicians who were clearly more strongly anchored to rhythmic designs of earlier music, and the result was a curious imbalance which I feel that Cage would have been more open to than Stockhausen.

Having suggested zero-gravity, packets of events and human motion as alternative ways of considering rhythm the next stage should be to consider how these events can be made to behave within the context of a larger scale composition. In part my compositions relating fractal art and music have considered both sides of the coin, constraint and freedom. It may be some time before I feel confident enough to lay down some guidelines, so the alternative is to get busy and explore how to leap, glide and slide your way through music.