Wednesday, 9 August 2017

The analyst’s toolbox

Making sense of musical information

In recent blogs we have considered the matter of complexity and simplicity in music. Most of us as well aware of the planning that goes into making a coherent pathway through a composition. Many of us have seen the Beethoven sketchbooks and his furious deletions in an effort to find the best of the possible routes to be taken to progress logically through a composition. A coherent pathway is for many listeners essential to the enjoyment of music, and coherence comes through repetition, sequences, confined tonal frameworks, adherence to convention and similar signposts. Without the guides that make a pathway a motorway many listeners give up on the music they hear, but some of us prefer the less well walked routes and a few relish the opportunity to take a risk and endeavour to encounter different landscapes.

Evolution has developed us into creatures that establish pathways through our daily experiences to guarantee our survival. In our time the evolutionary process empowers us by finding patterns of activity in finance, science, literature and the arts, and how we hear music. In order to clarify this let us begin with an extreme viewpoint and examine how we react to randomness. As randomness can mean different things to different people let us use the definition

The quality or state of lacking a pattern or principle of organization; unpredictability.

Randomness is the antithesis of the quality which for many is the essence of art and music, order. Cage’s introduction of indeterminate procedures marks for many the point where sound and music part their ways and all that was familiar becomes alien. However, if we examine our responses to the unpredictable without bias some fascinating aspects come to light.  

Humans are not well designed to accept randomness, we impose by instinct order onto random information and we filter that information to impose a narrative on our experience. One recent example of this revealed itself to the listeners of MP3 music on devices that offered random selection, the problem being that listeners considered the results insufficiently random. The selections were considered to be too closely linked. For an experienced listener one might argue that there may be limits imposed by the listener’s own collection of music, but whatever the restrictions because randomness lacks pattern clusters will occur. The problem for the technicians was how to identify sufficient differences to satisfy the listener’s notion of ‘newness’. The solution to the problem was to shuffle the entire content of the player so that each piece was played once and not played again until all others had their turn. If the owner had 10 different works he or she should have the full range during a run or car journey, but if, like me, you have a large collection you are likely to switch off your player and in this case the shuffle starts again.

So powerful is the characteristic of imposing order on events that it creates life-destroying characteristics, the “gambler’s fallacy”, the notion that a string of losses (denied expectation) means that a win has to occur. Translating this to the MP3 player is like saying “I should have the Tchaikovsky next” when there are several hundred composers on the device.

We are caught in an ancient dilemma, predictable patterns are desirable but we find rapid change both stimulating and worrying, we have to accept that there may be a tiger in the tall grass, as there is in Rousseau’s wonderful painting “Surpris!”

What happens when (if we chose to do so) listen to haphazard, unpredictable successions of sounds? The answer is that at times we will sense continuity which we will cluster into an event and at other times discontinuity, the alteration of these frames will impose over time a sense of order. If restrictions are placed on the origins or nature of the sounds the clusters will be more frequent, and the restrictions may be one of any of the musical parameters.

In the blog on simplicity diagram I marked a point at which musical matter (pitch, rhythm etc.) 'evolves' into information.

How does information arise, how do we recognise it, is it just a process of chunking different materials so that matter evolves into a more distinct form? The following table sets out to categorise the actions we engage in when encountering information, whether organised or not. To demonstrate its use let’s use the opening pages of Ligeti’s Etude No 4 for the piano, “Fanfares”.

Making sense of information

R.H. / L.H. activity
Main design
Rhythm / ostinato 3+2+3
Part to whole
Changing rhythm of dyads (constrained by ostinato)
Progressive change
Expansion of original dyad phrases (imbrication)*
Changes of register
Accents on dyads, suggested bar line downbeat
Bartok dances

*The imbrication or layering of the four lengths are shown by / marks:

First appearance                 

a 3+2+3+3 
b 3+3+2+3 
c 2+3+3+2
d 3+2+3+3 

Second appearance

a 3+2+3+3 
b 3+3+2+3 
c 2+3+3+2

b     3+3+2+3

a         3+2+3+3

Third appearance                b, c, d / c and then a / c / b / a

Though these combinations of 4 length units will not be noticed by the listener they are of some interest to the analyst at this point in that they may predict later events (or not).

We should notice:

A recurrence of particular clusters of information
Particular types of change
A pattern that reveals the rate of change
Some evidence in the opening pages.
Cyclical events
Stability or capriciousness to clusters of events
Lengths of longer phrases between LH / RH exchanges small scale changes

I found this particular approach useful when approaching the Feldman work for Bass Clarinet and Percussion discussed in a recent blog “Space and the composer’s toolbox” and with “Cartridge Music” by John Cage “What I do, I do not wish blamed on Zen”.

The difference between indeterminate music and organised music is that the regularity of what we perceive to be a cluster of events is going to be far higher in the latter. The process of articulation in performance plays a significant part in our perception of these clusters; articulation is part of the toolbox for both composer and performer, though in performance they may disagree with each other! Mahler’s use of markings on his symphonic scores should produce greater similarity, but experience shows there is great diversity. Articulation is so important in the process of creating information from text that it deserves a blog of its own.

I shall close with this observation on information:

A piece of information is considered valueless if, after receiving it, things remain unchanged.

Saturday, 29 July 2017

The problem of musical simplicity.

The first problem is the multitudes of meanings which are attached to the term simplicity, these include ease and uncomplicatedness, plainness, minimalism, modesty, artlessness and naivety. The second problem follows on in that if the music is plain or uncomplicated there is little to engage with, it does not support an often held view of art as profound, insightful, reflective.

We intuitively hold to the view that one requires a great deal of knowledge to appreciate the working of complex music and less to grasp simple music. I believe that the case can be made that the process of writing simple music is every part as challenging as composing complex music, and the degree of empathy and involvement for the listener is similar if not the same for both.  

For this introduction I shall make comparisons with texts and our skills of reading, speaking and listening in order to tease out the appropriate qualities of simplicity for musicians and artists.

The process of reading is recognized as complex, and it is one that every blog reader will be familiar with, so I shall be as brief as possible. There are five recognised stages of reading, emerging pre-reader, novice reader, decoding reader, fluent comprehending reader, and expert reader. The final stage relates to how much reading is integrated into our daily life and the type of literature we encounter as it has a direct bearing on our reading comprehension and interpretation. It is this process that liberates the reader and offers new insights outside the intention of the author.

Reading is a means of sharing information and ideas. It is complex because there is an interaction between the author and reader, and that process is itself creative and critical. The creativity is on different levels but may include image forming of characters and places, and in another context the association with ideas outside the text. There are as many interpretations as there are readers.

So let us compare a simple and complex text (with thanks to the plain English society):


Your enquiry about the use of the entrance area at the library for the purpose of displaying posters and leaflets about Welfare and Supplementary Benefit rights, gives rise to the question of the provenance and authoritativeness of the material to be displayed. Posters and leaflets issued by the Central Office of Information, the Department of Health and Social Security and other authoritative bodies are usually displayed in libraries, but items of a disputatious or polemic kind, whilst not necessarily excluded, are considered individually.


Thank you for your letter asking for permission to put up posters in the library. Before we can give you an answer we will need to see a copy of the posters to make sure they won't offend anyone.

This is a factual text, music is an expressive art form but one might associate it with a five finger exercise, a study in technique where we know there is a requirement to fulfil. However we can take something useful when we examine what makes the second simple, the differences that might interest the musician involve sentence and overall length, keeping to one topic per sentence, appropriateness of vocabulary.  

Differences between reading a language text and music are considerable, in no particular order words are used in either horizontal or vertical formations (depending on language), music regularly encompasses both; internalization of text is more common with words, the external voicing of music has articulations, length, speed imposed; group readings exist (in Welsh “Cerdd Dant” mix music and speech in a formal design) but are less frequent while group performance in music is common. The most obvious difference is that the musical parameters pitch, tempo / duration, dynamics etc. make the process of reading music more complex than text. There are historical reasons for the precision of scores involving communication (or the difficulties of communicating) between musicians in earlier times. The situation is of course far more fluid now.

Moving on, let us consider the listener and his or her understanding of simplicity and complexity through the use of language and speech. Every person has the capacity to acquire their mother tongue, grammar is integrated into understanding of speech from infancy. Early meaningful communication is key to language learning, and often the reason that acquiring a new language fails is that students sometimes use grammar guides and the like in place of significant and useful speech. What can be even more infuriating to the second language learner is that knowing rules of grammar does not result in speaking or writing well. Like reading one can make speech obscure by using a different vocabulary, even then context aids understanding. Practice with specific vocabularies can ease the process of listening, how natural the language becomes depends on the amount of time and effort put in to make communication possible.

Many musicians are brought up within a musical environment, but there are examples of non-musical families suddenly finding a musical cuckoo in the nest. This extract demonstrates how much works still needs to be done to understand the “naturalness” of musical ability:

Yet evidence for the contribution of talent over and above practice has proved extremely elusive. In another recent study, Ericsson and his colleagues studied young pianists and violinists in their early 20s at the Music Academy of West Berlin, Germany. They asked the music professors to nominate the best young musicians, those who they thought had the potential for careers as international soloists, as well as others whose potential they regarded as not quite so great, and a third group who were most likely to become music teachers. Hence, in terms of achievement, the first group comprises the most exceptional musicians, the second group the next most outstanding, and the last group the least exceptional.

If "talent" is the primary factor, we might assume that these three groups differ in their innate giftedness for music and that this explains their different levels of achievement. If a person is innately gifted, then he or she can very rapidly attain an outstanding level of performance once the basic skills and knowledge required have been mastered. Yet Ericsson and his colleagues obtained a surprising finding: the best musicians had simply practiced more across their lives than the next best ones, who in turn had practiced more than the ones likely to become music teachers. Each of the musicians was asked to estimate approximately how many hours a week they had practiced each year since the outset of their musical training, and these estimates yielded cumulative totals of about 10,000 hours for the best musicians, followed by 8,000 for the next best ones and 5,000 for the least accomplished. The musicians also kept diaries for a week, recording their exact amounts of practice, and these yielded comparable differences, suggesting that the retrospective estimates were roughly accurate.

this additional short extract reveals some surprising information:

Since the musicians were regularly taking musical grade exams, Sloboda, Howe and their colleagues were able to use this as a measure of musical progress and could therefore calculate the amount of practice that took place between successive grades. The surprising result was that the most gifted children required just as much practice as the less gifted ones: in fact, if anything, there was a tendency for them to require more. For instance, the most gifted group required on average 971 hours of cumulated practice to reach Grade 4, while a less talented group took 656 hours.

The document is available to read in full here:

Reading, speaking fluency requires considerable engagement to find that “expert” level, as does musical performance. This does not mean that the expert will only engage with a complex world only that they have the tools to do so.

Speech is not always on a performance level, it can be casual, music tends to be directed at an audience and as a consequence there is more planning, regularity and order, which should make it more direct if not simple. Prepared speeches will display many characteristics with composition (without the articulations written in), a speech requires order, a clear pathway is essential to retaining the audience. Can simplicity be ascribed to well-designed pathways, where the clearer the route the more immediate and powerful the resulting experience? Later on I will consider in some depth two works by Arvo Part from different periods of his life to make the pathway argument clearer, both compositions share characteristics but occupy different stylistic worlds, for some listeners it will be surprising that they share a similar degree of simplicity.

Having looked at reading and speaking let us turn to the process of writing / composing. I am certain that most composers have encountered music which communicates with a large audience in a style that is accessible to experienced and inexperienced listeners alike. The music may display full control over musical detail, style and convention without demanding that the listener is familiar with these. Such works might provide an immediate positive emotional response which in the best examples endures repetition and stands the ‘test of time’. In my mind Barber’s “Adagio for Strings” provides one example of simple and direct music. Even when playing the music in an ensemble one can appreciate the overall shape and progression of the musical argument, it is almost as if all the musicians are breathing together as the music progresses. While examples of simple and effective music like the Adagio turn up in every period of Western music they are in the minority of works.

There are many lovers of “Art” who hold a notion where age provides artists with a wealth of experience which is distilled into a spiritual elixir. This distilled essence makes their work ever more refined, accessible, even simple, and may offer a direct route to experiencing sublime or (near) religious experiences.

Is there a parallel level of expertise in writing which leads to being able to compose in a direct and accessible style? Is there some particular technique to simplicity? Our first question should be directed at recognising what is simple in music, and if we can identify what is simple, how long can this characteristic function in the context of a composition before it is ‘corrupted’ by complexity? Let us now examine two works by Arvo Part to explore a little further the nature of simplicity.

The spaces of “Perpetuum Mobile” and “Frates”.

“Perpetuum Mobile” presents a gradual crescendo and diminuendo, ranging from a calm pppp at the opening to ffff at the climax, marked by percussion. It is a very simple shape for the listener and for the analyst a complex web of organised detail. Within this detail instruments are given individual roles, there is a scheme for the duration of values, primarily with a process of reduction and repeated note values within prescribed lengths of time resulting in a mesh of cross rhythms. There are examples of association of values with texture, strings longer values, woodwind shorter, and vice versa. There are also controls on the number of instruments playing at any one time with associated dynamics. As is often the case with such complexity the ear picks out regular beat patterns and in the Neeme Jarvi recording I have one can pick out a regular four beat pulse towards the climax.

The main body of the music is composed of six sections each of twelve bars. Sections are articulated by changes of register and instrumentation, but dovetailing between sections make these less obvious to the listener so changes between higher and lower registers are less well articulated. To add to the pulsing character of the music in the opening sections each paragraph has a crescendo organised as initial dynamic + 2 increases and return to the initial value.

It would seem that “Perpetuum Mobile” is a work more concerned with dynamics, space and rhythm than pitch organisation, the work being of 12 tone construction. The row forms two identical hexachords of 0,1,2,3,6,7, but there is no symmetry within the pitch arrangement.

Just to recap the main point, “Perpetuum Mobile” has a simple overall shape, discernible on first hearing, general characteristics that require a good ear and some guidance, and complex detail that would emerge with study and a score to hand. Let us compare the situation with the later work “Frates”, which most would say uses a different musical language, e.g. the pitch formations have changed from 12 tone to far more conventional harmonies (key signature included). There are similarities between the works, let us explore some of these before determining if “Perpetuum Mobile” stands in the world of complexity and “Frates” in one of simplicity. There are a number of different versions of “Frates” but there are common features to each. A fine performance of the violin and piano setting is available on You Tube with a score:

At the first encounter the main features are the alterations between two musical events, one long and one short. In this violin and piano setting the short two bar section’s first appearance is dramatic. It emerges from the gradual violin crescendo with an accented bass register forte chord, echoed at the piano level, (both bars in 6/4). This event is regularly repeated at figs. 2, 3, 4 etc. in the score. While the piano repeats the same musical figure the violin part undergoes small changes. These two bar interjections form a frame for the second and longer musical phrase. This second event has a rhythmic design, the bars expanding from 7/4 through 9/4 to 11/4 x2. The listener can feel the music stretching and expanding in the process, and one may view it as either the frame being enlarged to accommodate the melody or vice versa.

The dynamic design is a process of gradual movement from quiet (ppp opening to climax – fig. 6 – to ppp close. In the version for percussion and strings the dynamics are progressive, the crescendo to the climax is shown in the table:

Perc. pp, Vlc./Cb. div. pp con sordino  
Vl. I div ppp c.s, Vl. II pp c.s.
Perc. pp, Vlc./Cb. div. pp c.s.
Vl. I div pp senza sord, Vl. II p s.s
Perc. p, Vlc./Cb. div. p c.s.
Vl. I p, Vl. II p, Vla. mp
Perc. mp, Vlc./Cb. div. p c.s.
Vl. I mp, Vl. II mf, Vla. mp, Vlc/Cb div. mp c.s.
Perc. mf, Vlc./Cb. div. mp c.s.
Vl. I and Vl. II div mp, Vla. mf, Vlc. section mp, Vlc./Cb. mp c.s.
Perc. mf, Vlc./Cb. div mp c.s.
Vl. I f, Vl. II div f, Vla. ff, section Vlc. f, Vlc/Cb div f c.s .
Perc. mf, Vlc./Cb. div f c.s.
Vl. I mf, Vl. II mf, Vla. f, section Vlc./, Vlc./ c.s.

Let’s take a quick look at the piano melody in the longer section, the design here can be reduced to three sets, the first (C’, B flat, D, C’) to 0,1,4, the second expands to 5 pcs, 0,1,4,5,7 (5-Z18) and the thirds to 0,1,3,4,6,8,9 (7-32), the second half of the melody has the same formation. I understand that there are other ways of looking at this melody but it is one type of logic that reveals the system. Each successive playing then transposes the phrases from C’ to A to F to D to B flat, G, E, C’. This taken as a collection forms 0,1,3,4,6,8,9.

If one takes the bass of the longer event, the most obvious detail is the drone on A, but take the 11/4 bar just before 5 and the chords form the collection 0,1,3,4,6,8,9.

We have two very different sounding works, one which we would immediately describe as complex and the other as simple, but in reality the approaches are very similar.

The gradual expansion of information in “Frates” enables the listener to anticipate events, as does the plan for the “Perpetuum Mobile”. Simplicity is deeply associated with anticipation. Let us compare the expanding cells of “Frates” with Bach’s C major prelude. Once we have heard the first bar we can anticipate some of the features of the second, and once we have the information on the first two bars we can anticipate even more events. Satisfying us by meeting our expectations and surprising us by offering a different harmony is part of Bach’s artistry. If providing information to create anticipation is the definition of simple then complexity is a state in which we require greater degrees of information over a longer period. The challenge for the composer is to offer the listener a pathway to experience the connections. If this is impossible then the listener can only experience chaos.

Let us expand a little on the idea of pathways for the listener. The first Prelude of the 48 Preludes and Fugues has been taken as an example of simplicity because we can anticipate much of its content. However, a well-educated listener might say the same of the Fugue or fugal writing in general, you may wish to refer to the table in the following link if you are less familiar with fugal construction:

Most of us would agree that the music of the fugue is more condensed and the harmonic rhythm and texture changes more irregularly than the prelude. The internal quantity of events (as in this example) produces no guarantee that we can describe the music as simple or complex. You may recall that the Part “Perpetuum Mobile” has a simple auditory structure arising from a complex and condensed web of detail, if you like it is content rich, while “Frates” has less content (particularly in respect of its use of repetition) presented within a structured lattice. The Part works demonstrate contrasts of expansive space and compressed events. Composers and performers can make use of tempo to temper the sensation of compression, if harmonically and rhythmically dense music is played slowly the result is a feeling of greater space, as an example listen to the Gould performance of the E major Fugue from book 2 of Bach’s 48.

Another outcome of compression is that we chunk events and may hear (unintended) patterns of regularity, this became clear to me when writing a set of pieces exploring a method of creating fractal music through sampling.

Touching again on previous blogs on this site Webern uses the words intelligibility, clarity and comprehensibility in his lectures to describe the necessity for unambiguous music. After Webern intelligibility and order become inseparable. If order and simplicity are the interwoven then serial music is simple, and in some respects it is, but it is not accessible to all. Conversations with other musicians on this topic showed that associations are sometimes made between tonal music as simple and atonal music as complex. There are conventions of style that make this argument seem convincing, but one could write a work based on two whole tone scales with a dance rhythm that demonstrates simple characteristics and still fulfil the term serial. Conversely one could write a tonal work with complex cross-rhythms and surprising dissonances that has all the characteristics of complexity.

Looking, for a moment, outside the musical universe I understand that St. Thomas Aquinas held the view that “God is infinitely simple”. The argument goes that as God is spirit he is not composite and therefore simple. The idea could be introduced into this blog that if a work of art expresses or realises a single intention it is simple, like the black canvas by Kazimir Malevich, but then the outcome, like Cage's 4' 33 is not as simple as it seems:

Do we perceive music as simple if it has one characteristic that we can refer to throughout the piece (pathway)? Take Bach’s “Goldberg Variations” which has a fixed bass with built in repetition. The music moves through various degrees of complexity with the outer sections standing as perhaps the most astonishing example of elegance in all musical history. Elegance, the process of being selective in order to achieve a perfection in workmanship and design should not exclude complexity but is more often associated with simplicity.

Let us return to the single characteristic or binding feature as defining simplicity and consider the process of variation. By convention the approach taken up to the last century has been to construct a melody and develop from it streams of differing degrees of complexity without losing the connection with the origin. The process can formally work with a reversal, moving from initial complexity to simplicity, but that was not the thinking of pre20th century composers.

When one considers the enormous number of possible choices open to a composer at the start of the creative process it isn’t surprising that order emerges as a key to mastering the elements, and why Cage’s re-evaluating of order is so revolutionary. I don’t want to overplay the importance of the variables available before writing a note of music but they have a part to play in moving towards a working description of what is simple and what is not.

Let us illustrate the development of options by analogy, matching the evolution of musical styles with three games, the early modal works are like a game of tic-tax -toe which taking into consideration possible symmetries has (so I am told) 26,830 different game possibilities. The tonal system might be compared to a game of draughts (or checkers to our US readers), here the number of legal positions alone is said to be 10 to the power of 20 and its ‘game-tree’ moves take it to approximately 10 to the power of 40. To complete the picture, chess, which may be compared in the analogy to serial music has not as yet had its possible legal positions calculated accurately, a range between 10 to the power of 43 and 10 to 50 suggested. The analogy seems to hold when we consider modes and the number of chords available, the major/minor scale system, along with key change, and the near 12 million different classes of tone rows. However the analogy weakens when we consider the other parameters active in performance, to say nothing of decorations and embellishments in different styles of music.

No matter which style we consider the amount of available information is huge and the strategies greater still. Having said that the greater the number of choices the easier it should be to find clear, unambiguous moves. Put another way elegance is possible no matter how great the content.

Some styles suggest simplicity in their stylistic descriptor, such as Primitivism. This art form plays with simple representations of figures and geometric shapes. In the music there are features which suggest stark and undecorated presentations but are they simple? Let’s begin with its use of tonality. This is usually heard as ‘substitute’ tonality through asserting a given pitch or pitches through repetition, duration and accent. Modes are employed along with octatonic / synthetic scales. These are often used as pairs of modalities or tonalities, and this is the first of the points where the music begins to move away from simplicity. The resulting harmony can form conventional triads but are more regularly stacked to create polychords and clusters. Where the complexity becomes most apparent is in the use of asymmetric meters, and similar sounding cells expand and contract with ever changing time signatures. Different meters may be presented simultaneously to create polymetric events. Taking this mix with additional use of accents, instrumental articulations and rapid alterations of dynamics etc. simplicity is not in play.

Is naivety a quality of art music? Silvestrov has music described as Naïve Musik which is (as most of his work) reflective, post Romantic. The third “Fairy Tale” is available on You Tube

This music is melancholic, has an affinity with Mussorgsky, makes liberal use of repetition, has the quality of a musician improvising around a partial scale of C, D, E flat, F, G. Then we have to consider the title, many composers still use myth as inspiration, but this in conjunction with the term Naïve brings us to child-like innocence rather than the hallucinatory and violent themes of say, the Mabinogion. Innocence brings us to another aspect of simplicity, the ability to create as if unaware of the multitudes of experience available, to deliberately restrain oneself. In the present age to adopt this approach without showing the capability and knowledge of the other could prove problematic to the composer in terms of unwanted criticism. Should you wish to assess your own openness to this situation play Lawrence Crane’s Seven Short Pieces:

Scanning through the various styles of the 20th century onwards one may say at first sight that minimalism is simple on several different levels, particularly when the music is determined by a numerical process, but techniques like the use of polyrhythms and phasing create a wealth of alternatives. As a blueprint minimalist scores may appear simple but they can emerge as multi-layered complexity. If you would like to test your responses to complexity with two works that use polyrhythm listen to Ligeti’s Etude No. 8

and compare it with Reich’s Drumming (part 2)

I would appreciate any comments on their relative merits in terms of the topic under discussion. The following grid outlines the use of polyrhythms in the Ligeti Etude:

This link offers information on the use of polyrhythms in Reich's "Drumming"

In my view the most important element to consider when considering complexity is time. Even in music where each parameter is notated the space occupied changes, sometimes dramatically. Listening to familiar repertoire we are always reappraising our expectations. An example that comes to mind is the final section of the final movement of Shostakovich’s 5th symphony, conductors have so many different viewpoints on this music that sometimes I think I am listening to an alternative version. Nurtan and I have been discussing Bernstein’s performances of Mahler after reading Murakami’s Conversations with Ozawa, and the first movement of the fourth is another example of stretched time. When the music has a degree of rhythmic or harmonic freedom, as it frequently does post 1950, the level of reappraisal increases.

Psychology has its own view of simplicity and concerns itself particularly with memory. Take the following patterns, 9, 11, 13, 15, 17, 19, and 5, 12, 19, 48, 20, 15. Which is simple? We would immediately select the first, but it is more complex than it seems. If the second pattern is seen as birth and death dates of a loved one (5th December, 1948 to 2015) the second is simpler, and over a period of time is more likely to be remembered (remembering which number started the sequence when it has no particular meaning makes it difficult to recall). Perhaps we should add to the notion of anticipation the ease of recall after the event, this doesn’t make the music any more simple but affects our perception of the music as being simple.

I would like to offer a table that could be used to classify works leaning towards or away from simplicity. Should you wish to use it I suggest this Schumann work (first section) for a first attempt:

and follow this with the Ligeti (web address above).

Now let us use the “Towards Simplicity” mind map to assess the complexity of the opening pages of Lemma – Icon – Epigram by Brian Ferneyhough:

There is a simplified breakdown of the material on the manuscript below. Double clicking will enlarge the image.

Observing the left hand side of the mind map we can place the pitch construction by pairs of dyads with a semitone characteristic as very simple. There are localised pitch repetitions which form a chain, there are two major repetitions in the opening pages. With repeated listening these are retainable so comparisons and a degree of expectation is established.

The rhythmic elements take us into the realm of the microscopic, grouping 64th notes into triplets, quintruplets etc. or adding a dot to extend the 64th to a 96th. For the average listener such changes do not register, for the composing intention it illustrates a world of stretching and contracting time, like trying to be aware of the changing size of individual snowflakes. This of course does not in itself make the music complex.

In terms of chunking material there is little evidence of imbrication, no counterpoint, transposed events (sequences), so that keeps us on the left hand, simple, side.

Flow is a little more problematic; any material that uses periodic repetition has flow, superficially the music flows, the pairs of semitones become trills, the sense of paragraphs of sound moving to a particular point are all there when one listens. The textures also move towards simplicity, one instrument, no sound processing, and historically we can make comparisons with jazz on one hand and Charles Ives on the other.

There is of course a lot of detail, many events, rather like the snowfall discussed earlier. It stands a whole world away from Satie’s Gnossienn No. 1, but the fact that there are multiple ways of considering simplicity should not take us by surprise.

Rather than closing the blog with conclusions I would like to offer the following table which suggests the flow of information from encountering sound to experiencing it as profoundly moving music. I hope to expand on the matter of “information” in a later blog.

 My thanks to the musicians who have offered views on the matter of simplicity and Anthony Littlewood in particular for highlighting the function of order on our perception of what is simple.

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!