The Rules
The Icon
The Clock



Blessed be all metrical rules that forbid automatic responses,
force us to have second thoughts, free from the fetters of Self.

Wystan H. Auden
(1907 - 1973)


I hate composition theories, philosophies of new music, recipies of how to compose, prescriptions of style. They're mostly unreadable, unintelligible, boring, and always humorless - whilst their necessity remains a big question.
From the great classics, the for-all-time-composers, with names such as Bach, Mozart, Haydn, Beethoven, Schubert, Chopin, no written composition theory is known - not a word. Their real answer to the question of how to compose is written in notes, on staves. At most they have a title (like Kunst der Fuge, Sonata Facile, or Forellenquintet) and an indication of the tempo, but for the rest no word about why these notes are written the way they are. Apparently they can explain that themselves. Just like Beethoven's Grosse Fuge that has now reached the interstellar space, beyond our solar system, with the first Voyager.
This majestic sovereignty stands in great contrast to the haystack of theories, bad philosophies and unmusical ideologies that precede most performances of 'modern music' today.
Why is this so and when did it start?

At the conservatory you learn that musical history can best be divided in periods of one and a half century (or roughly five generations - or 'Jesus-lives', if you like). The period that's now ending started with the opening-bars of Tristan and Isolde, in the eighteen fifties . It is called the 'romantic/modernist' period. I propose to let it end at Saturday Night in Stockhausens 'Licht'-opera (if that point will ever be reached in the foreseeable future.)
This period would cover the 'bloodiest century of human history' , as our era is often called. It would begin in the music of a megalomaniac monster that declared that not enough jews had died in a theatre-fire in Vienna in his days, and it would end, in perfect symmetry, with an even bigger monster that recently identified with the kamikaze-muslims of the 11th september, who 'chased 5000 people into the Resurrection' ("in die Wiederaufstehung gejagt"), as Stockhausen put it.
It is a period in which 'new music' seemed to be possessed by the sadistic desire to punish its listeners with dissonance,loudness and maddening repetition, as an ideal of terrorist music. Composed by well-to-do revolutionaries who pose as champions in the olympics of moral grandstanding - the games that open with every new crisis in the realm of human conflict.
The leading figure in that development, the Lenin of composition, is doubtless Arnold Schoenberg, who, in the beginning of the last century, decided to 'free himself of tonality', and of 'the dictatorship of the consonant', fighting for 'the emancipation of the dissonant' (which in reality was the discrimination of the consonant.)
To do all that, he needed to create new rules. They would also apply for 'other materials', like in the opera Die glückliche Hand, where 'apathy equals dirty-green equals violin', and rules like that.
But even more devastating for the art of composition was the work of the followers, the believers, influential local opinionpriests and music-manipulators of the media, like Elmer Schoenberger .(In Holland we say 'Erger dan Schoenberg is Schoenberger' - an untranslatable pun, meaning 'worse than Schoenberg is Schoenberger.)

People who know my work only superficially will tell you that I'm nevertheless also a propagator of a musical 'theory', which became known as 'the tone clock'
(a word that's now in the dictionary). But nothing is further from the truth - the tone clock is not a theory but a devise, a 'tool', or a 'map', as Jenny McLeod described it. I discovered it in the heart of the chromatic realm. It is an instrument to realize chromatic tonality, as I propose to call it.

'Blessed be all metrical rules' said Wystan Auden , the great poet who wrote the libretto for Strawinsky's The Rake's Progress. He referred of course to the art of writing poetry. The rules he meant originate from the material he was working with: language. And nothing else - not from mathematics, or biology, or religion, or mysticism, or whatever secret cabalistic source.
Because that simply wouldn't work.
Composers can expand the concept of ' metrical rules' to rules for rhythm, harmony and melody, provided they originate in music. And not in mathematics (like those of Allan Forte), or in catalogues (like Messiaen's birds), or in tyrannophiliac occultism (like Stockhausen's 'Formeln' in his wonderweek Licht.)

'…rules that forbid automatic responses…', is the poets clause for his rules. They're there to avoid cliché's, to fight the ' anaesthetics of familiarity', as Richard Dawkins puts it in Unweaving the Rainbow.
A contemporary composer would paraphrase (or at least should): to forbid the
' automatic responses of diatonic tonality'. In this fight (the fight against the leading notes in diatonic harmony) the tone clock can be a very effective weapon, as I discovered in nearly a Jezus-life of composing with it.

'…force us to have second thoughts…', force us to be critical, to take some distance, or even ' to kill your darlings', as Ingmar Bergman put it. Because ' a true work of art is the result of rejected ideas', as Picasso said. Which means to me that conceptualism is the demise of the art of composing.

'…free from the fetters of Self.' A truly buddhist insight: in the creative process we learn to see our Selves as an illusion, from which we escape - like Houdini from a straightjacket.

I drew the first 'map' of the tone clock, with great excitement, in a hotel near the Berlin wall in l982, and published it later that year in the leading Dutch newspaper NRCHandelsblad.

Jenny McLeod, who learned Dutch to be able to read it, was the first composer to recognize that
'the clock simply elucidates what our chromatic language is'
and it demonstrates
'that there is a natural chromatic order,
existing of itself, by virtue of the almighty power of Number.
It belongs to all, and it can work for anyone'.

These words were very liberating for me at the time they were written (1989), because they proved that the clock was more than an idea fixe that clung to my big Self, as some people preferred to think of it.
In 1998 I went for the second time to New Zealand, and stayed at Jenny's sweet charming house, admiring the possibilities of her romantic little garden, that only needed a helping hand. I cleaned it up and build some terraces - a healthy exercise. When it was finished, she said: ' What I did for the tone clock you did for my garden.'


Intellectual innovation often requires, above all else, a new image to embody a novel theory. Primates are visual animals, and we think best in pictorial or geometric terms. Words are an evolutionary afterthought.
Stephen Jay Gould - Dinosaur in a Haystack.

The first attempt to visualize the tone clock theory was recently discovered in the Blombos Cave in South Africa. It is 77.000 years old (7000 suncycles!) and the 7,5 cm piece of red ocre is regarded to be the oldest work of art in the solar system.
The triangular carvings on it are clearly a precurser of those on the clock face, be it that they are placed in a straight line, horizontally, and not in a circle. After all, the wheel had not been invented yet, so that the Book of Nature (which is written in circles and triangles, as we know from Pythagoras and the Orpheans) was still unknown.
But nevertheless: the Stone of Blombos is to me the archeological proof that the world is founded on music.

I published the harmonic theory of the tone clock at Christmas 1982 in the newspaper NRC Handelsblad, after I'd discovered its principles a few months earlier, staying at a Berlin hotel.
You can find this essay as chapter VI in my book The Tone Clock.

Here you see the icon with the 'dashboard' of the twenty 'steerings'.

Since this publication a lot of commentaries and misunderstandings came about, as they still do. The first consummate analysis was done by De Groot and Van Dijk in Key Notes. They found no errors.

In 1988 I started, in the NRCHandelsblad, a public discussion on the tone clock, with remarkable response, regarding the highly technical nature of the subject.
This discussion is reproduced in chapter VII of my book The Tone Clock.

Others prefer Alan Forte's 'set theory', but I think his 'operations' are too bureaucratic (like the old Darmstadt-serialists), and very uncompositional, if I may say so.
See Splijt de Leeuwerik! of my book Het Componeren van de Hemel.

The tone clock marks the transition in my musical thinking from the current romantic-modernist Dionysian expressionism to a more timeless Apollinian classicism, a transition that happens quite naturally to most serious composers when they grow older.

It takes a long time and ample training to really understand and absorb the principles of chromatic tonality. In my essays I've tried to communicate them with words and musical examples, but of course, the real proof of the pudding is in eating the notes: the scores.

The tone clock theory can only be judged when it 's heard!


by Jenny McLeod

Solutions to gigantic problems can sometimes be very simple and obvious. The musical triads are a case in point. Why, in almost a century, have our composers (with rare exceptions) not been able to make more palpable sense from the twelve notes of the chromatic scale? The answer, it would now appear, is that our analytical gestalt, our unit of perception, was too small: we had rejected the old triads and have been operating at the gestalt-level of the interval, the two-note group. (We worked with musical cells and 12-note rows, certainly, and Messiaen worked with his modes, but there was no common language to refer to those larger groupings.) This was actually a step backwards, not forwards, since Rameau and later harmonic theory had already pinned down three of the triads (the natural major, and its inversion the minor triad, and the diminished and augmented triads).
Now Peter Schat has isolated all the possible triads in the chromatic system and named them. What a very simple and sensible thing to do! Why did we never think of it before? We had named the notes and the intervals, and three of the triads: certainly the next step was to name all of them. The first surprise is that there are only twelve (allowing for inversions). The next surprise is that they all have a very distinctive and individual sound character - even more so than the intervals - and are easily recognizable in their own right. Thus, contrary to expectations perhaps, these more complex groups are actually a tremendous help as signposts in the chromatic territory. By means of the triads, moreover, all larger groups as well can be more easily identified and analyzed. It is a marvel, in fact, that such a simple thing can so change and illuminate the whole chromatic perspective.
Here are the twelve triads:

Now the major-minor principle is extended. There are now eight minor triads (smaller interval first, as shown above) and eight inversions, their major forms (larger interval first, not shown). The remaining triads are symmetrical and homogeneous: both their intervals are the same, thus they have no major or minor form (e.g. I, VI, X, XII).

Schat was then to isolate the musical equivalent of the life principle in the phenomenon he calls steering. Steering is not new: it is actually the principle at the heart of Rameau's harmonic theory, but we never before recognized it as such, or had a name for it. It is the combined musical principles of growth and reproduction: i.e. the idea that any given note has the power to generate, or give birth to, a group of notes (an interval, triad, tetrad, etc.) - coupled with the idea of transposition, the power of a group of notes to reproduce replicas of itself elsewhere in the system

The seeds of steering were present already in the medieval practices of troping (in which certain notes of the established chant would open up, so to speak, as birthpoints for a newly composed phrase) and organum (or parallelism: i.e. transposition in the vertical dimension), later in the cantus firmus principle, later still in the Bach chorale preludes.
In Rameau's diatonic system, the root of the triad is the steering note. The three roots of the three primary triads (tonic, dominant and subdominant) thus form a three?note steering of two perfect 4ths (or 5ths) symmetrical about the central tonic. The steering itself, we notice, is based on a triad (IX) different from those it generates (XI-major triads).
Just as the three primary triads (i.e. Rameau's major triad transposed twice) produce between them all the seven notes of the diatonic scale (with two note-repetitions) Schat now found that eleven of the twelve triads, transposed three times, produce all twelve notes of the chromatic scale (without note repetition) - the steering, now a tetrad, still always symmetrical and based on a different triad (2). Major and minor versions of the triad were both present (twice each) and there were over two dozen variants. Schat grouped these into twelve classes, or hours (chromatic tonalities, in fact) after the twelve triads.
This was really a remarkable discovery. Messiaen indeed had already isolated the principle of limited transposition, but Schat was the only one to see that it actually worked for the individual triads in a twelve?note chromatic format. Others had certainly come upon some of the 'hours' - I found a couple of them myself and used them more than twenty years ago - but nobody suspected there was a much larger series of natural symmetries hidden within the chromatic scale - a natural twelve-note harmony,
in fact. And we had no idea of the incredibly organic purpose this natural order could serve, by way of the steering process. It was pure genius on Schat's part that he latched on to this.
The real strength of steering is that it operates on different levels: notes grow from other notes, other notes again grow from these notes. The principle is actually so simple that we hardly noticed we were doing it in the past. In the classical harmonic theory we all grew up with, the seven notes of the diatonic scale, originating as I described from a deep-level 'steering', become in their turn the generators, or steering-notes, of a further series of triads: the seven diatonic common chords. From these harmonies, further scales emerge again, to form the melodic lines of the piece. Then the whole lot gets transposed into another key: steering at an even deeper level. Chromatic decorations emerge in the melody: more steering at the highest level - and so on. Doing this became second nature to us.

Likewise in the tone clock, the notes of any hour can then become the generators of new triads (intervals, tetrads, etc.) in that or any other hour. These notes in their turn can then become new steering-notes, and so on. It is a process exactly parallel to the way a tree grows or living cells multiply: that is, it is a natural process. Every single note can always be thought of as having the power to 'sprout' a new group of notes.

(1) The 'frequency multiplication' of Boulez is actually based on the steering idea, but here very complex in its end-result. It is impossible to deduce the principle from his scores (because he treats the 12-note series as a hidden generator of the actual notes) and his explanations, though 'sufficient' in fact, have been strictly minimal, with the result that next to nobody has ever understood it.

(2) The remaining triad (X) works as a tetrad, transposed twice.
[See figure below.]
material becomes phenomenal.
A triad is a 'sub-region' of the total field. As with any chord in the diatonic system, its notes may come in any melodic order, or may be subjected to vertical inversions. The conception is harmonic, in other words. A tone-clock hour is by no means the fixed melodic sequence of Schoenberg's twelve-note principle. It is a tonality. The triads themselves may come in any order: i.e. the total field is conceived harmonically as well. Thus we now have an extremely flexible chromatic tool, a twelve note harmonic language freed from the frustrating 'paralysis' of the old-style note-row, with its so often arbitrary vertical relations, a language in which both vertical and horizontal elements obey the same laws ? laws moreover which are in principle identical with those of the old harmonic tonality! No one ever dreamed that this was possible.

Consonance and dissonance have an equal place, as indeed they did not in Schoenberg's system: there the main consonancxes were 'forbidden', an unnatural prohibition which ensured that many composers would eventually turn away from it. Being in so many respects a reaction against the past, it was incomplete in itself, where the clock is now complete. Musical democracy is restored and fulfilled. There is no combination of chromatic notes that does not find a place in the clock.
So far I have focussed on the triads and their hours, since these are central. But there is a further large range of 'interval-hours', and another huge set based on the symmetrical tetrads. There are, moreover, endlessly varied 'combination' tonalitries possible. And when controlled note-repetition and larger groupings (quintads, etc.) symmetrical and asymmetrical, enter the picture, new vistas and principles appear again. There prove to be chromatic 'equivalents' for every known harmonic process which indeed are not 'imitations' of the old tonal processes, but are unimagineably richer extensions and developments of what we now see to be fundamental musical principles. The extraordinary logic, beauty, distinction and character of the chromatic territory thus revealed, and the sheer immensity of its potential, are quite literally beyond belief. There is material here to nourish composition for centuries. We have hardly begun to realize the possibilities that exist within the chromatic scale, just as most of us have hardly begun to know it.
The diatonic scale, and its proliferation into the chromatic, are what we are experts in. Despite Schoenberg (or perhaps even because of him) most of us still operate from a diatonic mindset, and that this is what the serious composer today still tries constantly to avoid! We are thus as incontrovertibly shackled by diatonicism as we ever were, only now in a negative instead of a positive sense. (The minimalists, on the other hand, like naughty children, have simply trotted right back to the 'forbidden ground', where they sit poking out their tongue and perhaps feeling faintly guilty.)

The possibilities formerly offered by diatonic tonality are now multiplied at least twelve?fold by the tone clock. Messiaen discovered his own small corner of it, and found there material enough to sustain a composing life prolific for over sixty years. Diatonic tonality is itself a tiny area of the clock, and that served us rather well. Indeed, it positively refuses to lie down and die: once the language of the few, it is now the language of the people. (So does the work of the serious artist eventually shape, even create, a mind-space for the many.) And for excellent reasons: as a musical system it is far richer than anything we were ever able to 'invent' as a common language to take its place - and indeed it was never 'invented': it evolved naturally from the purest and simplest of mathematical principles. The clock is the next step in this natural evolution: Schat did not 'invent' the triads or their harmonic properties, anymore than Pythagoras invented the harmonic spectrum, or Rameau the common chords. He was simply bright enough to isolate and name them all, and then to see that Rameau's principle also worked for the chromatic scale. These were the two strokes of genius - and simplicity - that make him the major musical theorist in this century after Schoenberg. And these are the musical (mathematical) facts, objective facts, capable of being taught, and freely available to all.
Knowledge of the clock does not mean, necessarily, that composers will suddenly start writing different music - although that is certainly an attractive possibility. But is does mean that they will have a far clearer idea of what the chromatic musical choices are. The 'few' may once again speak a common language. Indeed, clock analysis of the music of the past and present shows that we were speaking a common language all along: the clock simply elucidates what this language is, what its grammatical and syntactical principles are based in, what the common ground is. It explains the paradigm, provides a 'map'. We can now see, each one of us, where we 'fit in' as composers, and where every other composer fits in as well.
No longer need the perennial problem of the composer (what notes to use, and what shaky rationale to provide for them) be the soul-destroying dilemma it has been, ever since the collapse of the old system, for all serious practical purposes. A positive burgeoning of possible and logical inter-relations now emerges - so great that the problem now becomes rather how to limit them.

When listening to Schat's more recent compositions - such as Serenade and De trein (The Train) - you may pay close attention to his pure tone-clock melodies and harmonies. If you do, I feel sure that their very distinctive character and their rich diversity will strike you as they did me, when I first heard them. They have a profoundly riveting quality. I seemed to 'recognize' them somehow, as one sometimes recognizes a perfect stranger, as though one had always known them. The colours, the progressions, the atmospheres are peculiarly evocative. The notes have a 'rightness' somehow, they feel right. Even knowing nothing of the clock one can sense that this is the new tonality, that the music is permeated by some unknown but supreme logic and constancy: the harmonies have real substance, they move as a living tissue, worlds away from diatonic tonality, yet possessing the same authority and coherence. It is a language that makes sense at last!
The reason is, that they are solidly grounded in the chromatic deep structure. Yet no 'system' can ever guarantee that one will write good music: the reason is also that Schat is a wonderful composer.
But the real good news, for all the many composers who today are 'stuck', is that there is a natural chromatic order, existing of itself, by virtue of the almighty power of Number. It belongs to all, and it can work for anyone.
A vast and prodigious universe awaits us, and I, for one, am hailing it daily with shouts of jubilation!

The Zodiac of the Hours

Twelve small clock-faces (the Hours) combined into one large one
(the Zodiac). Each successive point around the small clock-faces represents a chromatic semitone (with C generally at one o-clock). The triangles are the triads. This is the face of the Tone Clock: a complete visual analogy which can be read accurately in musical terms.

Jenny McLeod
Pukerua Bay 1989

© Peter Schat
© design