Tom Johnson

The Chord Catalogue

All the 8178 chords possible in one octave

 

I like to think of The Chord Catalogue as a sort of natural phenomenon—something which has always been present in the ordinary musical scale, and which I simply observed, rather than invented. It is not so much a composition as simply a list.

 

Extreme and, one would think, extremely simple. A lesser man would have arranged those 8178 chords in some symphonically meaningful, or else quasi-random order, but Johnson proceeded methodically up the chromatic scale from two notes at a time, three, four, so on to 13. Before each section he would disconcertingly inform us, "the 715 four-note chords... the 1287 five-note chords..." His modest promise that we would "get the idea of the piece" within a few minutes wasn't really true. Two-note chords were predictably dull, three-note ones little better. But four notes began to sound almost like functional tonality in this denuded context: five sounded noticeably lusher, and reminded one of the era in which harmony was enriched by ninth chords and similar possibilities. By the time we reached 10-note chords, the information overload was such that differences were hardly perceptible, a situation reminiscent of serial music. Far from being heavy-handed minimalism. The Chord Catalogue was a pointed lesson in music history and the relativity of perception.

Kyle Gann, Village Voice April 14, 1987

Another transcendental experience was Tom Johnson's Chord Catalogue, which included all the 8178 chords possible in the octave c - c¹, from the two-note chords to the complete cluster. Johnson, who required only one hour for this, is the only pianist who can make his way through the dizzying multiplicity of colors present in the equal-tempered scale. Resonances in the space, and excitations in the ears, caused sheer psychedelic perceptions, that well surpassed the simple combinations game.

Matthias Entress, Berliner Morgenpost Nov. 24, 1998

 Musical Combinations in the Past

The Chord Catalogue is a rare and probably unique example of a piece using all the possible combinations of some musical phenomenon, but it is not the first time that musical combinations have been considered. Already in the early 17th century, the intellectual monk Marin Mersenne posed questions in the Livre du chant of his Harmonie universelle as to the number of possible melodies one could construct by changing the order of notes in a 22-note sequence. After much calculation he concluded that there was no point in actually carrying out this experiment, since one could never hear that many melodies in one lifetime.

An even more astounding group of possibilities was found when mathematicians began considering how many readings of Stockhausen's Klavierstuck XI were possible? Stockhausen's rule is quite simple: play the 1 9 motifs in any order, never repeat the same motif twice in succession, and stop when you play one motif for the third time. The shortest possible interpretation is

using only three motifs and five phrases, and the longest is of this type:

1 234567891O11   1213141516171819 1 234567891011   12131415161718191

using each of the 19 motifs twice and achieving the maximum length of 39 phrases. Of course, there are many variations of both the shortest and the longest solutions, and many additional readings have lengths of six to 38 phrases, and all together exactly 1 74239 35148 33295 81673 10127 28286 29013 34594 sequences are possible. But if you want to know how this number was calculated, we refer you to R.C.Read and Lily Yen: "A Note on the Stockhausen Problem" in the Journal of Combinatorial Theory, Series A 76 1-10(1996).

What's the Scare?

Even the best sight readers must go very slowly, and make many errors, when they try to read page after page of eight-note chords. The Chord Catalogue is simply not music to be read, and after many unsuccessful attempts to make traditional scores, Johnson realized that the best way to make a score has always been simply to explain what to do, as clearly and succinctly as possible for that particular situation, and to forget all about how music is usually written. Thus the score now is simply the verbal instructions given here in the box. Of course, one must study these instructions for some time in order to really see how they work with the sequences of two-note chords, three-note chords, four-note chords, and all the others, and to teach the fingers to follow the rules, and to teach the brain how and when to concentrate, and so on, but with this notation, there seems to be at least the possibility that someone other than Johnson will someday learn the piece.

Play all the two-note chords possible in one octave, then the three-note chords, four-note chords, etc., ending with the one 13-note chord. Always begin with the lowest possible positions and move gradually up according to this rule: The lowest voice that can rise a half-tone does so and any lower voices descend to their points of departure. Each time that the top voice is going to rise a half-tone, one leaves a pause and begins a new section. For example, in playing the four-note chords, one would play:

4 5555 6666666 … 13 13 13                                                                                                                                                                                                                                                                

Pascal's Triangle and The Chord Catalogue

by Jean-Paul Allouche

When Tom Johnson played for me a part of his Chord Catalogue, I knew the sequences of chords was written in a pure logical order, but I was more interested in the musical patterns that seemed to appear than in their combinatorial properties. Nevertheless, I began thinking about combinations as well, when the composer showed me the following array, which describes how the 8178 chords break up into groups

2 3	4   5	6	7	8	9	1O	1 1	12
1 3	6 1O	15	21	28	36	45	55	66
1	4  10	20	35	56	84	12O	165	22O
	1   5	15	35	70	126	210	330	495
	1	6	21	56	126	252	462	792
		1	7	28	84	21O	462	924
			1	8	36	120	330	792
				1	9	45	165	495
					1	1O	55	220
						1	11	66
							1	12
								1

where the first line corresponds to the two-note chords, the second line to the three-note chords, and so on until the last line, which represents the one 13-note chord, all played according to Johnson's rules.

Tom suggested with a very sure intuition that Pascal's triangle was certainly hidden in this array. Let me recall that the Pascal triangle is the array of the so-called binomial coefficients, defined as follows: given a finite set with n elements, how many different subsets of this set containing k elements can we find? Given seven people, for example (n = 7), how many ways can we select four for a game of bridge (k = 4)? The answer is denoted by (n k)(pronounced "n choose k") and can be proved to be equal to

 (nk) = n! =n(n-1)   =  (n-2) (n-k+1)

            k!(n-k)!                   k!  

where n! (pronounced "nfactorial") is defined to be the product 1.2.3...(n-1).n. In the case of the 7 people divided into foursomes, n = 7, k = 4, and the answer is:

The array for these numbers is given below: n is the row index, k the column index, and we find 35, the number of foursomes possible with seven people, as the fourth number of the seventh row. The array, called Pascal's triangle, can be continued to infinity:

1 1	1
1	2	1
1	3	3	1
1	4	6	4	1
1	5	1O	1O	5	1
1	6	15	20	15	6	1
1	7	21	35	35	21	7

The name "binomial" comes from the remark that these numbers are precisely the coefficients that occur when expanding the powers of (1 + X), as you can see already after only four lines.

(1 + X)²= 1 + 2X+ 1X²

Tom suggested that Pascal's triangle was involved, because when he looked at The Chord Catalogue array and turned the piece of paper by 9O degrees, so that the rows and columns were interchanged, then a significant part of Pascal's triangle appeared! In fact. The Chord Catalogue array was just the same as Pascal's triangle turned on its side, except that a row of ones was missing at the top. These missing ones exactly represent combinations that are not chords and that cannot be included in a catalogue of chords, namely, the one combination with zero notes and the 13 combinations of single notes.

Of course, a mathematician is happy only if he or she proves such assertions, and I want to prove that The Chord Catalogue array really is the same as Pascal's triangle, even though the casual reader may not be able to follow every step of the logic.  The binomial coefficients satisfy the following recurrence: each number in Pascal's triangle is the sum of two numbers in the previous row, situated respectively north and northwest.

Now let a(k,n) be the number of /c-note chords in The Chord Catalogue whose upper note is one of the notes 1, 2, .... n. In the case of the five-note chords, for example, the lowest possible top note is 5 and the highest possible top note is 13 (=n). To compute a(k + 1, n + 1), i.e., the number of (k + 1 )-note chords whose upper note is one of the notes 1, 2, ... ,n, n + 1, we observe that such a chord has its upper note taken either from the set 1, 2, ... , n, or else is equal to note n + 1. By definition the number of chords of the first type is a(7c+1, n). The chords of the second type can be exactly obtained by concatenating note n + 1 to a /c-note chord whose upper note is taken from 1, 2, .... n. Hence, the number of chords of the second type is nothing but a(k,n). In other words, we just proved that

a(k + 1, n + 1)   =   a(k + 1, n) + a(k, n ).

The numbers a(k,n) thus satisfy the same recurrence equations as the binomial coefficients. Since their initial values coincide, they must be equal: a(k,n) = "n choose k," and this proves Tom was right!

Machine Versions of the Catalogue

It is relatively simple to write a computer program to compute The Chord Catalogue, and several people have done so. The disadvantage is that listeners do not like to go to concerts simply to watch machines, and seldom buy records to listen to machines, but the advantage is that the machines go faster than musicians do, and they never make mistakes, as long as they are working.

The first person to make and present a machine version of The Chord Catalogue was Martin Riches, a sculptor living in Berlin, known for his walking machines and talking machines, as well as for his musical machines. His version of The Chord Catalogue is programmed for his Sound Machine, a small computer-driven pipe organ.

Clarence Barlow, a composer who alternates between The Hague and Cologne, programmed The Chord Catalogue for the Yamaha Disklavier, a computer-controlled piano, which Barlow has frequently used in his own music.

Wolfgang Heisig invents instruments and sound installations at his home near Dresden, but he also works with traditional player piano rolls, which he punches out with a self-designed computer-driven punching machine. He has produced several Johnson pieces, including The Chord Catalogue, as well as scores of his own, of Nancarrow, and of others.

Heisig's realization is the easiest to distribute, since working player pianos still exist throughout the world, but all three play at tempos about twice as fast as Johnson's, and they don't make mistakes. Of course, in a recording studio, with the advantages of retakes. Johnson doesn't make mistakes either, but there is one error in the editing. At one point Kenan Trevian was obliged to eliminate a couple of chords in order to make a flawless splice. We soon forgot where this flaw occured, and other listeners will have a great deal of trouble finding it, but knowing that it is there may give the listener something to search for while listening to the recording.

Anyone desiring further information about these mechanical versions of The Chord Catalogue may write to Martin Riches, Steifensandstr. 9, 14057 Berlin; Clarence Barlow, Merheimerstr. 214, 5O733 Cologne; Wolfgang Heisig, Grimmaerstr. 16, 04703 Leisnig (Germany).

Why   8178?

The total number of chords possible in one octave is 2¹³ -14. There are 13 keys in a complete octave, using the note C twice in this case, and for each chord, each of the 13 keys is either played or not played. Thus the total number of combinations is 2¹³, that is, 8192. The total possibilities can also be seen as binary counting from 0 to 8191:

etc. to 1111111111111=   8191

This is not at all the structure of The Chord Catalogue, and in fact, this binary logic has already produced four possibilities, indicated by stars, that are not chords at all, since only one note or zero notes are being played. Ten additional combinations must be eliminated for this reason, which is why we subtract 14 and end up with 2¹³ - 14, or 8178.

Explaining my Music: Keywords

by Tom Johnson

I have often tried to explain that my music is a reaction against the romantic and expressionistic musical past, and that I am seeking something more objective, something that doesn't express my emotions, something that doesn't try to manipulate the emotions of the listener either, something outside myself.

Sometimes I explain that my reasons for being a minimalist, for wanting to work with a minimum of musical materials, is because it also helps me to minimize arbitrary self-expression.

Sometimes I say, "I want to find the music, not to compose it."

Sometimes I talk about mathematics and formulas, and how these things provide a means of avoiding subjective decisions and permitting objective logical deductions.

Sometimes I quote my teacher Morton Feldman who said so often, "Let the music do what it wants to do."

Sometimes I draw a parallel with the way John Cage used chance, which was also an attempt to base his music on something outside of himself.

Sometimes I talk about all these things, and think that surely everyone will understand what I am doing and why I am doing it, but whatever I say there are questions: What am I supposed to feel? How can music be impersonal like that? Don't you want to express something? Won't this just produce a cold, meaningless music? Etc. etc. The idea of music as self-expression is so ingrained in the music education of almost everyone that people become totally disoriented when you try to take it away. Then one day, frustrated by my inability to communicate my esthetic goals to a group of students, I just said "I am not interested in autobiography."

This was the keyword, and suddenly everyone seemed to understand.

I think the reason my point of view was easier to understand when I said I wasn't interested in "autobiography" is because this word comes from literature, where it applies only to a small and relatively unimportant category. In literature it is clear that autobiography is subjective rather than objective, and that most of the time good literature involves characters that have an existence outside the writer's own life. This clearly implies that music too should be better when it is not centered around the composer's own subjective feelings.

Of course, the kind of objectivity I'm striving for goes quite a bit further than simply avoiding autobiography, but observing this parallel was helpful to the people who were trying to understand my music that day, and thinking about it later has helped me, so I thought it might also be useful to try to explain this in a written essay, even though it will require some autobiographical reflection in order to do so.

As I started to write about wanting to avoid autobiography in my own music, I realized that I have been using two other words lately to describe my music and how I make it, and it will be good to talk about these other two keywords as well: "truth" and " interpretation."

"Truth" is a lofty idea, and it is certainly not good for artists to think of themselves as distributors of truth. The term is meaningful for me, however, because I find it much less lofty and much more useful than the term "beauty," which is the one usually used to describe what artists attempt to distribute. I am not against beauty, of course, but I don't really know what it is or how to deal with it. "Beauty" seems to apply to all the best artistic products in all the centuries, things that are somehow attractive and agreeable, and that somehow have wonderful esthetic properties. But such things are very difficult to explain, while truth, on the other hand, is much clearer.

Something is true if it correlates to what one can observe in the real world, if it follows its particular logic, if we can somehow demonstrate that it is coherent and correct. Of course, true and false are often only relative, but often things are absolutely true, beyond any doubt, and sometimes there even seems to be a connection between truth and beauty. I would not say that the truth is always beautiful, but things are certainly not beautiful when they are untrue, as when musicians play wrong notes, or leave out a measure in the music, or insert an incorrect 5/4 measure in what is supposed to be 4/4 music. Composers, as well as interpreters, can miscalculate time proportions, or make bad judgments in orchestration, or leave inconsistencies that confuse the music, and make other sorts of mistakes, and a good music theorist or composition teacher can easily find such errors. In such cases no one seriously questions that the correct or true version is better than the incorrect or untrue one. Mistakes are not beautiful, and they are not inexplicable either.

In the case of my own music, this quest for the correct, the true, is particularly important. When I wrote Tango and Music and Questions, both of which consist of 120 permutations of sequences of five notes, I had to try many different sequences before I knew how I wanted the music to proceed, but there was no question of writing more or less than 120 phrases. Usually in such a piece one can also hear the logic. The truth becomes audible, which means that if a performer plays the same permutation two times in a row, or disturbs the six-bar phrases with a phrase of only five bars, the listener will probably hear that something is wrong, something is untrue. Of course, for this to happen, the music must be relatively simple, clear, transparent, which explains, once again, my preference for minimal materials.

Naturally, there is a disadvantage in seeking this kind of perfection, because performers are rarely perfect. There is usually some unfortunate moment, some mistake, some untruth that is quite audible in my transparent music, and that makes the performance somehow unsatisfying, regardless of tonal richness, speed, virtuosity, excitement, intensity and whatever other positive qualities may be present. Of course there is also an advantage, because occasionally, when a good performer is concentrating very well, one can hear a performance without a single mistake, a pure audible truth, and that is extremely satisfying.

The third keyword is "interpretation," but to explain that I must go back to truth, and then back even further to the source of truth. Where do permutations come from? Who decided that five elements can be permuted in 12O ways? Well, I believe that such things are given, that they would be the same on another planet or for another species. Such things existed before humans were on earth, and they have merely been discovered by us, not invented.

We have to be careful here, because mathematical things are abstract, and sometimes one can argue that they really are human inventions rather than discoveries. We know that a line, for example, is "the shortest distance between two points," but we also know that this is only a theorem of Euclid. It is very hard, if not impossible, to find visible lines in nature, or even to show that sounds and x-rays move in straight lines. We have no physical, empirical evidence for lines, and this can also be said of circles and triangles and geometry in general. We like this Euclidian way of explaining things, however, and it enables us to calculate hypotenuses and circumferences and lots of other things that are directly applicable to the physical natural world, the so-called "real" world, which platonic thinkers continue to find less real than the line and the hypotenuse.

I believe that many musical phenomena also have absolute existence, and were not simply invented by musicians, and the 8178 chords possible in one octave, as stated in my Chord Catalogue, is a case in point. Inversions and retrogrades seem to me to be additional examples of musical absolutes. It was not necessary to wait for a composer of genius in order to realize that one can turn something upside down, or backwards, or both. Mirror relationships and symmetries can be observed all over in nature, as well as in all sorts of music and visual art in all sorts of cultures. Somewhat more rare, but equally obvious and absolute, are additive procedures, where one note becomes a melody of two notes, then three, then four, etc. That too is something that has been discovered in many situations by many people, but one wouldn't say that anyone had really invented it.

More sophisticated phenomena like prime numbers, Pascal's triangle, and the Fibonacci series can probably not be discovered by someone who is not already rather advanced in arithmetic and geometry, but eventually people in quite a few cultures have discovered them, and eventually even musicians find ways of applying such things. Self-similar structures, formulas, automata, and other techniques have entered the musical language only recently because these are even less obvious, but 1 don't think we can claim to have invented any of these things. It was just a matter of time until somebody, or rather lots of people, found them.

Now that I have explained that most of what I am using when I write one of my mathematical pieces is things like combinations and formulas and symmetries, things we have simply discovered rather than invented, I can get back to the third keyword, "interpretation," and explain that my idea of composing is not so much composing new things, but simply interpreting things that already exist elsewhere. If you are simply discovering something already present in nature, you are just interpreting nature.

For example, the Indian mathematician Narayana came up with a curious infinite sequence in the 14th century. It is something like the better known Fibonacci sequence, and has curious properties, which I made use of when I wrote Narayana's Cows six centuries later. I was not composing a piece in any usual sense, though, but rather doing a musical interpretation of Narayana's interpretation of this phenomenon. I left this score simply as three lines of music, because it didn't seem right to lock the music into one specific instrumentation. I preferred to let this interpretation process go on, to allow Ugly Culture, MusikFabrik, percussion ensembles, saxophone groups, orchestras, and any other kinds of ensembles all do their own interpretations.

It is important to note that if Narayana hadn't found this series (which begins 1 2 3 4 6 9 13 19 28 41...), someone else would have, and the process continues today. Recently Jean-Paul Allouche demonstrated how the Narayana series and the Fibonacci series are related to one another, and to other series. As explained in our joint article, "Narayana's Cows and Delayed Morphisms," listed in the bibliography, there is a whole category of sequences that can be constructed by simply delaying the generation process, and all of these probably have applications somewhere in the physical world, and other mathematicians and musicians and artists and architects and biologists may want to interpret these in their own ways, and of course, there may be interpretations of these interpretations, and new related discoveries, and interpretations of these, and the process can go on and on. But no one will ever really create anything new, because at least in my opinion, those numbers and possibilities were always there, just waiting to be discovered and interpreted.

By now, I think it should be clear that if a composer can be content to simply interpret and discover things, instead of creating, and can focus on truth instead of beauty, then the result should be objective instead of subjective, and autobiographical music can be avoided, and at least a few people like myself will be happy with the result. Of course, it is also clear that I have been totally subjective in this present text, which speaks exclusively of my music and my personal experiences. But then, 1 didn't say that I wanted to avoid autobiography in every piece, or in every aspect of my life.

Paris, 1999

Tom Johnson, born in Colorado in 1939, received B.A. and M.Mus. degrees from Yale University and also studied composition privately with Morton Feldman. He is considered a minimalist, since he works with simple forms, limited scales, and generally reduced materials, but he proceeds in a more logical way than most minimalists, often using formulas, permutations, and predictable sequences.

Johnson is well known for his operas: The Four Note Opera (1972) continues to be presented in many countries. Riemannoper has been staged 20 times since its premiere in Bremen in 1988. The latest, Trigonometry, was premiered in Hamburg in November 1997. Non-operatic works include the Rational Melodies, Music for 88, and the often-played Failing: a very difficult piece for solo string bass. A theoretical book devoted to Self-Similar Melodies was published in 1996 by Editions 75, where all of his scores are also available.

His largest composition is the Bonhoeffer Oratorium, a two-hour work in German for orchestra, chorus and soloists, with text by the German theologian Dietrich Bonhoeffer. The oratorio was premiered in Maastricht in September 1996 and received its German premiere in Berlin in November 1998.

After 15 years in New York, he moved to Paris, where he has lived since 1983. His wife is the Spanish artist Esther Ferrer.

Allouche, Jean-Paul and Johnson, Tom: Finite Automata and Morphisms in Assisted Musical Composition, Journal of New Music Research, Vol. 24 (1995), pp. 97-108.

Allouche, Jean-Paul and Johnson, Tom: Narayana's Cows and Delayed Morphisms, Les cahiers du GFtEYC (Caen), 1996 No. 4, Journees d'informatique musicals, 1996.

Johnson,   Tom:   Automatic   Music, d'informatique musicale, 1998.

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