S.I.N : a music of numbers that count themselves
for Tom Johnson (1939–2024)
(May 2025)
Last autumn, I carefully read ‘Tom Johnson’, Gilbert Delor’s comprehensive and respectful book (in French) on the work of Parisian based American composer Tom Johnson. I knew Johnson mainly from his regular visits to La Générale Nord-Est in the French capital, back when this artistic, political and social playground cum laboratory was still located on the avenue Parmentier, not far from Johnson’s home on rue de la Roquette.
I remember the times when his works were performed at La Générale— most memorably during the festive concert on April 18, 2015, marking his seventy-fifth birthday. I also happily recall the cup of tea we shared with Magister Rébus at his home, together with his eega, the artist Esther Ferrer. He enthusiastically showed us the elegant diagrams he had drawn, based on the combinatorial and arithmetic principles that were occupying his mind at the time. He would always inquire whether we, as mathe-mates, might be able to help him answer the questions and solve the puzzles that were haunting him in the process.
Delor’s (French-language) tome appeared in 2021, and bears the subtitle ‘ou La Musique Logique’ (The logical music). It became the topic of my 40th Hoofd Stuk in the Dutch Gonzo (Circus) Magazine, entitled ‘Pakken van hetzelfde laken’, which appeared in the magazine’s November/December 2024 edition. ‘About Tom Johnson, music, numbers and logic’. I had wanted to hand him a copy of the magazine at the first opportunity.
But it never came to that.
Tom Johnson died.
On the last day of the year 2024.
He lived to be 85.
Delor’s book traces, in rich detail, Tom Johnson’s lifelong search for an essence — a kind of grail that he relentlessly pursued, a thing that might uncover the very core of count and number, and thereby of music. A quest that — by its very nature — was destined to remain unfulfilled. For such a core is a mirage, a phantom image: just when you believe you are standing before it, ready to grasp it, it slips away — receding into new and previously unsuspected far-off places.
It is not science that Johnson practices in his work, all ciphers notwithstanding. It is pure poetic speculation, and the art hides in the choices that he constantly made in the process. However inescapable they may then seem to be. Among all the properties and processes of numbers and structures that Johnson exploits so rigorously, there is not one that could not be rendered differently in countless other equally rigorous ways. Tom gives us the voicing that Johnson chose. Which, as a voicing, reflects the Tom that he was, and the world that he lived in. That choice, those choices, are the him, are the art and the artist.
Ha, ha, yes, whence that mirage effect!
Zelftellers
Reading Delor’s book, especially the parts on Johnson’s earlier ‘counting’ works, brought back memories of a bunch of number ‘games’ that I pursued in the mid-1980s, in my early ‘math’ days. One in particular re-caught my attention: the simple recursive recipe that (often, but not always) leads to numbers that, so to say, describe themselves. You will find the mention and descriptions of such numbers or sequences of digits here and there, in a number of variations (more or less similar, but often not quite equivalent), that go under names like ‘self-describing sequences’, ‘self-counting numbers’, ‘self-describing numbers’, ‘auto-counters’. The Dutch term is zelftellers. In French we might call them nombres auto-referentiels.
Here is again the picture that is also the header of this article: a photo of a page from one of my notebooks from back in the 1980s, on which I wrote down all the zero-less nombres auto-referentiels with a length of between 4 and 18 digits.
There are 58 of them.
There is only one smaller one, which also is the first evident example of this numeric self-reference. That is the number 22, read as saying: “I am made up of 2 2's”. Yes, the number 22 describes itself. It is the smallest self-counting number (the smallest ‘compact’ one, compact — my terminology — meaning that we exclude 0 as a digit that counts).
Playing around a bit, you will soon find that you stumble on a good many of these auto-counters by picking some number to start with, and then iterate the process of taking its ‘descriptor’, its ‘counter’. (For each of the possible ‘auto-counters’, there of course is a smallest number that generates it. The smallest — and only — generator for 22 is 22.)
Here is an example that shows how this works, starting from 123:
123 → 111213 → 411213 → 31121314 → 41122314 → 31221324 → 21322314 → 21322314 → etc …
This is were the process comes to a halt in a cycle (it will always end that way): in this particular case we hit upon a single cycling number, a fixed point, a ‘zelfteller’. The number that describes 21322314, its descriptor δ(21322314), is 21322314 itself.
(The smallest generator in this case is 1, and it takes 12 steps to find it: δ¹²(1) = 21322314, see the picture a bit further.)
Recalling these very rhythmically ‘expanding while repeating’ number patterns caused by the ‘digit-counting’ operation δ when reading Delor’s book, I thought writing them up in a score, that I might one day give to Tom, on a next tea visit maybe, or on some other occasion, as a present. Because it obviously felt ‘Johnsonian’, but digit-wise is also firmly rooted in a 9-ness, which links it to my sudokism, haha. (As for the sudoku’s, a fun little fait divers that I picked up from the listing of Tom’s ‘rejected and occasional works’ in the back of Delor’s book, is that in the same year that I first used a sudoku solution as a guide for dictaphone playing, in 2006, Tom Johnson did a thing he called Sudoku for musiciennes, printed as a business card for the salon Musicora in the Louvre Carrousel.)
I gave my effort a Latin title: ‘(Numerorum Musica Quae) Se Ipsa Numerat (1)’ (which nicely abbreviates, to S.I.N), a Music of Numbers that count themselves (there is a ‘1’, because maybe I will make some more later). It basically unfolds the digit-count re-writing up to an auto-counter for the nine non-zero digits in counting order (1,2,3,4,5,6,7,8,9), in two parallel voices: the second one does the counting of the digits in their natural counting order, and the first in the order in which the digits are encountered. E.g., ‘2321’ in counting order rewriting becomes ‘112213’, but in appearance order rewriting it leads to ‘221311’, because in 2321 there first occurs a 2, then a 3, and the last digit encountered is a 1.
(If we denote the counting in appearance order by γ, then γ(2321)=221311 , while δ(2321)=112213.)
The following pics show you almost all the information needed to produce yourself the score for S.I.N.
In the first one we see how each of the first four non-zero counting digits 1, 2, 3 and 4 γ-rewrites to a 2-cycle 23322114 ↔ 32232114 (permutations of the δ-zelfteller 21322314), while digit-counting 5 leads to 3322311415, 6 to 3322311416, and so on for 7, 8, and 9, where it ends in the γ-zelfteller 3322311419.
The second one shows how each of the first four non-zero counting digits 1, 2, 3 and 4 δ-rewrites to the auto-counter 21322314, while digit-counting 5 leads to 3122331415, 6 to 3122331416, and so on for 7, 8, and 9 ending with the auto-counter 3122331419.
The digits provide the parts of the piece, and for their musical rendering are assigned distinct nine-tone scales (like in the toy piano sudoku’s), as indicated in the pictures. Without going into all the details, the first ten measures, annotated with the corresponding numbers will give you the general idea:
For the one-before-final part of the piece I sneaked in a ‘zero’, by using 10 as generator, and interpreting the (single) recurring 0 as a rest. The piece then ends with a brief index, a summing up of the auto-counters that are generated in the piece.
One final little fact: when I recently re-checked the numbers, I found that in that one-before-final part, with 10 as its generator, I made a counting error. I decided to not correct the error, and leave it as it is, so that if one day you lay hands on the score, you’ll have a chance to find it :-)
So, no, unfortunately I will not be able to hand Tom this humble present/homage. Maybe though one day we can hear it being played. S.I.N can be performed basically by any two instruments, though I really would love to hear it being done by two saxophones, a tenor, say, for the γ-voice, and a baritone for δ ?
One day, maybe.
Originally published at http://www.harsmedia.com.
