Daniel Dockery

animî nostrî dêbent interdum âlûcinâri

Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.

Ex Bibliotheca, pt. II: A Proof of Concept

March 5th, 2012

Not long ago, in an earlier post, I wrote about Borges’ “Library of Babel” and some of the mappings that can be applied or removed to reveal it as pure number. In Borges, the mapping applied is textual so that the numbers take on the appearance of the written word, manifested as books, but thinking of an earlier experiment, the so-called Transcendental (Number) Études, where a sequence of digits was turned into a song, I began considering the idea of the Library manifested as music.

It occurred to me that reversing the Borges mapping, or a longer one supporting whichever alphabet or map one chooses instead of the limited one of twenty-two letters, any arbitrary text could be turned into a number, and that number then turned into a melodic line. As a proof of concept, I mentioned this and put out a request for a sample text on Tumblr, to which Mister Chu replied with this fragment:

A bag is packed. The man being left tries to save himself by allowing his life to tumble across his lips. All of his stories fall out, irrelevancies, laundry lists and places to park on the Southside of Chicago. This is everything he thinks, so little.

He subsequently posted the full text here. From that, I produced a number (using a 29-member mapping, the twenty-six letters of the Roman alphabet, the space, period and comma), and converted that into the following melody. The tempo, pacing, accompaniment and performance are, as before, at will, while the melodic line itself is an exact mapping from the number.

[This piece is not currently available online.]

Ex Bibliotheca

February 21st, 2012

On Tumblr, earlier, I saw this:

Here is a site that offers digital renderings of some of the books in the unimaginably vast universe of information (and un-formation) described in Jorge Luis Borges’ story the Library of Babel

Explore, but remember that the possibility of discovering your Vindication within the Universal Library “can be computed as zero.”

An interesting idea, but given Borges’ kabbalistic leanings, his remark that there is no form of capital lettering, no digits, and his insistence on the twenty-two letters and their fixed, symmetrical forms, it is most likely that he meant for the Library’s volumes to be expressed in the block Hebrew script, which he has admired in other places, though any language might be encyphered in that alphabet in the Library’s vast holdings, such as his “Samoyed-Lithuanian dialect of Guarani, with classical Arabic inflections” (“un dialecto samoyedo-lituano del guaraní, con inflexiones de árabe clásico“) mentioned in the story.

Other possibilities do present themselves, however, for clues suggest the language of the southern hemisphere of Tlön. How so? In “The Library of Babel,” Borges gives us explicit “titles” for three volumes: “trueno peinado,” “el calambre de yeso,” and “axaxaxas mlö“. The first could be the “Combed Thunder,” the second “The Plaster Cramp,” but what of the third? If one were to read “Tlön, Uqbar, Orbis Tertius,” from the same 1944 collection (Ficciones [Spanish, English]) in which “The Library of Babel” appeared, one would encounter a passage reading, “Surgió la luna sobre el río se dice hlör u fang axaxaxas mlö,” “‘The moon rose over the river’ would be said hlör u fang axaxaxas mlö.” In the additional text, he indicates all words are verbs in that language, and that “axaxaxas” would mean something resembling “flowing like a river,” while “mlö” would be “shining (or perhaps rising) like the moon”. Translate it however we will, there remains the point that at least one book in the Library is titled by one of the Tlönistas. Or by coincidence would appear to be.

This provides much of a proposed alphabet: in addition to the three titles, Borges also wrote that the combination of letters “dhcmrlchtdj” would appear in the Library and that there is one volume that consists of endlessly repeated “mcv”s. Taken together, sorted alphabetically with duplicate letters removed, this leaves us with 20 or perhaps 21 letters: a, b, c, d, e, h, i, j, l, m, n, o, p, r, s, t, u, v, x, y—and, as he does distinguish it, possibly ö. The alphabet in use on the above linked site adds f, g and q but omits the u needed for “trueno” and doesn’t distinguish the ö from o. If we accept that the line from the story of Tlön is of the same language or alphabet, as at least the title seems to be, then we also see evidence of the f and g in use. If we adapt those, but hesitantly combine the o and ö, we will have another set of twenty-two letters.

It likely doesn’t matter:

Un número n de lenguajes posibles usa el mismo vocabulario; en algunos, el símbolo biblioteca admite la correcta definición ubicuo y perdurable sistema de galerías hexagonales, pero biblioteca es pan o pirámide o cualquier otra cosa, y las siete palabras que la definen tienen otro valor. Tú, que me lees, ¿estás seguro de entender mi lenguaje? (Borges, “La biblioteca de Babel”)

An n number of possible languages use the same vocabulary: in some, the symbol “library” admits the correct definition: “an enduring, ubiquitous system of hexagonal galleries,” but “library” is “bread” or “pyramid” or anything else [in others], and the seven words that define it have other meanings. You who are reading me, are you certain you understand my language?

Many years ago (2003) my fascination with the idea of this library, from Kurd Laßwitz’s much earlier story (“Die Universalbibliothek,” 1904) on through Borges’ (whose 1941 variant of the tale I prefer by far) and similar themes (e.g., Arthur C. Clarke’s “The Nine Billion Names of God,” 1954), etc., had me working out the elaborate mathematics of the whole and being in awe at the staggering numbers. For instance, given that Borges explicitly defined the volumes of the library as containing 410 pages, each page containing 40 rows of text with 80 characters per line (including punctuation and spaces), defined there to be exactly 25 possible characters, and said that the library contained exhaustively everything it is possible to express with them and that no two books were identical, we can calculate that the library must have exactly 25^(80*40*410), or 25 to the 1,312,000th power volumes. To put that in perspective, that is 1.956 x 10^1834097 books, a number 1,834,098 digits long. If you were to write out five digits a second, it would take you more than four days of non-stop writing just to transcribe the whole number.

One of the most enchanting things about the library is, of course, that in its seemingly endless volumes you may find everything that it is possible to write down—including your own life story (your “vindication”), perhaps spanning multiple volumes in elaborate detail, as well as countless millions of erroneous copies whether differing by a single letter or missing entire events or with events that end otherwise than reality’s version, the answer to every mystery or riddle that it’s possible to answer, truly everything. The damning part is that these volumes can be scattered anywhere throughout the universal library and your chance of finding the one you seek is only 1 in 25^1312000, which is so incredibly small it is for all practical senses zero. Should you by luck find one good volume of a multivolume set, you have again those vast odds against your ever finding the next.

Yet perhaps the most damning, or tantalizing, aspect of it all is that you could calculate any or all of these volumes and discern the universal order of the whole by a task no more arduous than counting by ones.

In Tlön, etc., Borges mentioned in passing various numerical bases, touching on the base 12 system (duodecimal) of one of the Tlönistas and the base 60 system (sexagesimal) of ancient Sumeria and Babylon (and in its way even today in our system of minutes and seconds). This is the essential clue. While base 10 (decimal) seems to have conquered all others today, other bases have been in use elsewhere: the now infamous Mayan calendar system, for instance, uses base 20; and in computing, we sometimes use base 16 (hexadecimal), base 8 (octal) or at the lowest possible level base 2 (binary). We can conceive of the Library as being expressed in base 25 (quinquevigesimal) notation, but instead of mixed case, such as with base 16 (where the “digits” are 0 through 9, then A through F, such that the decimal number 190 is expressed as BE in hexadecimal), we can define our 25 digits to be the twenty-five symbols set out by Borges, the twenty-two letters of the alphabet, the comma, the period and the space. Each volume in the library then is merely a number, and the order of their seemingly entropic arrangement is numerical. Which is also suggested in the story when he writes how some have asserted that while the books are written in the “natural symbols” of written language, any appearance of meaning is coincidental, a side effect of using those symbols. To demonstrate, we can see the same thing in hexadecimal, such as above when I pointed out that the number 190 is “BE” which could be interpreted as an English verb; or the number 57,005 is DEAD in hexadecimal, again a “word” in appearance though a number in intent, like the dreadful notion of DECAF, which represents the number 912,559. Treating the volumes as natural numbers also explains the Library’s first axiom, that it exists “ab æterno,” just as the numbers.

An example, to demonstrate. If we take the numerical order of the characters to be that in which Borges gave them in his text—to wit, “the space, the period, the comma, the twenty-two letters of the alphabet”—and set them, in the natural arrangement, as being equal to our base 10 values of 0 through 24, then our first book, corresponding to 0 would be an entirely blank volume. The second, corresponding to 1, would be entirely blank but for a single final period. The next would be the same, but with a comma. Then an “a”, a “b” and so on for the first twenty-five books, one for each symbol. The twenty-sixth would be blank but for the last two symbols which would be a period followed by another space. The sequence would continue, almost forever. In the 9,752nd volume we’d find it blank but for the last letters “mm.”; the next would be “mm,”; the next, “mma” then “mmb” and so on again. At the end of the 19,370,996,558th volume, we would find “abula‘fia”, the name of the 13th century kabbalist renowned for his combinatorial ability. The numbers have grown enormous by the time we even get beyond single sentences. And it continues, letter combining with letter, line with line, page with page, growing and growing, as the early kabbalist text, the Sefer Yetzirah, has it (IV:16, dealing with permutations):

מכאן ואילך צא וחשוב מה שאין הפה יכול לדבר ואין האוזן יכולה לשמוע׃

“From here on go out and calculate that which the mouth cannot speak and the ear cannot hear.”

Unfortunately, even with the fastest computing systems in operation today, our sun would burn out before we could generate a fraction of the library, which is moot anyway since with even the best possible compression available today, the library would require more storage space than currently exists or will exist for likely ages to come, if ever. The most recent figures I’ve seen suggest that as of 2011, the global data storage demands for all digital media of any kind covers “only” about 940 exabytes (for those of you playing along at home, that’s roughly 985,661,440 terabytes), with that number essentially doubling every two years—and with it generally pushing the limits of the amount of storage capacity we actually have available globally. Another way to look at data amounts, if every one of the 6,995,685,817 people of the current estimated world population had a full terabyte of data, the total would be roughly 6.5 zettabytes, a vast number far, far beyond current global capacity (there are 1024 exabytes in a zettabyte). At the expected rate of growth, it should take us almost six years to reach that capacity. Yet a complete digital copy of the Library of Babel, at ideal, maximum compression, would require something to the effect of 4.24562416107 x 10^1,834,075 yottabytes—where a single yottabyte is 1024 zettabytes or 1,125,899,906,842,624 gigabytes! At that projected growth rate, it should take a measly 12 million years for us to reach that capacity.

For a countering thought, Seth Lloyd, in an article (“Computational Capacity of the Universe”) in the science journal Physical Review Letters, in 2002, asserted that the total information capacity of the observable universe is only 10^92: the capacity necessary for the complete Library is over 10^1,834,099. Borges equated the Library with the Universe, yet it seems the Library is greater in scope. By far.

So it seems very possible that while the mind can conceive of the Library, can understand it, can know how to create it, its actual creation is beyond the scope not just of human ability but also beyond the scope of universal capacity, beyond the time scale of our solar system. We have to be satisfied with only knowing that these things “exist” in some abstract sense, our vindication, the answer to every pressing problem, to every question that’s ever passed our mind or anyone’s, every great novel (and every awful one!), even this post I’m writing now, exist, waiting for someone to discover them, yet outside the possibility of truly being able to search or find them in any clear or orderly fashion.

Perhaps we should, after the fashion of the site at the beginning, pick up our own “forbidden dice cup”…

“A blasphemous sect suggested that” [rather than searching for meaningful volumes] “all men should juggle letters and symbols until they constructed, by an improbable gift of chance, these canonical books.” […] “The sect disappeared, but in my childhood I have seen old men who, for long periods of time, would hide in the latrines with some metal discs in a forbidden dice cup and feebly mimic the divine disorder.”

J.L. Borges’ “A quien está leyéndome”

June 14th, 2011

The poem:

Eres invulnerable. ¿No te han dado
los númenes que rigen tu destino
certidumbre de polvo? ¿No es acaso
tu irreversible tiempo el de aquel rio
en cuyo espejo Heráclito vio el símbolo
de su fugacidad? Te espera el mármol
que no leerás. En él ya están escritos
la fecha, la ciudad y el epitafio.
Sueños del tiempo son también los otros,
no firme bronce ni acendrado oro;
el universo es, como tú, Proteo.
Sombra, irás a la sombra que te aguarda
fatal en el confin de tu jornada;
piensa que de algún modo ya estás muerto.

The translation, “To whomever is reading me”:

You’re invulnerable. Haven’t they given you—
the powers that govern your destiny—
the certainty of dust? Isn’t your own time
as irreversible as that river in which
Heraclitus saw in his reflection the symbol
of impermanence? A marble stone awaits you
that you will not read. On it’s already written
the date, the city and the epitaph.
Others are also only dreams of time,
not enduring bronze or unblemished gold;
the universe is, like you, Proteus.
A shadow, you’ll go to the shadow waiting
at the fatal end of your journey;
know that in some way you’re already dead.

Twenty-five years ago today (Jun 14), Borges passed away.

The silent friendship of the moon

April 21st, 2011

In Borges’ “La cifra” and the obverse of his “Poema“, both translated here earlier, the poet refers to Virgil. In the “Poem”, among the things brought with the hexameters of Virgil’s Æneid is “[l]a amistad de la luna“, “the friendship of the moon”, and “La cifra” opens with “[l]a amistad silenciosa de la luna / (cito mal a Virgilio)“, “the silent friendship of the moon / (I misquote Virgil)”.

He is making reference to line 255 in the second book of the Æneid, from the scene where the Greeks slip out from the Trojan horse to sack the city of Troy, making their way “tacitæ per amica silentia lunæ“, “unnoticed through the friendly silence of the moon”.

Borges’ interest seems only in the expression quite abstracted from the story’s context through his own fondness for the moon as evidenced in many of his other works. He returns to the expression many times. In his brief essay on Dante’s Purgatorio, I.13 (from Nueve ensayos dantescos [1982]), he refers again to “el famoso hexámetro de La Eneida: «a Tenedo, tacitæ per amica silentia lunæ»” (“the famous hexameter from The Æneid: «from Tenedos, unnoticed through the friendly silence of the moon»”), comparing to it verse 60 of canto I of Dante’s Infierno, “mi ripigneva là dove ‘l sol tace“—”pushed me there where the sun is silent”.

In Borges’ defence, though he needs none, his misquotation isn’t as severe as that perpetrated by José Lezama Lima in Paradiso: “Pero con esperada frecuencia volvíamos al ternario, a unir sol, tierra y luna, aunque yo casi siempre me inclinaba a la luna silentiæ amicæ.” (In Gregory Rabassa’s translation, this reads “But with expected frequency we returned to the ternary, to unite sun, earth and moon, although I almost always leaned toward the silent friendship of the moon”, though his translation is incorrect for the Latin.)

Borges’ “La cifra”

April 8th, 2011

La amistad silenciosa de la luna
(cito mal a Virgilio) te acompaña
desde aquella perdida hoy en el tiempo
noche o atardecer en que tus vagos
ojos la descifraron para siempre
en un jardín o un patio que son polvo.
¿Para siempre? Yo sé que alguien, un día,
podrá decirte verdaderamenta:
“No volverás a ver la clara luna,
Has agotado ya la inalterable
suma de veces que te da el destino.
Inútil abrir todas las ventanas
del mundo. Es tarde. No darás con ella.”
Vivimos descubriendo y olvidando
esa dulce costumbre de la noche.
Hay que mirarla bien. Puede ser última.

Jorge Luis Borges, “La cifra”


The silent friendship of the moon
(I miquote Virgil) has accompanied you
since that night or evening,
now lost in time, when your restless
eyes uncovered her forever
in a garden or a patio that now is dust.
Forever? I know that someone, one day,
will be able to tell you this truth:
“You’ll never (again) see the clear moon.
You’ve already exhausted the unalterable
amount of time granted you by destiny.
It’s useless to open every window
of the world. It’s late. You won’t find her.”
We live discovering and forgetting
that sweet habit of the night.
Look at her well. It may be the last time.

A Poem by Borges

March 21st, 2011

On 2 October 1980, Borges published a small poem in two parts, front and back, which ran:


Dormías. Te despierto.
La gran mañana depara la ilusión de un principio.
Te habías olvidado de Virgilio. Ahí están los hexámetros.
Te traigo muchas cosas.
Las cuatro raíces del griego: la tierra, el agua, el fuego, el aire.
Un solo nombre de mujer.
La amistad de la luna.
Los claros colores del atlas.
El olvido, que purifica.
La memoria que elige y que redescubre.
El hábito que nos ayuda a sentir que somos inmortales.
La esfera y las agujas que parcelan el inasible tiempo.
La fragancia del sándalo.
Las dudas que llamamos, no sin alguna vanidad, metafísica.
La curva del bastón que tu mano espera.
El sabor de las uvas y de la miel.


Recordar a quien duerme
es un acto común y cotidiano
que podría hacernos temblar.
Recordar a quien duerme
es imponer a otro la interminable
prisión del universo
de su tiempo sin ocaso ni aurora.
Es revelarle que es alguien o algo
que está sujeto a un nombre que lo publica
y a un cúmulo de ayeres.
Es inquietar su eternidad.
Es cargarlo de siglos y de esterllas.
Es restituir al tiempo otro Lázaro
cargado de memoria.
Es infamar el agua del Leteo.

Jorge Luis Borges, “Poema”

I would render it,


You were asleep. I wake you.
The vast morning brings the illusion of a beginning.
You had forgotten Virgil. Here are the hexameters.
(With that) I bring you many things—
the four Greek elements: earth, water, fire, air;
a single name of a woman;
the friendship of the moon;
the bright colors of the atlas;
forgetting, which purifies;
memory, which chooses and rediscovers;
the habits which help us feel we are immortal;
the sphere and the hands that measure elusive time;
the fragrance of sandalwood;
the doubts that we call, not without some vanity, metaphysics;
the curve of the walking stick the hand waits for;
the taste of grapes and of honey.


To wake someone from sleep
is a common day-to-day act
that can set us trembling.
To wake someone from sleep
is to impose on someone the interminable
prison of the universe
of his time, with neither sunset nor dawn.
It is to show him he is someone or something
subject to a name that lays claim to him
and to an accumulation of yesterdays.
It is to trouble his eternity.
It is to load him down with centuries and stars.
It is to restore to time another Lazarus,
burdened with memory.
It is to desecrate the waters of Lethe.


December 24th, 2010

nec tamen, ut lauder, uigilo curamque futuri
nominis, utilius quod latuisset, ago.
detineo studiis animum falloque dolores,
experior curis et dare uerba meis.
quid potius facam desertis solus in urbe,
quamue malis aliam quærere coner opem?

Ovid, Tristia, V. vii. 39–40.

It’s not for praise I stay awake and toil for the future of a name best forgotten. I busy my mind with studies, sidetracking sorrows, trying to give voice to my concerns. What else can I do alone in this forsaken town, what other help to this should I seek?

I wish I had some such study tonight to drive myself to like distraction, that I would not now be sitting here brooding as I am, under full sway of this sad, sorrowful season’s selection of spectres. This flu business hardly helps.

Last night, I lost myself in music, or my fumbling attempts thereat. As they did not turn out well—they seldom do—I do not wish to pursue the same fruitless path tonight. The goal, as with Ovid, is to forget my frustrations, not encourage them. Yet running through my options now, I’m coming to a dreary conclusion: everything I know how to do is an equal frustration.

Writing, art, music, any of it, all of it—equally flawed, equally disappointing. I’m tired of this; I need a new outlet.

Fazer uma obra e reconhecê-la má depois de feita é uma das tragédias da alma. Sobretudo é grande quando se reconhece que essa obra é a melhor que se podia fazer. Mas ao ir escrever uma obra, saber de antemão que ela tem de ser imperfeita e falhada; ao está-la escrevendo estar vendo que ela é imperfeita e falhada—isto é o máximo da tortura e da humilhação do espírito. Não se os versos que escrevo sinto que me não satisfazem, mas sei que os versos que estou para escrever me não satisfarão, também.

Por que escrevo então? […] Tenho de escrever como cumprindo um castigo. E o maior castigo é o de saber que o que escrevo resulta inteiramente fútil, falhado e incerto.

Pessoa, Livro do desassossego, 231

To write something and recognize afterward that it’s bad is one of the tragedies of the soul. It’s especially terrible when we recognize that the work is the best we could do. But when we write something knowing beforehand that it has to be imperfect and flawed, seeing as we write it that it’s imperfect and flawed—that is the spirit’s maximum torture and humiliation. Not only do I feel that the verses I write do not satisfy me, I know that neither will the verses I’m about to write.

So why do I write? I must write—it’s like carrying out a punishment. And the greatest punishment is knowing that what I write will be entirely futile, flawed and uncertain.

As it’s said that “the truth is established” by “two witnesses”, these two botcheries of last night’s effort will be mine.

[This piece is not currently available online.]

The basic study of the intended melody, meant to be a throw-away piece, though it’s perhaps the better of the two in the end.

Black Keys

A more elaborate work on the same melodic material, passing through a number of modulations into other tonal areas than the basic F# minor of the original, though beginning and ending with it.

Further work and revision has—perhaps, or hopefully—improved this latter work, so I have removed the earlier recording; the new incarnation has been premiered on the Facebook “fan” page, under the title “A Waltz in Winter“.

He cometido el peor de los pecados
que un hombre puede cometer. No he sido
feliz. […] Mi mente
se aplicó a las simétricas porfías
del arte, que entreteje naderías.
No me abandona. Siempre está a mi lado
la sombra de haber sido un desdichado.

Borges, “El remordimiento”

I have committed the worst sin a man can commit: I have not been happy. I gave my mind to the symmetric stubbornness of art and all its webs of pettiness. It never leaves me. It is always at my side, the shadow of having been a brooding man.


July 28th, 2010

Borges, “El laberinto” y “El suicida”

Zeus no podría desatar las redes
de piedra que me cercan. He olvidado
los hombres que antes fui; […]
Sé que en la sombra hay Otro, cuya suerte
es fatigar las largas soledades
que tejen y destejen este Hades
y ansiar mi sangre y devorar mi muerte.
Nos buscamos los dos. Ojalá fuera
éste el último día de la espera.

No quedará en la noche una estrella.
No quedará la noche.
Moriré y conmigo la suma
Del intolerable universo.
Borraré las pirámides, las medallas,
Los continentes y las caras.
Borraré la acumulación del pasado.
Haré polvo la historia, polvo el polvo.
Estoy mirando el último poniente.
Oigo el último pájaro.
Lego la nada a nadie.

[This piece is not currently available online.]

A fragment, inspired by the preceding Borges and named “Nada a Nadie” for the last line, which I once meant to complete. I no longer have such expectations and present here only the surviving trifle, so truly lego la “Nada a Nadie”.


June 22nd, 2009

The poem:

Yo que soy el que ahora está cantando
seré mañana el misterioso, el muerto,
el morador de un mágico y desierto
Orbe sin antes ni después ni cuándo.
Así afirma la mística. Me creo
indigno del Infierno o de la Gloria,
pero nada predigo. Nuestra historia
cambia como las formas de Proteo.
¿Qué errante laberinto, qué blancura
ciega de resplandor será mi suerte,
cuando me entregue el fin de esta aventura
la curiosa experiencia de la muerte?
Quiero beber su cristalino Olvido,
ser para siempre; pero no haber sido.


I who am singing today, tomorrow will be dead,
the mysterious inhabitant of a magical, lonely
orb without a past, future or when.
So says the mystic. I think myself unworthy
of Hell or Glory, but I predict nothing.
Our stories shift like the shapes of Proteus.
What wandering labyrinth, what blinding brilliance
will be my fate when I surrender
at the end of this adventure
to the curious experience of death?
I want to drink its crystalline Oblivion,
to be forever; but never to have been.

Daniel Dockery

animî nostrî dêbent interdum âlûcinâri

Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.