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Fibonacci and the Tarot Trumps: a Conjecture
Roland Faber
[Please note that due to the length of this paper, it is divided into two separate online pages: Part A and > Part B.]
Introduction
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Why are there 22 trumps in the Tarot? This question has haunted Tarot-research since the historical quest began. Much has been written about the number 22 since occultists in the late 18th century surmised that the Tarot might represent an (or perhaps the) ancient “Book of Wisdom.” Right from the beginning of the occult preoccupation with the Tarot in the 1780s, Court de Gébelin not only sensed the Egyptian origin of the trumps, but also noted the coincidence of the “classical” number of 22 trumps with the 22 letters of the Hebrew alphabet. This observation occasioned a veritable flood of exchanges between esotericists, with Kabbalistic speculations on the 22 Hebrew letters and their relation to the ten sephirot, the Divine numbers from which, in the Kabalistic tradition, all creation sprang. This presupposition became part of the Tarot tradition, and investigations were undertaken to substantialize these claims.
Independent of this many suggestions were put forward by occultists and historians to explain the substructure of the 22 trumps. They were based on the proposition that one must account for a fundamental, intrinsic subdivision into 21+1 elements, with 21 as the number of trumps and 1 representing the Fool, which in the ancient game was not numbered or—later—was numbered either 0 or 22. They further suggested that the 21 trumps are divided either into a 3x7- or 7x3-pattern, thereby supposedly expressing the two (most) holy numbers three and seven. Such numerological suggestions are still held to represent a valuable instrument for a “real” division of the trumps, built-in by its “creators” to reveal their meaning through a (numerological) pattern. In suggesting the internal relations of the trumps to consist of a “rational” order based on their “mystical” contents, the Tarot was supposed to be an expression of the Divine (and, hence, could also be used for “divination”).
However, until today neither any occult proposal nor any (even profound) historic research has unearthed “rational” (or, for that matter, historically appropriate) arguments that can “explain” the fact of the trump-structure of the Tarot as necessarily implying exactly 22 (=21+1) trumps. On the contrary, the 22 trumps appear as a rather odd addition to a game otherwise structured into 4 suits of 14 cards. Indeed, nothing can be cited which explains the presence of the number 22 in the Tarot. Hence, the number of the trumps seems to be an accidental feature of the conglomerate that became the game of Tarot. |
In what follows, I will propose the (rather tentative) thesis that it is possible to produce arguments for the intelligibility of this number 22. Its “explanations,” however, are to be understood to provide theoretical, historical, and circumstantial reasons that may render the “mysterious number” 22 (21+1) if not necessary then at least probable. This “solution” will be based on a mathematical wonder: the famous sequence of numbers of the 13th Century’s mathematician Leonardo of Pisa, called the Fibonacci Sequence. As applied to the Tarot’s trumps, this mathematical sequence will account for the historical and theoretical probability of an interpretation of the trumps by some of its “creators” as manifestation of the Divine Proportion in the Tarot. Given the contention that the Tarot was set up as a game of cosmological depth (or in its development gained this dimension), this might even have been the reason to include the number 22 in the Tarot in the first place. Although this thesis will be suggested by inference—because there are no documents directly relating this mathematical discovery to the Tarot—it will show its value when it pragmatically gives reason to understand the presence of the number 22 in the Tarot-trumps. Moreover, it renders intelligible the substructure of the 22 trumps—especially regarding the order of the trumps that later has became the “standard Tarot,” known under the generic name of the Tarot de Marseille (TdM).
While the following arguments will reveal considerable mutual congruence between the structure of the Fibonacci Sequence and the grammar of the Tarot-trumps as related to their symbology, they may also, as a side-effect, “explain” some anomalies of the trump-sequence in relation to the suits with their 4x14 elements, each of which probably has its own origin. Contrary to many occult and numerological explanations, however, the Fibonacci-thesis is not in need of any recourse to secret teachings or the presumption of an early origin of the number 22 as applied to the trumps. It considers a mathematical procedure, that was known and used in both arts and sciences, to have become part of an ongoing formation of the order of the trumps—at the time of the invention and initial development of the Tarot: the 15th century, in Northern Italy.
1 - The Mysterious Number
Sunt enim 21 triumphi qui 21 gradus alterius scale in profundum inferi mittentis. Primus dicitur El bagatella (et est omnium inferior). 2, Imperatrix. 3, Imperator. 4, La papessa (O miseri quod negat Christiana fides). 5, El papa (O pontifex cur, &c. qui debet omni sanctitate polere, et isti ribaldi faciunt ipsorum capitaneum). 6, La temperantia. 7, L'amore. 8, Lo caro triumphale (vel mundus parvus). 9, La forteza. 10, La rotta (id est regno, regnavi, sum sine regno). 11, El gobbo. 12, Lo impichato. 13, La morte. 14, El diavolo. 15, La sagitta. 16, La stella. 17, La luna. 18, El sole. 19, Lo angelo. 20, La justicia. 21, El mondo (cioe Dio Padre). 0, El matto sie nulla (nisi velint). (Steele Sermon, circa 1470)
The so called “Steele Sermon,” quoted above, gives the first complete written list of Tarot-trumps in the 15th century. It counts 22 trumps; 21 of them “trumps” in a primary sense, adding a “nulla,” the Fool, which is counted as 0. So we have a list of 21+1 (= 22) trumps. This, in its entirety, is the “mysterious number.”
Traditional historical Tarot research as performed by G. Moakley, M. Dummett, J. Shepard, R. O'Neill and others, has held the seemingly self-evident proposition that this most basic structure, as stated by the Steele Sermon, was an invention ex nihilo. The Ur-Tarot, virtually from the beginning, would have exhibited a structure of 4x14+22 (or 21+1) cards, although maybe with another internal ordering of the trumps. Nevertheless, no reason could be given for this odd fact as such. It was inferred either externally from circumstances under which the “game with trumps” was played, or internally from the meaning of the symbology of the trumps and their relation to more or less contemporary humanist sources, such as Petrarch's Triumphs or Dante's Commedia and their transposition into pictorial series, or astrological conceptions like the Children of Planets. However, if we expect these sources to exhibit a 3x7-structure, supposedly patterning the 21+1 trumps, we would be disappointed, because we find nothing resembling this structure. Petrarch’s poem had only six Triumphs, Dante’s Commedia develops a 3x9-structure, and the “seven planets” of the Children of Planets do not fit the trump sequence (of any given order) at all. Although these structures, like many other symbolic patterns, may be regarded as a valuable sources for understanding the imagery of the trumps as well as their internal relations and suspected symbolism, they do not at all explain the number 22 or its subdivision as exhibited by the trump sequence and its symbolic content.
If the 22 trumps really were there from the very beginning, that is, from the first decades of the 15th century on, we must admit that we know neither why there are 22 of them nor where this number came from. Moreover, this mysterious number seems all the more odd when we consider it to be part of the design of the Tarot-game with 4 suits of 14 cards each. Nevertheless, it was this relation that brought a first hint as to a possible “rational” relation of the numbers in the form of two interesting clues—and they are mathematically based.
The first one, suggested by G. Moakley in the 1960s, pointed to the astonishing fact that there is, indeed, something related to both the 4x14 suit cards and the 21 trumps, namely the much older game of dice, from which they might have stemmed. If we take the total number of choices possible when throwing three or two dice, it adds up to either 56 (= 4x14) or 21, respectively. Although this may be a somewhat interesting connection, reaching back to the invention of “games of fortune,” as dice and cards where considered in the late 14th century when cards came into existence, unfortunately, the bare number of dice-throws does not at all account for the substructures of the 4 suits and the 21 trumps. Besides, the number of throws of two dice only sum up to 21, not 22. The number 22 remains mysterious, that is: the number 22 seems to be isolated from the necessities of the game, the 4x14 suit cards, and the possible symbolic sources for the imagery of the Tarot trumps (unless we accept the Fool to be an addition of the number 0, which, however, is not part of the game of dice).
The other theory by A. Bougearel relies on the indeed fascinating (neo-Pythagorean) relation of three geometrical figures—triangle, square, and pentagon—on the base of the number 4 (as expression of the four elements). They then produce a coherent relation between the odd composition of the Tarot-deck of 4x10 (=40) pips, 4x4 (=16) royals, and 22 trumps, because 16 is a square number of base 4, 10 is a triangular number of base 4, and, likewise, 22 is a pentagonal number of base 4. Nevertheless, the same problem as with Moakley’s solution persists, namely that, although this solution can account for the bare presence of the numbers and their interrelation as one entity, it can not help us with any further understanding of the internal relations and substructures of the 22 trumps as based on their iconography. Besides, while Moakley could find a rationale for the number of true trumps, namely 21, Bougearel’s theory gives the total number of trumps including the Fool, namely 22. In neither case is there reference to the inner complexity of the mysterious number 22 as 21+1.
Newer historical research, however, is on the verge of discarding the “all-at-once” thesis of the origin of the Tarot, instead proposing a gradual appearance of the patterns, later known as the Tarot. Indeed, it is obvious that we have no knowledge of any reference to 22 trumps in the first formative phase of the Tarot from about 1424 to 1461. Rather we find documents and cards that indicate that the early trionfi-game before and from around 1450 involved a 5x14- or 5x16-structure. Evidence comes from the most ancient extant Tarots themselves, namely the 14 original Bembo cards (as oldest part of the Pierpont Morgan-Bergamo Tarot) or the (probably) 16 cards of the Cary Yale Tarot. In fact—all the more mysterious—the first evidence for a 21+1-structure for the trumps happened to appear a poem of Matteo Maria Boiardo from about 1461 (or later), which however in its contents has nothing to do with the trump subjects, except for the Fool. The first extant card set offering a 21+1-structure was the Sola Busca Tarot from the 1490s, which, however, has also no standard subjects besides the Fool. The first 21+1-structure actually exhibiting a “classical” trump – imagery, although only contained in a written statement, is the famous Steele Sermon (sermo perutilis de ludo) of an unknown friar from the 1470s or later. The first extant Tarot pack with standard Tarot trumps that exhibits a 21+1-pattern is the uncut sheets of the Metropolitan Museum of Arts, New York, and of the Museum of Fine Arts, Budapest, as well as the Rosenwald Sheets (only the Fool is missing), both from the early 1500s. Finally, the first extant Tarot pack of cards with supposed 22 trumps is the Catelin Geofroy Tarot from 1557 (although only twelve numbered trumps survived). The first written poems, called tarocchi appropriate, that mention 22 trumps are Pomeran’s poem Triomphi, from1534, and G. Bertoni’s Poesie, from about 1550. G. Susuo’s poem from 1570 might be the first Tarot reference in TdM order; and another century must pass until we find the earliest extant deck in TdM order, the Jean Noblet Tarot.
These facts reveal an even deeper isolation of the number 22: On the on hand, it is no essential part of the symbolic representations of the trumps, which as a set of 22 trumps first occurred with another symbolism. On the other hand, the symbolism of the trumps came into existence at least half a century before we know of the first list of 22 trumps, or any extant Tarot, exhibiting it, as can be seen in the Visconti-Sforza Tarots, especially in the Bembo-14.
A third observation will underline this “splendid isolation” of the 22. The Tarot, in its early times known as ludus triumphorum, was first reported to have existed in 1442 in a note from the account book of the court of Ferrara in which we witness the ordering of some packs of carte da trionfi. Some say that the “14 figures” that were requested from the same court for a visit of Milan’s Bianca Maria Visconti in late 1441 might possibly be the first reference to a trump-set. While the trumps then, at that time, would have consisted of only 14 trumps, we can also only assume that these first known trumps exhibited the classic Tarot-subjects; but actually, we don’t know. Although the (presumably) first extant trionfi-deck, the Cary-Yale Tarot, may have been created in 1441, it came from Milan, not from Ferrara (besides their painter Sagramoro being in Florence). Furthermore, we know of the Cary-Yale Tarot-pack that it is not a standard Tarot pack—having included the three theological virtues that have been omitted from any later standard Tarot (except for the Florentine Minchiate) and consisting of 16 cards. The Milanese Bembo Tarot, on the other hand, which exhibits all the classical trumps, has only 14 cards and seems not to have existed before 1450. So, the first substantial Tarot that resembles a standard deck in number and contents might be assumed to have been created not much earlier than about 1500. As a résumé, we can say that in the earliest times of the Tarot we neither find a 21+1-structure anywhere nor can we be sure at all about the symbols represented by the early trumps.
This becomes obvious when we take into account the 1449-letter of Jacopo Marcello, sending some packs of trionfi to Queen of Isabella of Lorraine in France. One of them actually was the regained “Tarot” that came into the existence by the co-production of count Filippo Maria Visconti of Milan with his secretary Martiano da Tortona, an intellectual and the count’s librarian, and the painter Michelino da Besozzo, after 1424 in Milan. Although Marcello “defines” this game as ludus triumphorum, its trumps actually exhibit sixteen Greek gods (metaphorically referring to the Visconti family and their “mythological” origin). Given these facts, it really is the first (known) Tarot-deck! Marcello’s letter now adds a note stating that this “Tarot” is a “new kind of Tarot” (“novum quoddam et exquisitum triumphorum genus”). From this note, however, it is unclear whether the game of Tarot was still considered new in 1449, when Marcello’s letter was written; or whether the 1424-game, newly discovered in its astonishingly different design, was new in comparison with what otherwise was known as ludus triumphorum in 1449; or whether, in 1449, it was considered a new invention of 1424 in relation to the Tarots supposed to have been existed at the time. But in any case, this first known Tarot neither had 22 trumps nor did it exhibit any of the trump-subjects known to have become the standard identified as “Tarot” (and as it is recognizable as early as in the Bembo Tarot from around 1450).
From these facts, we may ponder the introduction of the number 22 into the trump-structure by imagining a “force-field” between at least seven paradigmatic Tarots of the 15th century—the Michelino Tarot (~1424), the Bembo Tarot (~1450), the Boiardo Tarot (~1461), the Sola Busca Tarot (~1490), the Metropolitan Museum of Arts/Museum of Fine Arts, Budapest Sheets (~1500) and the Rosenwald Sheets (~1500)—. Firstly, we can say that the number 22 played no role in the formation of the Tarot of the first fifty years of its existence (at least we don’t have any evidence that it did). Its first appearance is the Boiardo Tarot from 1461 or later. Secondly, the number 22 was not understood as a necessary measure to identify a “game played with the trumps,” as can be witnessed with the Bembo-deck from about 1452, which—if we take into account another Ferranese document from 1457 in which we have evidence of two trionfi-decks with 70 cards each—probably had only 70 cards and, hence, 14 trumps. Thirdly, the number 22 has no substantial relation to the symbols and figures used for painting and structuring the trumps, as can be evidenced by the Michelino-, the Boiardo-, and the Sola Busca-decks between 1424 and 1490. Fourthly, the number 22 is not very appealing for a game played with 4x13, 4x14, or 4x16 suit-cards, which evidence suggest was in use from the 1370s on (with also 4x12, 4x15, 5x12, 5x13, or 5x14 cards) when the Tarot began to develop.
If we take all of these facts into consideration, only one conclusion seems plausible: The number 22 appears as an isolated number in the context of the formation of the early Tarot. But how and why did this mysterious number then came into the Tarot? Why was a comparatively “rational” structure of 5x14 cards (Bembo) or 5x16 cards (Michelino, Cary-Yale), which correlates smoothly with the suite-structures in existence, driven apart into a double-faced pattern of virtually no (recognizable) internal relation—a deck with 4x14 suit-cards and a trump-suit consisting of 21+1 elements? Taking this “ anomaly” into account, this mysterious number might well have had its own origin, independent of the invention of Tarot, and then found its way into the formation of the Tarot in the late 15th century. At least it must be considered as an entity in its own right. Once it became an element of the Tarot-structure, however, the mysterious number 22 surprisingly reveals itself as a major force for the further evolvement of the trumps-suit and its interpretation.
On this historical basis, we ask again: Where did this mysterious number 22 come from? What reasons can be given to understand the “rationale” of its inclusion in the Tarot? Not many theories were put forward to explain this stunning development. One of them proposed what the occult interpreters finally have made one of the cornerstones of their interpretation of the Tarot-trumps as “Major Arcana,” namely that there may have been a Kabbalistic connection. Although we find no factual hint of any kind for such a connection prior to the de Gébelin-thesis 1781, we might, by inference, gather some information on the cultural situation of the second half of the 15th century, which might reveal some facts in favor of such a connection. Indeed, the Kabbalah used the 22 letters of the Hebrew alphabet to represent both numbers in general (as was a common feature of Jewish counting) and the relations between its 10 sephirot on the Tree of Life. Furthermore, they were used for spiritual purposes, be they mystical (e.g., in relating Divine attributes, represented by the sephirot, or in their numerology, called Gematria) or magical (as talismans or amulets). They were also understood to represent the cosmic order in its most elaborate way. Known to Christian circles from Raimond Llull (+1316) on, and still being present in Northern Italy through Pier Leoni, the physician to the Medici in Florence by 1475, as well as a circle around Nicolas of Cusa at the university of Padua at about the same time, Kabbalah was systematically studied and disseminated by the great Renaissance philosopher Pico della Mirandola at the Florentine Platonic Academy, also in the late 15th century. Additionally, conditions may have been favorable at this time for Jewish-Christian exchange between Florence and Ferrara where Boiardo, the creator of the first (as poem) known Tarot with 22 trumps, moved in 1461. He might have had Jewish teachers and certainly knew “oriental languages.” Furthermore when we take into account that Boiardo was the uncle of Pico della Mirandola, it is not impossible to assume that Boiardo, in fact, invented or at least used the mysterious number to structure the Tarot-trumps by reference to the Hebrew alphabet mediated by its Kabbalistic connotations.
Yet, it is highly improbable. The reasons for this are many. Firstly, the number 22 of the Tarot-trumps does not correspond to the alphanumeric theology of the Sefer Yetzirah (circa 5th century), the basic writing adopted by the Jewish and Christian Kabbalah, because its 22 letters were neither associated with 0 to 21 nor with 1 to 22. They were read as connecting the 10 sephirot (in the form of the Tree of Life). While the sephirot were counted 1 to 10, their interconnections represented the numbers 11 to 32. Secondly, the Hebrew numbering system as associated with the 22 letters of the alphabet did not count from 1 to 22, but represented the numbers from 1 to 400. The 1st to 9th letter being 1 to 9; the 10th to 18th letter being 10 to 90; the 19th to 22nd letter being 100 to 400. The earliest correspondence between the 22 letters and the first 22 numbers 1 to 22 dates from an invention of Michael Stifel in 1532. So, we may assume that the number 22 that Boiardo used for his poem in relation to the trumps was not borrowed from the Kabbalah, not even from the Jewish counting system as based on the Hebrew alphabet to which it otherwise would hold only extremely external relations, namely only the incidental resemblance of the bare number 22 stripped of all the mystical, magical, or alphanumerical connotations.
There is, however, another source for the number 22 we may assume to have been used for the trumps: the Book of Revelations. The Tarot historian R. O’Neill, for example, has suggested that it is no coincidence that the number of chapters of the last book of the Christian Bible is the same as the number of Tarot-trumps - 22. Indeed, there are reasons to believe that the Book of Revelations was an important source for the imagery and, even more, for the semantic and syntactic formation of the symbolism of the Tarot-trumps. As some researchers, like R. O’Neill, T. Betts, M. Hurst, and myself, have put forward, the Tarot-imaginary as, for example, represented by one of the earliest Tarots with only classical trumps, the Bembo-14, might have been a product of eschatological considerations of a time and culture, arising after the impact of the Black Death that reached Europe between 1347 and 1352 (and from there on struck on average every 12 years). In the wake of its devastating passage, removing a third of the population of Europe, the specific way of depicting “(Black) Death,” namely as a decomposing corpse or skeleton—as invariably seen in the Death-card—arose only from about 1360 on to remind people to live their lives under the icon of the immediacy of death. Furthermore, the apocalyptic movement, which had been present from the times of Joachim of Fiore (1202+) and through the inheritance of the Spiritual Franciscans, must have gotten a major boost by this “event.” By everyone from the Pope downwards, pestilence was seen as the wrath of God, visited on the sinful world, and a sign of the imminent end of the world, initiating the Last Judgment right now. Whichever way this might have been seen—as a reminder of the inevitability of death and judgment, or as heralding the imminent arrival of the Judgment Day—the early Tarot reflects this situation in the last portion of the trumps, the “eschatological” trumps.
When we look at the earliest extant, (probably) “complete” Tarot deck, the Bembo-14, the imagery “ends” with Death, the representation of the Black Death, and Final Judgment, the depiction of the Resurrection to Judgment (be it individually or collectively). Later, when another six trumps where added to the original 14 Bembo-cards, this now so called Pierpont Morgan-Bergamo deck exhibits some additional “cosmological” subjects grouped around Death and Judgment, like the Star, the Moon, the Sun, and the World (which all were missing in the original Bembo-14). What they do is to introduce an additional “apocalyptic” flavor to the overall picture so that we are now firmly drawn to the conclusion that these early Tarot-trumps wanted to tell the story of the Apocalypse, which of course was a focus of interest at the time through the Book of Revelations and its numerous depictions in the 15th Century’s Books of Hours, Apocalypses, and illuminated Bibles. As the Bible text itself and many of these book illustrations reveal, we have good reasons to interpret Star, Moon, and Sun as the apocalyptic signs of Rev 6:12-13—the falling stars, the reddened moon and the blackened sun. We may look at the Star and read it as either another sign of the freed demons in Rev 9:1 or as Christ as morning star of Rev 22:16. We may see the World and might envision the New Creation and the New Jerusalem of Rev 21:1-2. And if there really existed an intimate relation between the early trump symbolisms and the Apocalypse (at least in the eyes of the people of the time when this imagery was invented), we might also assume that structural features have been transferred. The number 22, then, would be a revealing and affirming sign of this transference not only of the contents of the Apocalypse, but also of an overall structure, as exhibited by the Book of the Apocalypse, the Book of Revelations—and its the 22 chapters.
Nevertheless, a similar problem already recognized regarding the Kabbalah-connection arises: We cannot find any similarity of the micro-structure of the 22 chapters of the Book of Revelations and the 22 Tarot-trumps! Nevertheless, considering the wider Apocalyptic background, we could refine the thesis by adding that the number 22 might not have been transferred from the Bible directly, but rather from the living Apocalyptic tradition, e.g., as represented by the Spiritual Franciscans, who revered the vision of Joachim of Fiore. Indeed, his apocalyptic numerology of history envisions three stages of Divine revelation, subdivided into seven phases. So, this apocalyptic approach, present at the time of the invention of the Tarot, instantiated a 3x7-pattern similar to one of the most “rational” divisions of the trump-series. Even so, this order, again, if considered regarding its micro-structure, does only make sense for the TdM-order, which I consider not to be in place before the 16th century. Although such a relation might be a real possibility, the character of the 16th century—after a hundred years of cultural humanism and Renaissance “rationality”—makes it highly improbable that, when the number 22 became a formative part of the Tarot, the Apocalypse should still be a source for Tarot formation. Times had changed, so had the contents: The New Jerusalem of the Visconti-Sforza Tarots gradually began to transubstantiate into the World-Soul of the TdM; biblical contents changed into cosmological and spiritual reflections with neo-Platonic overtones, made possible by the intense cross-cultural transfer between the East and the West in the second half of the 15th century. In other words: The number 22 comes too late for the Apocalyptic transfer. Again, we are left with the isolation of the number 22 as a seemingly independent source for the trump formation.
Besides the Kabalistic and the Apocalyptic transfer, however, there is another possibility brought forward by Huck Mayers (and the autorbis-team). It is built around private coincidences at the Visconti court from which the first extant Tarots stem: Galeazzo Maria Sforza was 22 years old in the year 1466, when his father died and he became duke of Milan; Maria Bianca Sforza, Galeazzo’s mother, was 22 years old when her father, Filippo Maria Visconti, died in 1447. 22 years old was Galeazzo Maria’s father, Francesco Sforza, when his father, the famous condottieri Muzio Attendola, drowned in a river near Aquila. So in 1466, three 22’s accidentally accumulated from father, mother, and son. Galeazzo Maria, in turn, was stated to have been very superstitious. How should this family-related “triple 22-event” not have caught his attention? From these coincidences, then, the number 22 could have come into the Tarot, which was already in use in form of patterns of 5x14 (Bembo) and a 4x16 (Cary Yale) at this court (and not counting the Michelino Tarot). In favor of this thesis one may argue that there are some hidden hints that the extant, exquisite Tarots that were commissioned by the Visconti-court were actually symbolic accounts of court events, disguised by the developing standard imagery.
Nevertheless, this argument remains problematic in light of three other considerations: In the first place, it reduces court matters to private events of highly random coincidence instead of taking into account the otherwise highly archetypical world-view of that time. Numbers were part of a mythological realm of relations that were operating on levels of cosmology, theology, philosophy, and astrology. To reduce them to private numbers appears to be an anachronistic projection of the “new subjectivity,” achieved in the age of Enlightenment two centuries later. While the 17th century made a distinction betweeen the scientific use of numbers and any private subjective use, this was not the general attitude in the (early) Renaissance. Evidence of this is found in the Michelino Tarot, which on a mythological level created an archetypical nimbus around the genealogy of the Visconti-family by relating it to Greek gods, and sixteen of them. In the second place, this thesis reduces the process of the Tarot-formation to a matter of the courts alone in which we first find documents linked to the (birth of the) Tarot. In other words: It does not take into account the urban context in which the Tarot was created and of which it was part, that is, the social triangle between court, university, and market place—or between courtiers, intellectuals, and merchants. It was this new economic structure that allowed the urban agents, like card makers, artists, and rich merchants, to create and commission “art” independently from court interests. So, I would propose a wider cultural context to be applied for the formation of the Tarot. This is missed by the highly randomized 22 of this thesis. In the third place, in common with the other theories discussed, it does not contain any reference to, or explanation of, the substructure of the trump set; it is opaque in relation to the (supposed) “story” of the trump – imagery.
Apparently, none of the suggested theories concerning the origin of the number 22 as related to the structure of the Tarot-trumps has succeeded either to explain its appearance or its incorporation into the Tarot-tradition or, for that matter, has been able to account for the substructure of the 22 trumps as found in the Tarot. The Tarot, obviously, appears as a fusion of three essential, but essentially independent, elements, namely the imagery (with the structurally relevant symbolism of the trumps), the function (relating the trumps structurally to the four suits) and the number (between 14 and 22, or, as in the Minchiate, even 41) of trumps.
Regarding the number 22, this leads us to the following paradox: If we allow for a theory of the gradual development the Tarot, being a fusion of these three elements to what finally has become the standard Tarot, probably between 1500 and 1650 (= TdM), the Kabbalistic connection would be right in place at the right time, while the Apocalyptic connection obviously comes too late for substantially informing the Tarot. If, however, we prefer a theory that proposes that the basic structure of the Tarot (and, thereby, the “fusion” of imagery, function, and number) virtually has been in existence from the invention of the Tarot on, approximately between 1410 and 1450, then the Apocalyptic relations would be in place at the right time, but it would be definitely too early for a substantial Kabbalistic connection. However, both solutions lack any significant match between their points of reference for the number 22 and the early trump-imagery. Besides, both theories are devoid of any power to account for the substructure of the 22 trumps in any reasonable way.
Imagining an ideal theory based on these facts of mutually independent elements joining the Tarot-composition at a certain time that is not necessarily identical with the supposed “origin” of either the Tarot or each of its three formative elements, we might like to think of it as having the following characteristics: In the first place, it should allow for a gradual development of the Tarot-structure. In other words: It should not be based on any worldview—be it Apocalyptic or Kabbalistic—that might not have been in place to be integrated with the emerging Tarot at the right time—namely the late 15th century. This would also include a preference for an interpretation that is not based on a secret doctrine, but a doctrine that was available in a wide cultural context to be included (independently from any secret group) in the urban space into which the Tarot was born. In the second place, it should be a theory that can explain the number 22 as an independent entity, but also one that came into the Tarot through more than an idiosyncratic decision by some of its “creators,” even if it were the “holy will” of a superstitious count of Milan. In other words: This theory must account for the essential appeal of the number 22 in relation to the Tarot to render feasible its use for the Tarot (without, again, supposing secret doctrines). In the third place, it should be a theory that “explains,” or at least fits, the (probable) substructures of the 22 trumps as suggested by their symbolism and imagery—at least better than any other theory already discussed. This endeavour, I believe, could be achieved by the mathematical object called the Fibonacci Sequence, which in what follows is proposed to be a probable formative element of the mysterious Tarot-entity “22.”
2 - The Fibonacci Sequence
Fibonacci was a mathematician. Born as Leonardo of Pisa between 1170 and 1180, he was the son of Bonacci, a diplomat for the Republic of Pisa in Algeria—hence his name “Fibonacci,” an abbreviation of filius Bonacci. He was educated in Algeria, and like many intellectuals of the middle ages, he traveled enormously—Syria, Greece, Sicily, and the Provence—to learn all known methods of computing. Incidentally, in the same year in which Joachim of Fiore died, in 1202, Fibonacci, back in Pisa, published his major opus Liber abaci, a 450-pages long treaty that introduced the Indian art of counting and the Arabic numeral system to European mathematics. With these new elements, we are on the verge of a new era in European history, not only for mathematics but for the new world-view that grew to its full eminence in the Renaissance and finally led to the scientific revolution in the 17th century.
Indeed, all began with a revolution: Fibonacci was the first (Western) mathematician to understand the sign 0 not to be an equivalent to “nothing,” but to symbolize a number. The Liber I of his Liber abaci began with the words:
These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0, which in Arabic is called zephirum, any number can be written, as will be demonstrated.
Besides the importance of this decimal system for merchandizing procedures (even the use of negative numbers for depths, which was soon recognized as very valuable) of which the number 0 became an essential part, Fibonacci transmitted Arabic science to Europe and, thereby, made modern mathematics possible.
Back in Pisa by 1200, he reached out to other men of learning in Europe. He began a spirited correspondence with Michael Scotus, the court astrologer of Emperor Frederick II of Naples; with Theodorus Physicus, his court philosopher; and with Dominicus Hispanus, who suggested to Frederick to meet Fibonacci when Frederick’s court resided in Pisa around 1225. Because at that time books were handwritten, only a few copies remain, and several works were lost, e.g., Fibonacci’s commentary on Book X of Euclid's Elements that contained a numerical treatment of irrational numbers, which Euclid had only discussed from a geometric point of view. After 1228, only one document exists from 1240, when the Republic of Pisa granted Fibonacci a yearly pension. Despite the importance of his number theory, he was virtually unknown in later times, until he was rediscovered in the 19th century. Although he was perhaps the most sophisticated mathematician of his times and his achievements were clearly recognized, it was only centuries later that he was granted the status of the major contributor to number theory before the 17th century’s French mathematician Pierre de Fermat.
In 1202, Leonardo solved a certain mathematical problem by inventing a number-sequence that later became famous under the name “Fibonacci Sequence.” The original problem that Fibonacci investigated was about how fast rabbits could breed in ideal circumstances. Suppose a pair of rabbits (always being male and female) is able to mate at the age of one month so that at the end of its second month a female rabbit can produce another pair of rabbits. Further suppose that the rabbits never die and that the female rabbits always produce one new pair every month from the second month on, the question is: How many pairs will there be in one year? At the end of the first month, there is still only one pair; at the end of the second month one new pair was produced—so there are two pairs; at the end of the third month, the original female produces a second pair—making three pairs; at the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also—making five pairs. Hence, the family-tree of the rabbits exhibits the following interesting sequence: 
As can be examined by this the diagram, it is the levels of the reproduction rate of the pairs of rabbits in this family-tree that generates the exponentially increasing sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …
This is the Fibonacci Sequence.
The structure of this sequence is that of a recursive series of numbers. It has a point of departure, a generating pair (1+1), and a formula for addition, namely that every number is added to its precursor, to produce a new number that is added again to its precursor:
1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21 …
Stated as formula with the initial two numbers being a1=a2=1, the Fibonacci Sequence reads as:
This sequence, in which each number is the sum of the two preceding numbers, has proved to be extremely fruitful. It appears in many different areas of mathematics and science. Today, the Fibonacci Quarterly, published by the “Fibonacci Association,” founded in 1963, is solely devoted to the studying of the mathematics related to this sequence, seeking and discussing fascinating applications, virtually without end.
One of the most interesting features of the Fibonacci Sequence was found by realizing that the ratio between successive Fibonacci numbers approaches the famous Golden Ratio, and a geometrical construction using the Golden Ratio as the width to length ratio for rectangles in a nested configuration leads to the so called Golden Spiral.
Consider the fractions formed when we divide the first Fibonacci number by the sceond, the second by the third, the third by the fourth, and so on, ad infinitum. The succession of fractions (or ratios) is as follows:—1/1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34,.... ad infinitum.
It is easier to appreciate the size of these fractions if we express them in decimal form (correct to 4 decimal places):
1.0000, 0.5000, 0.6667, 0.6000, 0.6250, 0.6154, 0.6190, 0.6176,...
We see then that successive ratios are drawing closer together. They oscillate around an implied fixed value, gradually getting nearer and nearer to it. This number which the sequence of ratio converges towards is known as the Golden Ratio, named Ø', which is has a value of about 0.61803. As an exact equation, the Golden Ratio reads as: 
Inversing the ratio in the Fibonacci series (taking the higher of successive numbers and dividing it by its predecessor: 2/1, 3/2, 5/3, 8/5, 13/8, etc) brings us to the more common expression of this number also referred to as phi(φ or f). It has a value of 
Ø and Ø' have some unique properties such as 1/Ø' is the same as 1+Ø'=Ø. Above the eighth Fibonacci number, the ratio of the subsequent fractions closely approximates the Golden Section, ie, Ø. 
Because the Golden Ratio is an irrational number, that is, a number that cannot be stated as a ratio of natural numbers, it approximates indefinitely to the Golden Section. And like pi (p), its digit-sequence never repeat.
Geometrically, the Golden Ratio produces a very special proportion, known as the Golden Proportion. If we take a line with its two endpoints A and C and than divide the line between A and C at the point B so that the ratio of the short part of the segment (AB) to the long part (BC) equals the ratio of the long part (BC) to the entire segment (AC), we have the Golden Proportion: 
The ratio of the lengths of the two parts of this segment is also called the Divine Proportion, because it was considered to be of most natural beauty, exhibiting a Divine harmony. Stated as an equation, it reads as: 
Restated in two dimensions, the Divine Proportion produces the Golden Rectangle, a rectangle in which the ratio of the length to the width is the Golden Ratio. First constructed by Pythagoras in the sixth century BCE as allegedly aesthetically most pleasing rectangular shape possible, it is defined as the rectangle that, when squared within, leaves another Golden Rectangle behind. 
The Golden Rectangle has the Golden Proportion x/y. Section a is a square drawn in the rectangle with the proportion x/x. Section b, however, is another Golden Rectangle, with its own Golden Proportion (y-x)/x, that is, the ratio of the lengths of the sides of section b is the same as the ratio of the length of the sides of the entire large rectangle. This is the characteristic of a Golden Rectangle exhibiting the Golden Proportion. It is recursive like the Fibonacci Sequence, that is, when another square with lengths the same as the length of the short side of the rectangle is inscribed, we create another rectangle with the same proportions as the original.
When we go on to divide the rectangle in this way, we produce the Golden Spiral as a series of quarter-circles drawn into the successively produced squares that are inscribed into the rectangle to produce the Golden Proportion. The Golden Spiral is constructed in a similar way as the Fibonacci Spiral, produced by the Fibonacci Sequence, except that the latter approximates the Golden Spiral with the same degree of tiny inaccuracy previously shown (with the Ratios). 
Although the Divine Proportion as generated by the Fibonacci Sequence seems to be an abstract concept, it actually, as many scientist (and esoterists alongside) believe, can be found in many places in nature. At least, reality seems to be such that it can be reconstructed as expressing this ideal proportion. Phi manifests, e.g., in the structure of the human body: If we take the length of the hand, then the combined length of hand + forearm has approximately the value of phi. The same may be said for the proportion of the upper arm to the compound of hand + forearm. Furthermore, the human head seems to form a Golden Rectangle with the height of the eyes at its golden mean. Mouth and nose are each placed at Golden Sections of the distance between the eyes and the bottom of the chin. Also, the human body may be supposed to be based on phi; also the DNA cross-sections. Phi is understood to be manifest in fauna and flora, too: Eye, fins, and tail of a dolphin’s body, for example., all fall at Golden Sections of its length. It is further believed that phi and the Fibonacci Sequence are illustrated in the numbers and arrangements of petals, leaves, sections and seeds of many plants, such as pinecones, pineapples, and sunflowers, which are formed in spirals, or plants that produce new branches in quantities that exhibit the Fibonacci Sequence.
Besides its natural instantiations, it is assumed that phi and the Fibonacci Sequence are present in classical architecture and art in an unexpectedly vast amount of structures and constructions. The Great Pyramid of Giza, the Parthenon of Athens, or the proportions of Notre Dame in Paris are all said to conform to these mathematical entities—be this a coincidence of unconscious expression of natural harmonies, or be it by conscious application through mathematical or, moreprecisely, geometrical knowledge. Since the Medieval builders of churches and cathedrals approached the design of their buildings in much the same way as the Greeks, who built their constructions on the Golden Section, they are also supposed to have used this Divine geometry.
With Renaissance humanism, artists like Leonardo da Vinci (1451-1519), Michelangelo (1475-1564), and Raphael (1483-1530) are understood to have deliberately applied the Golden Ratio to construct their compositions. We know that the proportions in (some of) Leonardo’s works reflect a deep interest in the mathematics of nature. Earlier still Pythagoras had made a close study of the human body and its proportions and had shown that its various parts were related to the Golden Section. Leonardo’s unfinished canvas “Saint Jerome,” as another example, shows the great scholar with a lion lying at his feet, where a golden rectangle fits neatly around the central figure. So it is often said that the artist may deliberately have painted the figure to conform to those proportions. So did Michelangelo with his “David” which smoothly conforms to the Golden Ratio (from the location of the navel with respect to the height to the placement of the joints in the fingers). This may also be true for many other works of Renaissance-painters around 1500, and from this time on.
Where, however, had the Renaissance knowledge of the Golden Ratio come from? We know that a certain mathematician of the Italian Renaissance, Luca Pacioli (1445-1514), made it known. Being a close friend of some of the great Renaissance painters, he also studied the Golden Ratio very closely by grounding his knowledge on the theories of none other than—Leonardo Fibonacci. Hence, he may not only be accorded the rediscovery of the Fibonacci Sequence for the times of the Renaissance, but also with the acknowledgment of its fundamental importance and the potentialities for its application in Renaissance science and art. Furthermore, he realized the metaphysical value of this number. When he published his mathematical considerations in 1509, he states its utmost importance by referring to it as the Divine Proportion. All of the sudden we are also in the cultural “space” of the development of the Tarot. And if this was not enough of “coincidence,” we may consider this stunning fact: At the beginning of his treaties on the Divine Proportion, where Pacioli places a Compendium de divina proportione, which is a discussion of the fundamental value of the Golden Ratio, we read a very familiar dedication from December 1498—namely to the Duke of Milan, Lodovico Sforza.
In summarizing these findings regarding the Fibonacci Sequence, the Divine Proportion, and their historical “situation” in the Italian Renaissance, we can say that we are, indeed, at the right time, the right place, and the urban atmosphere, altogether allowing for the suggestion of a possible connection with the Tarot. By its rediscovery in the Renaissance, the Fibonacci Sequence not only introduces us to an interesting recursive series of numbers, but to a mathematical entity that is virtually all-present in nature and classical architecture and, even more important, was considered all-present in nature and art in the Renaissance—at least from the end of the 15th century on. Since it generates this very special proportion which was called Divine Proportion and since this, incidentally, was known to Renaissance Italy, I find it worthy of being considered to be of real influence on the development of the Tarot. While Renaissance art and architecture began to reinitiate its deliberate use, thereby associating itself as part of a long-standing tradition that can be conceptualized as “Divine Architecture and Art,” the Fibonacci Sequence obviously must have been recognized to be exhibiting a law of nature so profound that it virtually could not haven been ignored in matters of aesthetics, humanism, and studies of cosmology, all of which should be understood also to have influenced the formation of the Tarot.
In fact, the Fibonacci Sequence is known to have been taught, practiced, applied, and revered in the urban, intellectual, and increasingly cosmopolitan atmosphere of the growing Renaissance-consciousness in the finest Italian cities of the late 15th century—Padua, Florence, Ferrara, and Venice—where Tarot seems to also have been invented. If this was not enough motivation to study a possible influence on the Tarot, we may also take into consideration that this mathematical knowledge was even brought to the attention of the Dukes, the court, and the educational institutions of Milan—one of the primary “loci,” we must suspect the Tarot to have been invented and where we see it flourishing from the 1450s on. Given the substantial independence of the appearance of the number 22 in the final Tarot-composition (as has been suggested in section one) and given the strange fact that the mysterious number 22 began to influence the trump-sequence earlier or later in the same century (and not later than at the beginning of the 16th century) in the same places the trump-patterns were evolving to finally create the TdM—all of this is, in my opinion, reason enough to not ignore the Fibonacci Sequence (and the Divine Proportion) to be considered a possible source for the incorporation of the number 22 into the Tarot.
> continue to Part B
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