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A phylogenetic interpretation of the canonical formula of myths by

Levi-Strauss

Marc Thuillard

independent scholar; St-Blaise, Switzerland; e-mail: Thuillweb@hotmail.com

Jean-Loïc Le Quellec

Institute of African Worlds (IMAf), National Center for Scientific Research; Paris, France;

e-mail: JLLQ@rupestre.on-rev.com

Abstract

Lévi-Strauss’ canonical formula and Mosko’s narrative formula have a simple interpretation

within the theoretical framework of phylogenetics. The canonical formula represents a complex

but by far not unique way of combining 2 and 3-states characters in a phylogenetic tree

description of myths. The canonical formula describes an instance of myth’s evolution that can

be described exactly by a perfect phylogenetic tree. Mosko’s formula describes a completely

different scheme of evolution. Mosko’s formula is typically the result of a fast evolution of

mythemes resulting possibly in all combinations of binary characters. The evolution of myths

corresponds quite often to intermediary situations. This observation may explain why the

canonical formula has been identified only in a limited number of instances.

Keywords: Lévi-Strauss; canonical formula; narrative formula; myths; phylogeny

1. Introduction

The discovery of the connections between many apparently unrelated myths or folktales has

been one of the most significant advances in their studies (Propp 1928; Lévi-Strauss 1984: 115-

45, 1964-1971). Lévi-Strauss defined a myth as the set of all its versions and suggested that the

different versions can be described by a general rule, called the canonical formula. (Lévi-Strauss

1955, 1988; Godelier 2013). The canonical formula has interested as well mythologists and

anthropologists (Maranda 2001; Desvaux 1995; Mosko 1991; Scubia 1996; Mezzadri 1988;

Berger 2013; Burkert 1979) as mathematicians (Racine 2001; Côté 1995; Solomon 1995;

Morava 2005; Darányi et al. 2013; Petitot 1988, 1995, 1997, 2001), Despite Lévi-Strauss’

clarifications and the demonstration of the predictive power of the formula (Lévy-Strauss 1988)

one is today in a somewhat paradoxical situation. The canonical formula is perceived as

capturing fundamental, non-intuitive relations on how myths transform, but is neither

completely defined nor understood (Godelier 2013). The predictive power of the canonical

formula has been verified only in a limited number of examples. Mezzadri (1988) applies it to

the famous Greek myth of the ‘different races’ described by Hesiod. The canonical formula can

take into account the entirety of the myth and, above all, allows the prediction of the existence

of "anti-kings" of which the myth itself does not speak! Now the existence of the "anti-king" is

verified well in the ritual context of the myth, with the’ pharmakos’, that is to say, the man who

served as a scapegoat in the Greek cities. Let us also mention here the recent work by Berger on

the analysis of Malagasy circumcision rites (Berger 2013)

In recent years, the combination of aera studies and phylogenetics has emerged as a particularly

meaningful approach to the study of myths (Le Quellec 2014; d’Huy 2013). We present here an

interpretation of the canonical formula in terms of transformations on a phylogenetic tree.

According to the canonical formula, a myth is reducible to an expression:

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The left part of the canonical formula contains 2 functions acting on two terms . The

different terms can be typically persons or animals. The right part of the formula contains the

inverse of the term as an index of the function. Maurice Godelier (2013) furnishes an

excellent introduction to the state of the art on the question in his recent book: “Lévi-Strauss”.

During Levi-Strauss’s lifetime, several critics have questioned the meaning of such a formula.

Responding to those critics, Levi-Strauss furnished more explanations and examples in “The

Jealous Potter” (Lévy-Strauss 1988).

The Jealous Potter is a Jivaro myth about the origin of pottery (see location of the Jiravoan

territory on the map).

Map: Approximate location of the Jivaroan territory

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A woman called Goatsucker is married to both Sun and Moon. After a quarrel caused by

jealousy, all three rise to the sky. At some point, Goatsucker falls from the sky with her basket

full of clay. Goatsucker dies and is transformed into a goatsucker bird. A curious aspect of the

myth is that it relates jealousy to the origin of pottery. Among Jivaros, potters are often thought

of being jealous. The goatsucker lives a solitary life and is mostly active at night and among

Jivaros the bird is also associated with jealousy. Lévi-Strauss explains the connection between

the goatsucker and pottery using another bird, the ovenbird. The ovenbird is a very social bird

that builds its nest out of clay and appears in other myths. It can be considered as the inverse of

the solitary and gloomy Goatsucker. According to Lévi-Strauss, the connection between jealousy

and pottery is given by the canonical formula expressed as:

.

The expression can be translated as: ‘the function jealousy of the Goatsucker is to the potter

function of the Woman what the function jealousy of the Woman is to the function ovenbird of

the Potter.’ The canonical formula relates typically different components of a given myth version

to other myths or versions of a myth. In the Jealous Potter, the last component is not part

of the original Jivaro myth but is found in other myths. In recent years, the canonical formula

has been shown to be related to quaternions (Morava 2005), an extension of complex numbers,

and also possibly to quantum mechanics (Darányi et al. 2013). The interpretation of the

canonical formula in terms of quaternions or within the framework of quantum-mechanics is

certainly very appealing, but still quite far from an intuitive explanation of the canonical

formula. In section 2, a phylogenetic interpretation to the canonical formula is given and the

predictive power of our new approach is shown in section 3.

The narrative formula, a variation of the canonical form, is believed to better describe the

transformation of some myths (Mosko 1991). We show in section 4 that the narrative formula

can be interpreted in terms of transformations on a phylogenetic network, a generalization of

phylogenetic trees.

2. Phylogenetic Interpretation of the Canonical Formula

We suggest below a new, more intuitive interpretation of the canonical formula. Let us first

transform the different functions and terms in the canonical formula into a binary character B

with states {0,1} and a 3-state character T with states {-1,0,1}:

B:

T: .

For the Jealous Potter, one has

Let us note that the term Woman with state ‘0’ occupies an intermediary position at the same

distance to the state ‘-1’ (Goatsucker) as to the state ‘1’ (‘Fovenbird). The indetermination between

is simply removed by requiring that each member contain a function and a term. Applied

to the Jealous Potter, one obtains:

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The different terms and functions of the canonical formula are represented in Figure 1 as a

transformation on a phylogenetic tree. The canonical formula expresses a simple relation

between the nodes of the trees: The first node is to the third node what the second node is to the

fourth node.

Figure 1: The different terms and functions in the canonical formula are represented on a

distance-based phylogeny (top) and in a character-based phylogeny (bottom). The distance

matrix is obtained by summing on each character the absolute value of the difference between

the state values.

3. Applications of the Phylogenetic Approach to the Canonical Formula

a) Identifying 3-state characters semi-automatically

The identification of 2 characters fitting the canonical formula requires a deep knowledge of the

different versions. Let us describe for the canonical formula

a method for reducing the number of possible characters that have to be analyzed by a specialist.

Let us first define binary characters based on the presence/absence of a character. The reader

can verify that no perfect tree describes exactly the data after binary coding. The Dress-Steel

procedure (Dress and Steel 1992) may be applied in order to discover the 3-state character as

sketched in Figure 2. Applied to the canonical formula, the procedure simply consists of

removing iteratively one unique binary character and check whether the data fits exactly the tree

in Fig. 1. One can show that the unique solution consists of removing the binary character

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corresponding to the ‘b=0’ state (i.e. “Woman” in the Jealous Potter). The two characters with a

unique ‘1-state’ can be identified to the ‘-1’ and ‘1’ states. By using this procedure, a human

operator has to verify only one proposal among the 10 possible combinations of one 3-state and

one 2-state character (For different forms of the canonical form, the procedure may lead to more

than one instances that have to be verified by a specialist).

Figure 2: The different terms and functions in the canonical formula can be identified by i)

binary coding the characters according to their presence/absence in a version of a myth; ii)

eliminating one character so that the resulting tree is the line tree drawn below; attributing the

state ‘0’ to the removed character and the states ‘-1’ and ‘1’ to the 2 characters with a unique ‘1’

state.

b) An example showing the predictive power of the phylogenetic approach

More complex formulas can be obtained, for instance by combining two 3-state characters. The

two canonical formulas and can

be represented on a phylogenetic tree with 5 end nodes after transformation:

If the term is interpreted as ‘trust’ then the last component corresponds to

(or better as and the two related canonical formulas

make quite some sense. (Lévi-Strauss describes the ovenbird so: “Unlike the goatsucker, the

ovenbird likes to be close to humans…and constantly chats with his female” (Lévi-Strauss

1988)). This example shows the predictive power of our phylogenetic interpretation of the

canonical formula. Let us demonstrate the power of the phylogenetic approach with a second

example.

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Figure 3: Tree structure obtained on two 3-state characters using the characters of the ‘Jealous

Potter’.

c) “The Hare/Rabbit in the Moon”

If one drops the distinction between terms and functions, other forms of the canonical formula

can be identified. The combinations of two characters may describe a double inversion similar to

the canonical formula. As an example, let us discuss the ‘Rabbit/Hare and/or the Frog/Toad in

the Moon’ tale, one of the oldest tales already recorded in China more than 2000 years ago

(Shaughnessy 2014). One finds the ‘Hare-Rabbit in the moon’ character in Africa, Eurasia and

America (Bascom 1981; Harley 1885; Berezkin 2015). In Africa, more than 30 versions of a

Hare-Rabbit in the Moon have been recorded mainly in Namibia (Bushmen), Nigeria (Hausa),

South Africa (Hottentot and Bushmen), Botswana (Bushmen). The Botswana version is quite

short: Hare scorched his garment in the fire and threw it at Moon’s face. Moon said, ‘Men shall

not die’, but Hare objected, ‘Men shall die.’. Whereupon Moon took an axe and split Hares’

mouth. Le Quellec (2015) has analyzed myths about messengers announcing death. Quite

interestingly the geographic repartition of myths in which a message is perverted by the

messenger has a complementary distribution to the myths in which death is represented as a

person. The myths may have two competing messengers or a single one. In the latter case, the

messenger is in large majority a hare or a rabbit (40%). In India, the Sanskrit name of the rabbit,

śaśī, is also used to designate the moon (śaśadhara, śaśabhṛit "bringing the rabbit"). The

relationship between moon, rabbit and sacrifice in a fire is found in India in the Śaśa-Jātaka

(rewritten in Sinhalese around 1200 under the title of Śaśadāvata and widely diffused), which

tells the story of a rabbit that would have been transformed into the Moon by the Buddha, in

thanksgiving for having thrown himself on a fire to offer him his flesh in order to nourish him,

one day when he asked alms disguised as a brahmin. This is the reason why the Kalmuk call the

hare Śākyamuni, the Buddha (Le Quellec 1993; Gubernatis 1874; Frédéric 1987). In Southeast

Asia and China, an ancient myth relates a Hare-Rabbit, a Toad and the Moon to the elixir of

eternal life (Berezkin 2015). In Japan, lunar hare Tsukoyomi killed Ukemoši the Goddess of rice,

and is often depicted pounding rice in a mortar.

The African version is centered on the status of men in relation to death, the Indian version is

connected to the value of sacrifice, while the Chinese version deals with the topic of eternal life.

The Hare-Rabbit in China is often depicted in the moon with a pestle in the hand, providing

possibly a connection to the therapeutic properties of the frog/toad.

Figure 4 shows that the combination of a 2-state with a 3-state character leads to a tree identical

to the canonical formula. The binary character is defined by the presence/absence of the

Rabbit/Hare while the 3-state character has the 3 states (announces) ‘Death to Men’, ‘Self-

Sacrifice’ and (Elixir of) ‘Eternal Life’.

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Figure 4: The different versions of the Rabbit/Hare in the Moon and Frog/Toad in the Moon tale

can be represented using one 3-state and one 2-state character. The resulting tree has the same

topology as the tree in Figure 1 for the canonical formula.

Figure 4 permits to relate a trickster Rabbit/Hare announcing death to men to a Frog/Toad

related to an elixir of eternal life. At the phylogenetic level this example is also quite interesting.

While phylogenetic networks (see next chapter) are often more appropriate to describe evolution

through mutations and lateral transfers, it would be wrong to conclude that a tree structure, like

the ones in Fig. 1 and 4, results from an evolution through mutations only. Actually, the

‘Frog/Toad and Rabbit/Hare with a pestle in the Moon’ version probably results from the

combination of several myths versions. The story of the Goddess Chang flying to the Moon

(Shaughnessy 2014) was associated with the elixir of eternity possibly during the Han dynasties

(Dai et al. 2005). It is believed that the Rabbit was included at a later time than the Frog (Zhang

2004, Sun and Chen 2009). In the phylogenetical language, this means that lateral transfers have

played a central role in the evolution of the myth. The presence of several different tales

furnishes a reservoir of terms and functions increasing the probability of associations between

different terms and functions.

Further branches may be added to the tree in Fig. 4. Each branch results either from a new

character, as for instance an opposition, or a new state of an existing character. As an example,

let us discuss a possible new character, the character rain/water. In some Chinese versions, both

a frog/toad and a rabbit/hare are in the moon, the rabbit/hare carrying a water pail. The water

pail is not anecdotical as it reminds us of another widespread myth: the ‘Girl in the Moon with a

Water Pail’. This tale is mostly found in North Eurasia with a western limit coinciding with the

border between Balts and Slavs (Berezkin 2010) with some versions in North America. The

association between a rabbit/hare and water is also found in some North American versions of

the ‘Rabbit/Hare in the Moon’. Several Mesoamerican versions tell of a rabbit warning people

from a flood and taking refuge into a box. As the box reached the sky, the rabbit jumped onto

the moon (Berezkin 2015). The connection between a rabbit in the Moon and a flood is not

completely obvious except if one postulates that the rabbit did inherit some properties of the

frog. Frogs are known to be quite active around or even during rain time and the connection

between a frog/toad and rain is almost universal. Was the association between water/rain and the

frog/toad transferred to the rabbit in Eurasia (an instance of interaction between myths or in the

language of biology a lateral transfer) and brought into America through migrations? The Toad

in the Moon is prominently represented on the Northwest coast of North America slightly below

the ‘Girl in the Moon’ versions and in the Northeast of South America. In conclusion, a branch

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corresponding to the ‘Girl in the Moon with a Pail’ versions of the myth may possibly be added

to Fig.4. Our best guess is that the branch may be anchored at the edge connecting the second

and third nodes (Rabbit Rabbit and Frog) in Fig.4.

Let us complete the analysis of the canonical formula applied to the ‘Rabbit/Hare and/or

Frog/Toad in the Moon’ with a few comments that will serve as an introduction and illustration

to a more formal description of the narrative formula in next section. The Rabbit in the Moon is

quite present in Mesoamerica (Berezkin 2015). The Moon Goddess (Metztli) is represented in

Aztecs codices as a disc carrying a rabbit. Some Mesoamerican creation myths remind of

Eurasian versions. According to Sahagún, the gods wanted to choose which ones of them would

sacrifice themselves in order to become the luminaries of the world; The candidates were

Teccistecatl and Nanauatzin. A great fire was lighted where they were to throw themselves.

Teccistecatl, after having hesitated four times, finally followed Nanauatzin who had not

hesitated. To punish Teccistecatl (Moon) of his cowardice, one of the gods threw a rabbit into

his face, which is always seen there (Bonnefoy 1981; Sahagún 1975; Soustelle 1940; Cancik

1988-2001). In the Indian version, Rabbit is sent to the moon as a reward. In the Aztec version;

Teccistecatl is punished by throwing the rabbit in the Moon bringing an opposition between

cowardice and courage. Let us point out that in both versions, a fire plays a central role.

The Frog in the Moon is also discussed by Levi-Strauss in his book “The Origin of Table

Manners” (Levi-Strauss 1968, Dorsey-Kroeber 1903: 321-323). In an Arapaho version Sun had

Frog as a wife while Moon had a ‘human’ wife. In this version of the myth, Moon humiliates

Sun by giving some meat to both his wife and Sun’s wife, testing which one will eat most

elegantly to the ear. The frog tried to cheat by replacing the meat by some piece of charcoal.

Moon found out about the trick and scolds the Moon. As a kind of revenge, the frog jumped on

his brother-in-law’s chest. From that day on, a frog can be seen on the Moon with some water

jar. A very strange detail puzzles the reader: in the Arapaho version (Dorsey and Kroeber 1903),

the frog chews some piece of charcoal. Why should a frog chew some charcoal? Let us give a

tentative explanation. In both North and South America, there are numerous myths (Wassén

1934; Frazer 2016) explaining how the fire was brought to the humans by a frog eating glowing

charcoal (Some frogs and toads’ species have luminous insects in their diet and may be lured by

a glowing charcoal (Woodland 1920)). The association of fire and frog is nevertheless quite

surprising considering that the frog is often associated with water and rain. Is it the result of a

possible association between rain and lightning? (Tlaloc the Mesoamerican god of the rain, was

regarded as a God bringing water and fertility, but was also feared for his ability to send

lightning)? We suggest that the charcoal is associated to fire, heat and indirectly to the sun and

the moon (The charcoal produces light if very hot and is very dark when cold).

These examples show that the character ‘Fire’ may appear in very different forms in the

Frog/Toad and Rabbit/Hare in the Moon. Almost all combinations of these elements (Frog/Toad,

Rabbit/Hare, Fire) are found in the various versions of the myth. The characters ‘Frog/Toad or

Rabbit/Hare’ and ‘Fire’ are neither compatible with the canonical formula nor a perfect tree

description of the different versions of the myth. Let us discuss this central aspect of

phylogenetics in the following section.

4. Mosko’s Narrative Formula

A modified version of the canonical formula was proposed by Mosko (1991), originally to

describe transformations in non-mythical materials identified in the Bush Mekeo culture of

Papua New Guinea. It is called the narrative formula and takes the following form:

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The last member is the only difference to the canonical formula, making the formula symmetric.

The narrative formula can be coded with 2 binary characters as

. No

phylogenetic tree represents exactly the narrative formula as the latter breaks the so-called 4-

gamete rule. The 4-gamete rule states that for each pair of binary characters, at least one among

the 4 gametes (i.e.

has to be absent in order for the characters to be exactly

represented by a phylogenetic tree (Hudson and Kaplan 1985). Figure 4 shows that the narrative

formula can be represented exactly by a phylogenetic network (a so-called outer planar network).

Outer planar networks (Bandelt and Dress 1992) permit the simultaneous representation of

alternative trees and are thus generalizations of trees. Generally, phylogenetic networks are

better suited to describe the evolution of myths than phylogenetic trees if some myths combine

into new versions. In genetic studies, phylogenetic trees permit to describe well evolution

through mutation while phylogenetic networks are suited to the description of both mutations

and lateral transfers (Thuillard and Fraix-Burnet 2015). In many instances, the combination of

two myths’ versions can be modelled quite similarly to a lateral transfer. A well-known result is

that a set of binary characters can be represented exactly by an outer planar network if there

exists a circular order of the end nodes so that the consecutive-one’s property is fulfilled by each

binary character (Thuillard and Fraix-Burnet 2015). For the two characters in Figure 5, starting

from the node labelled one has the (clockwise) circular order: (0, 1, 1, 0) and (0, 0, 1,1).

The two characters fulfil the consecutive-ones’ property, meaning that all ‘1’ states of a given

character are consecutive on the circular order).

The canonical formula is consistent with a phylogenetic tree representation of a myth whereas

the narrative formula is consistent with a phylogenetic network. The canonical formula is more

complex than the narrative formula in the sense that it requires one 3-state character and one 2-

state character to represent the formula, whereas the narrative formula requires only 2 binary

characters. In the evolutionary framework, the narrative formula describes pairs of binary

characters having both high transition probabilities. In real applications one may expect many

characters fitting neither a perfect phylogenetic tree nor the narrative formula. In summary, the

perfect tree description (with the canonical formula as an example) and the fully combinatorics

description represent two limit cases in the evolution of myths.

Figure 5. After coding the narrative formula can be exactly represented by a phylogenetic

network.

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Conclusions

This study shows that phylogenetic approaches may help us to discover complex evolutionary

processes in myths. Many myths are believed to have originated long before the invention of

writing. The study of myths may therefore furnish a precious window on mental processes as

well as provide clues on the mechanisms of cultural evolution in prehistoric times.

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