Review of Freemasonry
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An interpretative and Masonic hypothesis constructed on the relationship between Man and his environment.
by Bro. Oskar Dudo



Does a connection exist between the geometric vision of a particular territory (geometric system) and the music it produces? Can music be considered the sonorous expression of a territorial space? Does an instrumental and functional relationship exist between Geometry and Music?

The leading idea of this brief paper is the hypothesis that an intimate connection exists between making music and doing geometry.



1. Geometry in science and music.


From the Greek géa (earth) and metria (measure), geometry is the method by which we measure the earth. It represents the idealisation and study of the properties of physical  space and of the figures contained in it. The ideative process, a characteristic at present attributed only to our own species, allows the formation of ideas which are interpretative visions of the world. No dominant ideas exist, because every human being and therefore every culture is able to manage several visions simultaneously (Mercadante F., 2005). The term geometry comes to us from the Greek via Latin. In the III-IV century B.C., the Greek mathematician Euclid conceived it as an abstract science whose object is space, considered as a togetherness of points and figures.

The new discipline (but not new practice which was already present in ancient Egypt) emerges therefore as a study of space. Actually useful for essentially practical purposes (such as measuring land), with Thales (still earlier then Euclid) it encountered a fundamental transformation. In fact, Thales introduced a new cognitive methodology, thanks to which it is possible to imagine points, lines, planes and spaces. The imaginative process results in considering these elements no longer as material entities but on the contrary as abstract ideas observable only through the eyes of the imagination. The modern neuro-sciences have identified in a particular zone of the human brain a population of neurons which allows the mind to operate inventively, that is to say “to imagine”. Some years ago Giacomo Rizzolatti of the Parma University of Studies discovered some neurons which he defines as “in the mirror”. Thanks to this discovery cognitivism attains the confirmation of a solid neurological base. The neurons of the F5 area are objects of  great scientific interest to many scholars of the neurosciences, because they are part of a population of neurons which carry out important tasks connected with sensorial and motor activity. During an experiment with macacos monkeys, Rizzolatti’s team noted that the monkeys undergo a variation of electrical potential, corresponding to the area F5, in the presence of motor activity being carried out by other subjects present in their visual field. This population of neurons, unknown until then, is called “in the mirror” precisely because it represents in a mirrored fashion the motor activity of a subject in the brain of another subject (Bertirotti A., 2004:17). In other words, the cells of the F5 zone are activated both when an animal carries out a specific action finalised with respect to an object, as when it observes other individuals carry out the same type of action. Obviously, these cells are present in Man who can therefore manage to project himself into a  fellow creature, becoming an ingenuous psychologist thanks to a process of incarnate simulation[i].

Consequent to this definition tending towards idealisation are the words (really quite difficult to understand unless accompanied by a strong effort to think in the abstract) that Euclid pronounces as follows: “a point is that which has no sides while the (straight) line is a length without width” (Euclid 1996, Book 1).


By the term geometry today, we mean the discipline which deals with the abstract form of matter as well as the science of proportion (in this case we mustn’t forget the famous 47th proposition of Euclid, motive of the jewel of the Ex-Venerable and defined by the same Anderson, in the constitutions of 1723, as “surprising”) and measure. If the object of geometry is the abstract form of matter, then geometry is a first physical theory of reality. One thinks of the link between mass and geometry in the frame of general relativity, which will be mentioned later.

Modern geometry has as its object all that is measurable such as lines, surfaces, solids, etc. This science-method utilises mathematics as the form, and numbers as the language. It is based on axioms (abstract principles considered true without demonstration) and on the theorems descending from them.

In Masonic language the term geometry is synonymous with order and harmony. For example, it can be asserted that the great men of literature possess their own particular geometry of exposition; that geometry is present in musical and architectonic works, etc.

The ancient Greeks tell us that geometry is “solidified music” (Geymonat L., 1970), meaning by this expressions that in geometry we find an evident fixity of theoretical structure, together with a rational  explanation (the mathematical root) of the musical phenomenon which is anything but motionless. For their teachers (the Egyptians), sacred geometry and music were inextricably connected inasmuch as  geometric laws govern the diatonic and mathematical intervals which form the notes in the Western musical scale. Centuries later Goethe himself was to say that geometry was “frozen music”.



2. Darwinian and Masonic evolutive harmony.


From the point of view that we are proposing, the idea of Harmony is originally associable with the relationship our species has established with Nature; and consequently with  its territory and its rhythms. Its territory, that is the earth, is the primary expression of Nature and her will. It is possible to participate in her will insofar as man’s energies permit. These energies are not usable in a generalised way, simply because some aspects and phenomena of nature cannot always be controlled by Man. A constant rationalisation of this relationship with the forces of nature is necessary to resolve the resolvable. Harmony with Nature thus takes on the form of Utopia i.e. Nowhere, or in other words fulfilment that is possible but not guaranteed[ii].

The relationship Man-Nature is expressed in the west by attitudes of power, both in small everyday resolutive interventions and in large projects of dominion. Nevertheless, there is a widespread belief that it is impossible to carry out sufficient effective change to alter the course of the collective’s daily existence. One of the possible cultural reactions to this belief is manifested in a collective/individual tendency to social de-responsibility, involving a passive attitude to the will of Nature. This takes the form of “strong individualism” which in turn leads to a mystification of the concept/attitude of solidarity.

In short, the concept of harmony with Nature is often reduced to a kind of passive acceptance  of harmony itself, with the consequent incapacity to rationalise, or rather to find an acceptable explanation for events. The collective’s relationship to its own territorial space is in direct function with the extent of its absolute necessity to establish such a relationship, and with the concept of harmony with Nature as a collective economic/productive necessity. From this point of view, it becomes evident that there can be at least two contrasting attitudes toward the space of Nature that is the territory. 

At this point we need to understand the role and function that the territory has had in the development of the human species. The territory is an area which presents well defined characteristics of the ecosystem and which must be defended from entrance by other individuals of the same species. Conquering a territory signifies being able to reproduce. Every animal finds both its vital space and place of refuge in the territory. Furthermore, a better distribution of the population is possible within a well defined territory, preventing excessive ecological exploitation. Inside the territory there is a zone which has no need of any particular defence, and indeed is also frequented by individuals from neighbouring areas. This is defined as the familiar or home range. Within this zone the individuals present in a given territory move about to perform their own activities. Apart from the nomad species of animal, nearly all animals carry out their daily activities in the territory and on the home range. In some special circumstances they may possibly go beyond the home range and venture into neighbouring territories. In this case the venture space is defined as the area of exploration.

Animal behaviour, including that of Homo sapiens sapiens, is visibly modified according to the zone occupied. In fact, aggressiveness diminishes when passing from a territory considered one’s own to an area of exploration. The nearer an individual approaches a territory that has to be defended, greater is the aggressiveness shown him, while it is almost completely absent if the individual approaches the boundary area. The final element to remember is linked to the evolutionary function that the area of exploration carries out. Inside this area reproduction exchanges among individuals of different territories can take place. In this way inbreeding, the mating among relations within the same territory (a harmful consequence of territorialism), is reduced by the possibility of meeting outsiders in neutral areas.

The mirror neurons to which we have briefly referred, intervene in all spheres of human knowledge exactly because they constitute the foundation of the imitative processes and because they induce the formation of awareness of self. In fact, the moment one realises that one’s actions are similar (because imitable) to those of other individuals, a sense of collective belonging develops and one becomes aware of one’s own volition. For these reasons the crossing over from the territory to the area of exploration (via the home range), which our species continually does even in the present state of evolution, is a complex cognitive phenomenon thanks to which we conquer space, trying to harmonise it with what we previously occupied and lived in.

To sum up, Man searches for spatial harmony through the temporal management of his own actions which are mirrored in those of others. This is an anthropological valuation of a purely Masonic thought. It is a matter of the search for harmony expressed in Jacob’s ladder, Mosaic Law and the Spiral Staircase. The ascent is always a spatial conquest, both in the metaphorical sense and in the more typical profane territorial sense. One also ascends horizontally i.e. elevating our similars and working in the area of exploration (Cedrini R, Masali M. Chiarelli B., 2005).

It seems clear that, notwithstanding the fact that anthropological science is erroneously held to be absorbed in toto by the Darwinian concept, the Masonic concept of earthly existence is especially linked to the cognitive evolution of Man independently of the scientific contributions of Charles Darwin[iii]. In the Origin of man (1979) Darwin wonders whether man, and all other species, descends from some pre-existent form of life. Secondly, he wonders in what way Man eventually evolved and, thirdly, what significance there is in the differences we find among diverse human races. The main concept dominating Darwin’s work is linked to the conviction that there is no solution of continuity between biological evolution and social-cultural evolution. Moreover, some behaviour traits attributed to man can also be traced (by careful analysis) in other animal groups. However, the following  question then arises spontaneously: is the social human dimension also subject to the same selective laws as biological evolution[iv]? As a matter of fact, Darwin establishes that “in Man a change of direction took place as the process of civilisation gradually progressed. The combined course of progress (selected) of rationality and development (also selected) of the social instinct; the correlated growth in sentiments of sympathy; the emergence of moral sentiments in general; and the combination of behaviour and institutions which characterise individual life and community organisation in a civilised nation, enabled Darwin to observe that Natural selection at this stage of evolution is no longer the principal force governing the formation of human groups, but that it has relinquished its place in this role to good behaviour. (Tort P. 1998:60). It is indeed “good behaviour” that opposes the eliminatory effects of natural selection and inversely directs an apart of social activity towards the protection and safeguarding of the weak of body and spirit and assistance of the have-nots. Natural selection has thus selected social instincts which in their turn have developed good behaviour and favoured ethical tendencies as well as institutional and legal devices, both anti-selective and anti-eliminatory (Tort P., ibidem:60). In this way natural selection has selected itself because it has eliminated an antique form of selection substituting it with a more suitable one. A reversive effect of evolution is established in which competition tends more and more towards ethicity, morality and altruism, together with the values of intelligence and behaviour. It appears evident that every form of biological sociologism attributed to Darwin is absolutely contrary to the theory of natural selection itself, and that his ideas have instead been distorted and instrumentalised.

Equally evident is the Masonic functionality of the Darwinian doctrine: the constant formation of a Masonic identity is linked to the evolutionary principle which pushes men towards selecting the highest degree of both social and biological harmony. In the present writer’s opinion, the metaphor which best explains the socio-existential action of the Mason is the Mobius Strip, whose torsion manifests didactically both the revolutionary continuum and the idea of the infinite.



3. Ernest Ansermet and music.


Music has both a globalising and globocentric role (A. Bertirotti 2005). Music leads to transcendency. Ernest Ansermet, in the first page of his essay, Les fundaments de la musique dans la conscience humane, begins by setting “auditory awareness” into the frame of an auditory Horizon. He then goes on to describe the appearance of music in the action of sounds, through which the auditory image is transformed into musical image[v].

This passage to imaginative behaviour is made possible because affective activity (neuronal population of the limbic system) is linked to that of hearing, thus giving sounds and image which has conscious meaning for Man. Mans musical consciousness produces a precise emotion to the musical image formed by sounds that reflect a singular existential course. On the one hand, auditory awareness, understood as a particle of the species’ existence, identifies the sound; on the other, the sounds heard circumscribes the existential course of consciousness. In this way a close relationship is established between the sounds heard and real life, because the former are inserted into human existence and set themselves up as milestones of experience. The dichotomy between the outside world and the intimate human one, between the abstract and real life, is resolved in man’s consciousness. The moment he hears the sounds, a dual action takes place, neuronal and cognitive. Both a simply two ways in which this unique structure, the perception of the mind, is lived as an event of consciousness. Music resembles God because it says things that are infinitely diverse, in affective situations that are infinitely variable, and through human subjects that express there own infinite needs through concrete every day ones. For this reason its creative activity can be endless. But it can only be endless in its tonal and rhythmic conditioning.

These in synthesis are the guidelines to Ansermet’s thoughts on music. Music originates in human affectivity, in those emotional depths of the individual that are impossible to explain fully in words, and which are expressed by utilising what Ansermet describes as the only means possible i.e. Western tonal and rhythmic language. (In this perhaps resides the philosopher’s limits). A further element underlined by Ansermet is Man’s desire to be in the likeness of God. Ansermet is profoundly religious and the whole of his philosophical thinking expresses this conviction more or less clearly. He also maintains that the dialectic medium which links Man to transcendency is the tonal system, precisely because it is so rooted in human nature that no other ways of expression can exist beyond it.

The concept of creativity is inserted into a philosophical-individual dimension and linked to wider concepts such as: talent, ingeniousness, inspiration, imagination, form and style, etc. These terms, even if they denote apparently quite distinct spheres, are in fact very often inter-connected. In creativity – according to Ansermet – resides the purest musical talent of the composer, even if his personality is manifested in the personal style, original form and affective pattern, all translatable into human sentiment. In the first dimension which Ansermet defines as pure reflection, the composer finds himself  in direct contact with the musical image and with his own imaginative powers. For the author, pure reflection is almost a conditional reflex. The Great Architect of the Universe (defined as the Creator by Ansermet) comes into direct contact with the musical image and He Himself  - as the impulsive consciousness of self – is implicated in the act of expression. In pure reflection of the imaginative act, the composer places his faith in intuition only. This input towards transcendency, part of the psychic dimension, is integrated, transcended and perfected by another dimension: the second reflection, i.e. the dimension of thought or abstraction. This dual route – first reflection/second reflection – is important, since the Grate Architect of the Universe is continually found in equilibrium between the two poles of spiritual affectivity and absolute values. Both contribute to forming a work of art.

From this point of view, creativity expresses the musician in his entirety, his emotional life included. The work of art transfigures and transcends what is performed, and contains a whole world and way of being in the world. The musician, though starting out from his individual dimension, moves away from his real and contingent mental situation to enter a world of spiritual affectivity and of absolute values.

Music (as a creative-cognitive activity) becomes a human expression of universal architectonic transcendency, and therefore propounds a metaphysical cognition, even though rooted in the neuronal map of our brain.



4. Geometry and Music.


The relationship existing between geometry and music has remained essentially the same as conceived it Pythagoras in his time. He observed in fact that the hammer blows of the blacksmith at work are sometimes consonant and sometimes dissonant, tried to find out the reason. He weighed the hammers that produced the sounds, and at once noticed that the sound produced depends on the weight of the hammers. From there it was a short step to the stretched nerve strings of the ox and to stringed instruments in general, such as the lira. Pythagoras discovered that the same rule applied: if one string is twice the length of another, the two produce sounds at an interval of one octave[vi]. When on the other hand one string is one and a half times as long as the other, they produce sounds at an interval on one fifth.

It was thus that Pythagoras discovered the musical intervals. In the temperate tonal System (fig. 1) each note or degree is signed with Roman numeral: I II III IV V VI VII VIII.

Figure 1
The temperate tonal System with relative degrees


The first note of any scale is called tonic precisely because it defines the tonality. In this System, the tonic note is the most important of the scale (unlike the dodecaphonic System). Immediately, after the tonic in importance comes the dominant, because it carries out a fundamental and central role from the aspect of harmony and melody. Then come the sub-dominant that carries out a slightly inferior role to the dominant. The leading note is the VII degree of the scale in the tonal System and has the extremely important role of leading the melody and harmony to the tonic[vii].


Pythagoras numbers the whole universe and it is difficult to move away form such a brilliant and portentous product of the mind. For years Geometry and music began to speak to each other only through the harmonic intervals or, at best, through the play of symmetry and movement of J.S. Bach or the Flemish counterpoints. More recently, the amazing fractals are making way.



5. Living system and music.


“Consciousness is a process,

not an object”

(Edelman G.M., Più grande del cielo).



Diverse animal species manifest complex co-operative behaviour. Glow-worms that light up in synchrony, crickets that sing in harmony. Animal behaviour from which a definite scheme seems to emerge of auto-organisational systems.

The interpretation of living structures as open systems (exchanging matter and energy) and of the work Ilya Prigogine on the Theory of dissipative structure, the co-existence of structure (the scheme of living being) with change, enable us to see organisms as living system that operate far from the point of equilibrium (death) by means of myriads of irreversible processes, the chemical reaction of life[viii].

A living organism is structurally linked to its habitat, in the sense that it continually carries out structural changes (adaptation, learning) in reply to interaction with the environment. Ontogenesis outlines the history of change every living structure through the process of the Piaget theory: assimilation, adaptation and equilibrium. Analogously, the structure of the living organism determines its ontogenetic behaviour.

From this point of view, communication is not intend as an exchange of information, but only as behaviour co-ordination between living organism in a reciprocal structural combination. This kind of mutual interaction applies to the simplest organ as well as to more complex ones.

Humberto Maturana, Chilean scholar of neuroscience, was struck by one clear example of behaviour organization, a non-human case of communication that he found in the song of the African parrot during courtship and the formation of the pair. The two parrots are enable to see each other through the thick forest and to the ear it seems that only one bird is singing. But in fact both partners are singing one melody in unison, and the melody is unique to each pair of birds. The song is a duet in which each bird begins a refrains of each pair. And naturally the melody is a not passed on the offspring (Capra, 1996).

Still on the same theme of behaviour co-ordination, a group of Ditch biologists has discovered that titlarks (Parus Major) sing a higher frequency in town in order to overcome the noise of traffic and attract a mate. Probably the first case reported of a wild bird modifying its mating call to adapt to an ecosystem altereted by the human species. Other bird species lack this singing flexibility and could lose the opportunity to mate in extremely noise conditions, victims of the changes introduced by urbanisation. It emerges from this research that birds living in very noisy area choose to sing at a higher frequency noise of cars, aeroplanes and factories (unlike those living in quieter streets). This would explain why some species survive without difficulty in the large cities of Europe compared to sparrows, for example, which are less able to modify their singing.

Duncan Watts was able to observe on hot summer evenings that when crickets populating the countryside emit sound by rubbing their wings together, they do so in amazing unison. Since the intellectual capacities of crickets are not particularly well developed, now does each insect known in advance when they emit a sound and when to stop? Similar phenomena have been observed in glow-warms that emit light in synchrony by means of the sinoatrial node, the natural pacemaker that regulates the heart beat.

We are once again in the presence of an organisational tendency in nature. When a crickets sings it induces the crickets nearby to emit sound a little earlier than they would otherwise have done if they had not heard the signing. If this process is repeated with other neighbouring insects, in the end they will all sing in harmony[ix].

What kind of geometrical diagram can best explain the interaction between these insects? Just as in a stadium we hear the voices of those nearest to us more loudly than those furthest away, so it is possible that the principal harmonic interaction takes place amongst crickets nearest to each other. All the same, to explain the speed with which the harmony speeds we also need to consider a small number of casual links amongst crickets further away.: some crickets sing more loudly so as to be heard by others more distant. In this way we obtain the small world network illustrated in Fig. 2 in which a few casual connections also enable elements further away to connect quickly.

Figure 2
On the right, the small world network


 It is quite likely that the function of this kind of arrangement is to ensure greater cohesion un the group, and as such guarantee greater control over the territory and frighten away eventual predators.



6. Anthropology and music.


“And what do you say about

the message which says we

should look for the truth?

It makes us forget that life

without mystery is arid

and that certain things – for

example our friends – should

be loved more than happens

deep down”

(Feyrabend P., Dialogue on Method).



The evolution of Homo Sapiens Sapiens is shaped above all by continuous and constant adaptation to the spatial-temporal. This characteristic, which is truly cognitive ability, enables us to colonise (anthropomorphis) the entire planet, including areas with difficult climates (poles and equator). In fact, every environment in which Man is present is witness to the immense flexibility and enterprise that distinguishes our participation in the ecosystem.

On the long course of evolution (which is still not complete), every migration has carried us towards newer and pleasanter regions. That course, however, has also necessitated adapting to changes in the time zones in order to get used to different rhythms in the day. Here too Man has met with complete success, revealing his second great gift of flexibility, that of the temporal.

Studies have revealed the existence of special internal biological rhythms lasting roughly 24 hours. The rhythms called circadian (from the Latin circa, about, and dies, day) are present in most living organisms, from unicellular to more complex ones like Man. These regulate numerous vital functions (from photosynthesis to cellular division, from the blossoming of the flower to baby’s sleep). The circadian rhythms are not the effect of an external phenomenon (such as day alternating with night or the ebb and flow of the tides), but of internal biochemical mechanism. What is surprising is that the human biological clock has 25 hours and not 24, and that a complex biological mechanism called dragging provides for bringing it into alignment with day alternating with night. This discovery (1972) was made by the French speleologist Michel Siffre who spent 7 mouths of solitude in a grotto in Texas. At the end of the experiment he found that his organism was synchronised to 25 hours. The circadian rhythms in the behaviour, psychology and biochemistry of mammals are controlled by the central internal clock of a brain structure known as the superchiasmatic nucleus (SCN). The clock is synchronised to the environmental cycles of light and darkness. It is well known thanks to everyday experience that a number of days are needed to adapt to a new time zone. Nevertheless, the details are not yet altogether clear how this adaptation takes place. In a paper published in the magazine Current Biology, a group of researches led by Johanna H. Meijer of the Dutch University of Leiden, reported their discovery that this process does not necessarily involve the gradual adaptation and synchronisation of the neurons comprising the central circadian clock. The authors studied the behaviour of specimens exposed to a 6-hour delay in the normal cycle of alternating light and darkness (a change covering the transition of the time zone from the eastern Unites States to western Europe). Thanks to electrophysiological analyses of cells constituting the central circadian clock, the scientists made a surprising discovery: a part of the cerebral mechanism (represented by a dorsal group of cells) shows oscillations of activity corresponding to a slow regulation of the clock to changes in rhythms, while another part (represented by a ventral group of cells) shows a distinct scheme of activity that corresponds to a much faster regulation. It is probable that all this due to the effects to the neuro-transmitter GABA which, the researchers found, stimulates the dorsal SCN and inhibits the neurons of the ventral SCN. Since GABA transmits information to the two part of the SCN, these differences could in fact have a complex influence on the way in which the two groups of cells adapt to the change of light (Albus H., Vansteensel M.J., Michel S., Block G.D., Meijer J.H., 2005:886–893).

A highly evolved animal, therefore, creates geometry around itself of both a simple and complex nature. Hunting (in its social and co-operative organisation) the perimeter of villages in which it is organized, its very movements in search of new areas to exploit and know are expression of this important and refined existential geometry.

In observing its own territory, both the inhabited area and the one to be domesticated, mental geometrization (also defined today as mental maps, or conceptuals) is the basis of complex and fundamental evolutive behaviour. The mental maps allow us to codify and decode everything that reaches our brain, through the senses. Every event typical of daily human life, every reaction and observation, depends on these maps. They are veritable windows on the world which often we confuse with true reality. In fact, mental maps always are untrustworthy in greater or lesser measure. An absolute reality simply does not exist exactly because reality is absolute for each individual, and is the architectonic result of one’s own mental maps, or rather those which reveal themselves and continue to reveal themselves as useful strategy in personal adaptation. The communication of several maps takes place in the cultural system, thanks to which diverse and personal ideas are compared to each other[x]. The retina absorbs less than a trillionth of a quantity of electromagnetic energy reaching it. It is consequently impossible to see-constructs the world as it actually is or could be, since each individual automatically excludes a part, even before the whole is elaborated by the primary visual cortex.

It is possible therefore that a given geometric system, characterised by mental maps, has led to a consequent codification of expression (hendecasyllables, sonatas, various communicative styles, music) and produced precise anthropological results, as well as enlightening roots. In other words, the telemetrical apparatus of Man (constituted by the synergic action of sight and hearing) is closely linked to the walking mobility of our species. When an individual  moves from one place to another, he creates the time for covering the distance by the adverbs before and after. These two adverbs are really tow sounds which acquire a linguistic significance only when our species ascribes a precise semantic value to them. The terms before and after do not refer to real objects present in nature, nor to living subjects or organism. They describe a temporal process that occurs in a geometric space. This exact space is that of the walking movement. In this way, the sound of the morpheme before and after become expressions of an existential segment which is geometric, because inserted in the space-territory where it was born and developed.

Figure 3

Pankepp (Altenmuller, 14:42-53) maintains that music was born from the cries of a few who, moving away from the group for hunting or other necessary purposes, kept in touch with the others by signalling their position in the territory (Révész G., 1983). The “Call Theory” shows exactly how much these cries can depend on the perception of space, and so on the geometric interpretation that one has of one’s own territory. We are dealing with a truly geometric system which each individual shares and that matures and consolidated within the social group the individual belongs to. The sound, in the form of a cry (call and recall), becomes a signal of individual specification in space (territory) and time (day). On this subject, we recall the presence of a symbolic isomorphism between the spatial temporal dimension of making music and geometry and that represented by musical notation. The musical score, when not referring to a monodic instrument, extends along the two Cartesian axes of space (vertically, from low to high, i.e. harmony or sonorous density) and time (horizontally of melody i.e. of the chronological sequence of a linear series of sounds).

The present author maintains that the sonorous measure of this geometric spatial-temporal system is music. It takes shape therefore as a geometric perception of territory, of its particular vital organisation, and is meant to be a mirror of its culture through the formation of semantic sounds, .i.e. the verbal language.

The geometry that every culture creates is the formal expression in signs a particular of ecobiological system of life, because it is functional to survival in that particular ecosystem. In revealed and formalised geometry, every individual reflects his own territorial identity, an evident necessity of onto-filogenesis[xi].

While on this subject it is useful to remember the revealing contribution of Karl Abel (the German glottologist) given in 1884 on the opposite meanings of primordial words (Abel K., 1884). He sustains that “in the Egyptian language, the only relic of a primitive world, we find a considerable number of words with two meanings, one indicating the exact opposite of the other. Try to imagine, if one can manage to visualize anything so obviously absurd, that the German word for strong signifies not only strong but also weak. That the substantive light is used in Berlin to indicate both light and darkness. In face of this and several other similar cases of antithetical meaning there can be no doubt that at least in one language a large number of words signified one thing and its opposite contemporaneously” (Abel K., ibidem:4-7-). Certainly this linguistic tendency is not yet a matter of chance, particularly considering the number of examples cited by the glottologist. Furthermore, “Egypt was not at all a country of the absurd. On the contrary, it was one of the first places where human reason was developed. They posses a pure and highly dignified morality and had formulated a great part of the 10 commandments at time in which people to whom we attribute civilisation today were still in the habit of sacrificing human victims to gods thirsty for blood. Of all the eccentricities of Egyptian lexicon, perhaps the most extraordinary, apart from the word that contain in themselves their opposite meaning, is the inclusion of other compound words in which two terms of opposite meaning are united in a combination in which only one of the two terms has meaning. There therefore exists in this extraordinary language non only words that mean strong as well as weak, command as well as obey, but also compound words like old-young, far-near, join-divide, outside-inside, words which although they join extreme opposites, intend the sense of only one of the two words young in the first case, near in the second, join in the third, and inside in the fourth. Initially, therefore, conceptual contradictions were united in these combined words not in order to create a third concept, as sometime occurs in Chinese, but simply to convey with these composite words the meaning that one of the two contradictory words would have indicated on its own” (Abel K., ibidem:15). The question however is relatively simple, since our concepts are formed by comparison, i.e. through the relationship that an individual establishes between numerous aspects of nature. On our planet all things are relative, especially with respect to the place-time position in which the individual finds himself. “Since every concept is in some way the twin of its opposite? (…) Seeing that bit was not possible to conceive the concept of strength if not bi opposition to weakness, the word that signified strong a simultaneous reminder of weak, and only in this way could its existential meaning be reached. This word defined in fact neither strong nor weak, but rather the relationship between tow concepts and their difference, which created a likeness in both (…). In fact Man could not acquire his simplest and most ancient concept except by comparing them with their opposites, and has only learned gradually to separate the two parts of the antithesis and to think of one without consciously measuring it against the other” (Abel K., ibidem:18).

In communicative writing is antithetical lemmas (such as strong-weak) had to be included in some way to respect the intention of the writer. To deal with the problem the primitive Egyptians added determinative images to the writing. “When the word ken is meant to signify strong, the image of the straight, armed man is placed after the sound written alphabetically; when the same word is meant to signify weak, the letters represented the sound are followed by the image of the crouched, indolent person” (Abel K., ibidem:18). In the spoken language quite probably gestures took the role that the images express on the papyrus.

The stylized geometry written besides the word indicates its true meaning in the context in which it stands. In order to stimulate the formation of the idea of strength in individual I must be able to associate it with the action that expresses it. Every action is sign-design in space. Every action is a geometry which is codified in the syntax of the language so that it is more easily administer to and understood by the whole of humanity.

In this way, music becomes a key to reading and an expression of geometry. In fact, both time and space reside in music because they are its constituent elements, obviously subject to the perception of these two categories that the people who develop them posses (Bertirotti A., 2003; Bertirotti A., 2004).

For example, the measurement of space can be in the oral tradition by means of the time employed to traverse a given space while producing sounds. The sounds therefore become a musical expression ante litteram of the given space inside which the same sounds move and are reproduced. In this way, both time (understood Aristotelian accession of measure of movement according to before and after) and movement become components of sonorous space (Aristotle, Physics). The measurement of a space can therefore be expressed by the sounds that Man attributes to the movements in the course he covers, inside the space itself. There can be many examples, of play in this behaviour. Take, for example, the games linked to the reproduction of songs that mark the time in which one must carry out particular actions.

In addiction, music offers the possibility of quantifying space (simply by passing from a single melody to a polyphonic one we sense intuitively a growing fullness of volume). As a further example, think of the noise of the heard of buffalo which different from the noise of the single animal and immediately coveys the intuitive knowledge of the numbers of element involved, and therefore of the space that they occupy.



7. Geometry, sound and action.


“The hands are the soul of man.

They are our character.

Educating a hand means

teaching it how to be


(Merini A., The easy life).



A child crawling on all fours in a straight line is already constructing a rudimentary form of geometry. The first fortress he builds with material to hand to defend himself from childish fears is also geometry. The games he creates on his on with the help of building blocks (or clay etc. for children thousands of years ago) express the child’s consciousness of the space around him and of the relationship resulting from it.

Actions of this kind are important for forming a sense of personal identity which is the result of an endless process of ontogenetic, exactly because every individual experiments with daily life as an imperfect approximation to himself. No human being who has the fortune to live (not to survive) to the end of his days can claim that he has achieved a sense of knowing who he really is. There always is some difference between what we are or what we believe we are and what we would like to become. If this were not so, no individual would have projects in his mind for living. From this point of view, the sense of personal identity is not a state so much as a continual process of architectonic construction of what we would like to be. It is a process which develops during the time of one’s existence, during the history of evolution of Man, and within the space of one’s own action.

Within the bounds of spatial perception, the very presence of sound and of its interpretation plays a significant role. The lateral changes in the source of sound are in fact more easily perceived that the vertical changes. It would therefore be easier for a child to understand the concept of segment or the straight line in general when this is represented by horizontal lines rather than vertical ones. The sound of this representation would reconcile a profound spatial interpretation in the brain and a fundamental medium. For these reasons, a knowledge of the Set square and of the Masonic compass demands a high level of abstraction from the neuronal point of view which young evolutive ages cannot attain. The symbols,(particulariy masonic ones) and their comprehension require not only the initiation and conclusion of neuronal myelination, but also the connected and refined action of neuronal maps in the frontal, prefrontal and associative cortex.

To return to the subject of sound perception in the spatial ambit, the position of the ears helps us to localise the source of sound just has the frontal position of the eyes enables us to evaluate distance. The direction or provenance of a sound is identified thanks to the sensibility of the scolopidian cells and to the difference in time of the latency of activation between the two sides (Giulio L.F., 1991).

The decodition of sound does not take place exclusively on one definite, precise side of the brain, it has been shown that the difference in auditive elaboration of sound already originates in the ear. “From birth – explains Yvonne Siranger of the University of California at Los Angeles – the ear is structured to distinguish between different types of sound and to transmit them to the optimal side of the brain for elaboration. The different treatment of sound on the part of the two sides of the brain is well known. The left side dominates the decipherment of language and of other signals of rapid change, while the right side deals with elaborating the tonality of music”.

After studying the tiny cells of the internal ear which pact as vibration amplifiers, Sininger and Barbara Cone-Wesson of the University of Arizona discovered ~in a sample of 3000 new-born baby) that the left ear provides greater amplification of musical sounds while the right aniplifies more loudly the spoken language (Sininger I., Cone-Wesson B., 2004). This asymmetry between the two sides turns out to be fundamental adapting to the universe of sound, consenting to its better scansion.

0ne’s own geometric vision of the world can be defined, interpreted and explained by the expression vital geometric system. From this thesis it follows that Masonry configures a cognitive process as a form of visual-auditory architecture of the world.

Euclidian geometry was sufficient, to a Limited degree, for a vision of the pre-relativistic world. Einstein’s theory and his Weltanschauung introduced a new geometry: a non-euclidian geometry even more suitable for explaining reality compared to the traditional method. This variation can also be considered as a radical change in the vital geometric system. Einstein sustains in fact that space is not linear but on the contrary curved, due to the multitude of celestial bodies. The presence of the Sun has the effect of deforming the surrounding space, and time muovement of the planets round the sun takes place along the “natural lines” (the geodetics) of space, which of course have also undergone this curvature. In other words, space becomes curved, and in this curved space the shortest distance between two points, which previously one imagined to be a straight line, is deformed into a curve.



7.1 The geometry of action.

In the history of western philosophical thought one often comes across diatribes about how much the body influences the psyche and vice-versa. According to a cognitive-connexional prospective (to which the writer makes reference) the question put in these terms constitutes a false problem (Siegel D., 2002). The development of human thinking, always configured as thought, comes through action. Even in remote times, Man began to explore his environment. He moved, acted and projected new actions. These always were penetrative, i.e. linked to the necessity of moving to achieve the objectives and aims to be found in his ecosystem. This matter has already been dealt with in the question on the home range.

From actions directed to the outside, one then passes on to the formation of awareness of the actions themselves[xii]. Man learns to understand that in order to plan a course of action he must imagine himself moving, both in space and time (Rizzolatti G., 1998). This is extremely refined form of cognition can usefully be expressed in a code that formalises it exactly. This code is called language, whether it be numerical, verbal or geometrical.

For us masons it is called ritual. Movement (therefore action) becomes the prerequisite of verbalisation, of numeration and of geometrical music. A really true system of spatial –temporal codification begins to take shape based on existential geometry, in which every form of movement constitutes cognition of the action. For this reason, theory cannot exist without practice, which precedes the former, ab origine. Clearly, both act as reciprocal carriers at the present stage of our evolution: the actions stimulate the formation of conceptual maps (it is no coincidence that they are defined as maps) and these in turn stimulate the actions, both present and future.

7.2 The geometry of sounds.

Listening to sound often induces the evocation of images. A high frequency sound can evoke (with a formal symbol) a shape with few details, visualized as a texture with smooth internal transitions. A bright, acute and highly articulated sound, on the contrary, will evoke a shape rich with indented and well outlined details.

In the same way, well defined sonorous articulation can be born from images, contexts, habitat, or from the unconscious visualization of them in relation to the geometry attributed to them or invented for them.

But what motives can there be for inventing a geometry?

Herodotus and Aristotle did not believe that the birth of geometry (and of mathematics in general) was possible in any age before the ancient Egyptian civilization. Aristotle assumed that its origins should be looked for in the well-to-do priestly class present in Egypt, who used it for ritual ceremonies. Herodotus dates geometry back to the particular necessity the Egyptians had to restore territorial boundaries after the periodic flooding of the Nile. The fact that the Egyptians surveyors were also known as “layers of cords” might mean that they used them both for relaying the territorial boundaries, and for tracing the plans for religious buildings[xiii].

The graffiti of Neolithic Man reveals its own peculiar interest for spatial relationship, probably propaedeutic to the birth of geometry far from the rock face, even though it could have simply been aesthetic pleasure for its own sake that give origin to the pictographic exertions of our ancestors. Further evidence derived from India also induces us to think that the need to construct buildings and measure land played a part in the origin of geometry. Arabesque appears later, probably originating in the taboo placed by the Islamic religion on depicting the realistic (in terms of human or animal figures with the purpose of avoiding ill-omened idolatry). This prohibition induced artists and decorators to direct their creative imagination toward a characteristic abstract dimension.

The attention to geometry, and sensibility to the different kinds of geometry, comes about through the basic urge to resolve practical problems – like the position of horses and of men during a hunt or a battle – or through social coercion.

We also observe that a parallel can be drawn between the dimensional development of geometry and the anthropological development of music. Geometry of a linear dimension that exists on a straight line can be compared to the first indistinct repetitive sounds made almost accidentally by Man. The extension to the plane, to two dimensions, can be compared to the emergence of the stratification of space, and of the classification and arrangement of sounds, this time differentiated. The passage to three dimensions, space in its stricter sense (codificable in forms of superposed lines, just like the future pentagram), gives origin to move complex scores, inside which can be identified the precise it is musical tasks assigned to them. As already mentioned, it is the manifold variety of timbre that creates the sensation of increased volume.

Music is linked not only to the geometric-spatial dimension, but also to expressig the perception of time, obviously through the duration which is one of the four fundamental parameters of sound.

In Hebrew tradition, the Jubilee year (shenat yobel: Leviticus 25, 8-17), proclaimed every fifty years on the tenth day of time seventh month by the trumpeting of the ram’s horn (yobel), signified a fundamental turning point in the relationship between the Lord and his people, and among the people themselves.

“And ye shall hallow the fiftieth year, and proclaim liberty throughout all the land unto all the inhabitants: it shall be a jubilee unto you; and we shall return every man unto his possession, and to his family. Ye shall not sow, neither reap, that which groweth of itself in it, nor gather the grapes in it of thy vine undreased... in this year of this jubilee ye shall return every man unto his possession. Ye shall not therefore oppress one another; but thou shalt fear thy God” (Leviticus 25, 10-11.13-14.17).


The sound of the horn takes on a precise meaning: it transmits the announcement of freedom from slavery, a sociological turning point. The necessity to transmit that meaning immediately, reveals how sound is a more than efficacious method for spreading the word. Furthermore, the sound of the horn is useful in giving solemnity and prestige to an important occasion, no doubt already present in the minds of the faithful. The sound of the horn also reaches far distant places. Cheosing such an instrument probably indicated the people’s knowledge oh the long distances involved, i.e a fairly clear image of the geometry of their boundaries and of their own vital space.

The slowness and substantial emphasis of each sound is typical of traditional Chinese music as a source for philosophical meditation, and through these passages of sound we return once again to the spatial-temporal perception inherent in this kind of culture.

In the novel The virgin in the garden, A. Byatt writes: “The iambic pentameter embodies, if one can put it that way, the number of heart beats between inspiration and exhalation, Shakespeare’s verse is the human heart beat,” claiming that his observation was inspired by Music practice by Franchinus Gafurius (l496). An observation that clearly arises from the assumption, that the human heart was the first clock on which Man placed the time.

Just as metrics unveil the perception of time, so a piece of music can do the same.




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[i] Ingenuous psychology is the study of what is usually called “good common sense”, thanks to which a culture system manages to control the social dimension of its human relationships. Every culture, in fact, develops special adaptive strategies within its society to guarantee its own survival. The explanation (or rationalisation) for these strategies comes from the core of good common sense.

[ii] The term Utopia is used for the first time by Sir Thomas More, in his Libellus vere aureus nec minus salutaris quam festivus de optimo reipubblicae statu deque nova Insula Utopia (Lovanio, December 1516). The term emerges from the myth of Cronus, from the Golden Age and from the Garden of Eden, and systemized in Plato’s great political-philosophical opus The Republic. In the time of the Romans, characterised by pragmatism adapted to governability, Tacitus touches on Utopia, comparing the barbaric strength of the invasions to the corruption of Imperial Rome. The Hebrew view-point, on the other hand, links Utopia to the awaited coming of the Messiah which is then transferred to the Catholic Christian eschatological perspective, from the Apocalypse to the Fathers of the Church. The Middle Ages meet Utopia through the pauperist heresies and the palingenesian visions of Gioacchino da Fiore (millenarianism or chiliasm); via tracts of social communism we come to the Reformation with the Lollards, the Hussites and the Anabaptists. From the zeal of the Puritan concept in Law of Freedom (1652) by G. Winstanley, through the modern era, we come to the flowering literature of the Enlightenment. However there are many cases in which , to use Sorel’s expression, Utopia becomes a work hypothesis. We are reminded of the technological visions in New Atlantis (1627) by Bacon and the Commonwealth of Oceania (1656) by J. Harrington. The Enlightenment vision of history confers on Utopia the task of resolving the glaring contradictions between Man and Nature, between the town and the country. In the Code of Nature (1755) by Morelly and the Social Contract by J.J. Rousseau the myth of the “noble savage” becomes synonymous with social progress, spanning the French revolution to arrive at the Restoration and the challenge of romanticism. We witness the advent of the technical-religious work of Saint Simon (in his Utopian Socialism), of the egalitarianism and experimentation of R. Owen and C. Fourier. In 1888, E. Bellamy with his book Looking backward introduces into Western philosophical thought the idea of a sociological Utopia, which blossoms with Nietzsche and Spengler into negative antipositivistic thought. In the 20th century the Utopian question acquires particular importance in the Frankfurt School, especially with E. Bloch and his Spirit of Utopia and The Principle of Hope in 1918. Agnes Heller in her Radical Philosophy of 1978 brings to our notice how the Utopian idea at this period is intimately connected to the Rousseauian and Kantian concept. In fact, Rousseau himself was the first, in Emile, to attack the realists, who in order to protect what is realizable end by defending what they do. With Kant, Utopia acquires value because associated with the regulating function of transcendental ideas. Even if the ideas of reason do not constitute the basis of experience (unlike those of the intellect), they direct human actions towards noumenonic concepts, thanks to which positive law imposes itself on brute force, and peace on war. Thus we reach H. Marcuse who, in his The end of Utopia, reminds us that the social sciences of today must rediscover the Utopian attitude in order to elaborate hypothesis and conduct theoretical research directed towards the well-being of the whole of  humanity.

[iii] Charles Darwin sustains the all animal species evolve and are not immutable. The evolutive process is determined by: a) spontaneous variation, i.e. by individual differences in a species produced by chance; b) the struggle for existence, originating in the fact that more individuals are born than is the number that can survive; c) natural selection or survival of the fittest, in which case the favourable variations are maintained and the unfavourable ones eliminated spontaneously. Darwinian biology exerted an influence on the psychology of the time, with particular reference to the theory of a possible continuity in the mental processes animal-man. This resulted in considerable change in the objectives of psychology. William James, for example, took into consideration the eventual functions carried out by the conscience, and no longer the individualisation of its components. Again thanks to Darwin, the concept of individual differences is introduced into psychology with the consequential study of the differences between various human minds, above all from the functionalistic point of view. In his Great Man and the Environment of 1880 William James maintains that Darwin was the first to distinguish between causes of production and causes of conservation, with reference to the extraordinary qualities which characterise human beings and animals. In other words, Darwin, in isolating the causes of production defining them as tendencies to spontaneous variation and relegating them to a physiological cycle, places his attention on the causes of conservation and, with the expressions natural selection and sexual selection, regards them as functions of the environmental cycle.

[iv] The answer to this question dominated the speculations of Spencer and Haeckel, who promptly formulated the theory of “Social Darwinism.

[v] This cognitive effective course is possible precisely due to the synergic action of mirror neurons, and the intervention of the neuronal populations in the frontal and prefrontal lobe and in the limbic system.

[vi] The interval is the difference of pitch between two notes within a scale, so it concerns the relation of frequency.

[vii] If the frequency in Hertz, of the fundamental note do is indicated with n, the other notes of the natural scale have frequencies as shown in the following table:



























where the last line gives (in decimal form) the relation of frequency for each note with respect to the note C, assumed as fundamental (we indicate this note with C0): i.e. the connection between the frequency of each note and the frequency of central C. The note C1, which has double the frequency of C0, initiates another series of seven indicated thus:











These seven notes have the same relationship of frequency between them as the previous ones. Another C follows which indicate as C2, and which as double the frequency of C1 (The frequency of C2 is therefore equal to 4n). With C2 another series of seven notes is initiated and so on… The interval of one octave is always equal to 2.























The sharp of a note has with the note itself an interval of 25/24. The flat of a note has with the note itself an interval of 24/25. Until now of course we have been talking about the natural scale and its seventeen notes. In the Temperate Scale the interval of a semitone, i.e. the relationship between the frequency of any two successive notes among the 12 existing ones, is equal to:


 = 1,06  4


Clearly in the temperate scale all the intervals and therefore frequencies of the same notes are altered from those of the natural scale. For example:


D diatonic

D temperate

9/8n = 1,125n

(1,06)2n = 1,123n


Up to here we have been talking about the relationship between music played by only one instruments or sung by only one voice, the absolute values of these frequencies would be of no importance. However to perform pieces with more that instrument or one voice, clearly the instruments would have to be tuned to the same note, one with a well established frequency: this note is also called normal note or diapason. For this purpose the A0 has bee chosen with an established frequency fro 440 vibrations a second (440 Hz). For the international standard the normal diapason or pitch is used, oscillating with a frequency of 435 Hz, and deposited at the Office of Weigths and Measures in Sèvres (Paris).

It is simple to deduce from the tables above that if A0 has a frequency of 440 Hz, C0 has a frequency equal to:           

The frequencies of do in the diverse octaves are therefore:















C-3 with a frequency of 33 Hz is the lowest C0 that the ear Can hear (cf. Bertirotti A., 2003: 24-27).

[viii] “Life is the combination of functions that resist death. A tautological definition, it contains basic truth: the resistance of living creatures to any form of aggression, and special reactions of tactics or flight; the productions of antibodies or toxins; phenomena of anabiosis or slowing down of the chemical functions; the production of protective membranes, of cists, spores, or shells; reflexive paralysis in face of predator, etc. Natural examples are practically infinite” (Bertirotti A., Larosa A., 2005:221).What is more “life, at least as we know it on this Earth, is linked to the recombination of 5 nucleotides (the pyramiding and purine bases) linked to each other by a sugar (deoxyribose) by means of a phosphoric bond. As is known, the combination in series of these three elements constitutes the genetic code; a polymer endowed with a self-duplicating capacity able to codify the sequence of amminoacids in the proteins and so function as a matrix for the production of basic biological material. This polymer with self-duplicating capacity known as deoxyribonucleic acid (the famous DNA) is life” (Chiarelli B., 2003, vol. II:454).

[ix] Also at biological level, the proximity of a cell to other cells determines social behaviour. It is thanks to cellular redundancy that an adult staminal cell, placed next to for example, an hepatic cell, will fall under its influence and begin specialising as an hepatocyte.

[x] The term idea has the etymological meaning of vision, thereby preserving a subjective and partial dimension. It follows therefore that the concept of dominant idea is without foundation because visions are always contemporary. The believe that a dominant idea exists in the diverse cognitive processes is because it has been confused (in a process of assimilation) with that selected by the perception as the more significative at that precise moment of time and space. Cfr. Mercadante F., 2005.

[xi] Human symbolism is a combination of signs whose comprehension is fundamental to interpreting what they describes. The sign is a possibility and cognitive necessity (real-ideal) for adapting to the environment. The sign and its utilisation become a module through which it is possible to identity the communicative code and semantics of a more general group of collective descriptions. The use of signs is founded on the real capacity they have as instrumental objects, i.e. adapted to following the objective that the user has established beforehand. If the sign is adequate it can be used to achieve the objective in a given situation. “Sign can be used as a means of making money, obtaining social prestige, control over others, to deceive, inform or amuse, to comfort, reassure or stimulate; to refer, describe, or predict; to satisfy some needs or create others, to resolve problems objectively and obtain partial satisfaction in a conflict that the organism can not resolve completely; to secure the help of others and strengthen its own independence; to show itself and to hide itself” (Morris C., 1963:96).

[xii] The creation of awareness of an even is the result of a complex cognitive process. At the moment that an action is curried out our brain is directly linked to the activation of automatic mechanisms which enable the execution of the action itself, while a part of it is linked to the planning necessary to bring the task to completion. The formation of awareness of an action that has taken place is typical of linguistic elaboration which depicts the action just completed. In other words, the greatest degree of awareness of finished actions comes through the verbalization of the contents characterized the action, not during execution of the action itself.

[xiii] In the Old Testament the stretched cord (or measuring cord) is often mentioned. For the Free Masons the stretched cord is liked to the “sisaro” one of the work tools of the Third Degree. It represents undeviatingly upright conduct that leads to the search for immortality. The “sisaro” is a cord support turning on a central pivot. This instrument was given its name only after the United Grand Lodge of 1813 and became a Masonic term not diffuse outside the Institution. When the cord is unrolled quickly the “sisaro” emits an acute sound.


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