THEOLOGIA ARITHMETICA TRIADUM Arithmetical Theology of triads John Opsopaus Here are some observations, mostly Pythagorean but also Alchemical, about triads, both theological and musical. I. TRIADIC GENESIS A. The Monad The first principle is the Monad or Unit. It is prima materia (first matter), an undifferentiated unity, an unstructured wholeness, the principle of stability. The Monad corresponds to the first stage of many cosmologies, the Primeval Waters, the Confused Mass (Massa confusa), Tiamat, etc. B. The Dyad 1. Undetermined Dyad The second principle is the Undetermined Dyad (aoristos duas), the principle of change, separation, differentiation, conjunction and mediation. The Indefinite Dyad is an active/energetic principle, as opposed to the Monad, which is passive/material. 2. The Dyad Itself The Undetermined Dyad becomes determinate by operating on the Monad. As the principle of separation (Separatio) it splits the Monad, so that Chaos (gap) opens in it. In this way the Indefinite Dyad creates the Dyad Itself (auto to esti duas), an actual separation into opposites (Separatio oppositorum). It creates a polarity between male/female, form/matter, stability/change, etc. They are the passive forms of the alchemical Sulphur and Mercury. The Dyad corresponds to the second cosmological stage, in which some active agent (e.g. Marduk, Shu, "Destructive Strife" -- oloios neikos, etc.) opens a gap between Earth (Gaia, Keb) and Sky (Ouranos, Nut); it's the splitting of the Cosmic Egg. C. The Triad The Undetermined Dyad operates again, this time as a principle of mediation, to create a Mean between the Extremes of the Dyad, thus producing the Triad, which is a Harmonia (joining), a structured unity, the Rebis (Two-thing), a synthesis of the opposites (Coniunctio oppositorum) of the Dyad. This is the Alchemical Wedding, in which the alchemical Mercury attracts alchemical Sulphur towards itself so that they can be joined be means of the alchemical Salt; the result is the Living Gold or Philosophers' Stone. The Triad corresponds to the second cosmological stage in which some active agent (e.g. Eros) causes Earth and Sky to mate and beget the Gods. These are some aspects of the Polarity and its Harmonia in the Triad: Male Female Androgynous Form Matter Formed Matter Stability Change Lawful Change Fixed Volatile Fixed-Volatile Unification Separation Harmonia Emission Reception Reciprocation Sulphur Mercury Salt D. Higher Orders The Undetermined Dyad can operate, through Separatio and Coniunctio, to form Tetrads, Pentads, and so forth in a cyclic evolution, but the first three are the most important. II. THE SACRED FAMILY (Synchronic Triad) The Triad can manifest in the Godhead in many different ways, but one of the most common is as three archetypal genders, masculine, feminine and neuter (or androgynous). (Needless to say, they are only loosely correlated with biological sex.) Most commonly the Triad appears as a Sacred Family of three Gods, a Father, a Mother and Child, whose characteristics often derive from those of the Triad. A. Structure 1. Father The Father is the form-providing and matter-forming element of generation, the seed, and represents stability and rigidity, conscious soul (anima), emission, creation, and isolation. His element is Air (or Air and Fire) and His planet is the Sun. 2. Mother The Mother is the matter-providing and form-receiving element of generation, the womb, and represents change and fluidity, vital spirit (spiritus), reception, redemption, and the possibility of relation. Her element is Water and Her planet is the Moon. 3. Child The Child is the formed matter resulting from generation (through the mediation of Harmonia, the Mean), the offspring, and represents controlled change, equal exchange, the spirit uniting the soul with the body, the actualization of relation, resurrection, and balance in all the oppositions of the Father and Mother. Its element is Earth (and perhaps Fire) and Its planet is Mercury. B. Examples 1. Capitoline Trinity: Jupiter-Juno-Minerva Note that Minerva (like Athena) is one of the more androgynous Goddesses, since Her personality integrates both archetypally masculine and archetypally feminine characteristics. (Of course, in the Greek version Juno is not Her Mother, though Juno does provoke Jupiter into begetting Her.) 2. Egyptian Trinity: Osiris-Isis-Horus 3. Wagnerian: Wotan-Erda-Bruenhilde Bruenhilde is also androgynous, exhibiting archetypally masculine characteristics in Walkuere, transforming in Siegfried, and becoming archetypally feminine in Goetterdaemmerung. 4. Christian: Yahweh-Mary-Jesus 5. Chthonic Trinity: Hades-Demeter-Persephone (Hecate) This Trinity requires some explanation, since Hades-Demeter-Persephone do not fit the Father-Mother-Child pattern. However, Demeter is Persephone's Mother, and Hades is Persephone's Uncle (since Zeus is Her Father). Furthermore, since Hades represents the World of the Dead and Demeter the World of the Living, we see that Persephone becomes the Harmonia that joins the two, and reconciles the two adults; the two realms are "related by marriage." Hecate plays an interesting role in the myth, for She is both an Underworld Goddess and a Moon Goddess and so transcends the divide. She seems to fill the role of the Undetermined Dyad, since She is the mediator, first when She tells Demeter where Persphone has been taken, and second when She becomes Persphone's guardian during Her time in the Underworld. III. THE SACRED LIFECYCLE (Diachronic Triad) Whereas the Sacred Family is a static relationship within a Triad of Gods, the Sacred Lifecycle is a Triad in the life of an individual, which manifests as a Triple God or Goddess; I'll call the stages Virgin, Parent and Sage. In many cases these Triads are cyclic. A. Structure 1. Virgin The Monad corresponds to the first stage of life, the Virgin, and is characterized by self-centeredness, individuality, lack of discrimination and ignorance of norms. The Feminine Virgin is the archetypal Maiden, and the Masculine is perhaps the archetypal Hero. The characteristic activity of the Hero is protection; that of the Maiden is perhaps inspiration. 2. Parent The Dyad corresponds to the second stage of life, the Parent, and is characterized by other-directedness (i.e., caring, supporting, self-sacrificing), excessive competition or cooperation (interaction with others), inflexible discrimination, and rigid adherence to norms. The Feminine Parent is the archetypal Mother, whose characteristic function is nurturing; the Masculine Parent is the archetypal Father, whose characteristic function is providing. 3. Sage The Triad corresponds to the third stage of life, which integrates the preceding, and is characterized by balanced devotion to self and others, balanced competition/cooperation, flexible discrimination, and transcendence of rules. It is the synthesis of the thesis and antithesis of the first two stages. The Masculine Sage is the Wiseman, whose characteristic function is "professing" (philosophers, priests, statesmen); the Feminine Sage is the Crone, whose characteristic function is perhaps healing (counsellors, witches, healers). B. Example 1. Olympian Cycle: Dionysus(=Son of Zeus)-Zeus-Kronos 2. Chthonic Cycle: Persephone-Hades/Demeter-Hecate Persephone is the archetypal Maiden engaged in self-centered enjoyment, when She unwittingly inspires Hades to abduct Her. Demeter reacts as the archetypal Mother. Hades of course is not Persephone's Father, but He does at least obey the norms of the time by securing Her Father's consent and making Persephone His legitimate wife and queen; in this sense He fits the Parent archetype. Hecate is a archetypal Crone, working for a practical resolution of the situation. IV. GENESIS OF MUSICAL HARMONIA In this section I'll show how the same triadic structure produces the Harmonia (tuning) of musical scales. A. Triadic Genesis 1. Unison The Monad is the Perfect Unison, two sounds of the same pitch (C-C). The Perfect Unison is the most consonant interval. 2. Octave The Dyad is the Octave, since a string divided in half will sound an octave above the undivided string (C-c). Further, if a string is divided in the ratio 1:2, its parts will sound an Octave apart. The Octave is the next most consonant interval after the Perfect Unison. 3. Perfect Fifth The Triad is the Perfect Fifth, since if a string is divided in the ratio 2:3, the parts will sound at the interval of a Perfect Fifth (C-G), which is the next most consonant interval after the Octave. B. Tetraktys The Tetraktys, a triangle of units in rows of size 1:2:3:4, was a central mystery of the Pythagoreans: "What is the Oracle at Delphi?" "The Tetraktys, the very thing that is the Harmonia of the Sirens." (Iamb., Vit. Pyth. 18) The Tetraktys embodies the first three consonant intervals after Perfect Unison: the Octave (1:2), the Perfect Fifth (2:3), and the Perfect Fourth (3:4), e.g. C-F. Because the interval between the Perfect Fourth and the Perfect Fifth is a whole tone, it provides the basic unit for constructing a scale. Ancient lyres were tuned as follows: the tonic (hypate), the octave (nete), the fifth (trite or paramese), the fourth (mese). The tuning of the remaining strings would determine the Harmonia (tuning or mode) of the lyre. (The earliest music was limited to just the four tones, "the Harmonia of the Sirens.") C. Oblong Numbers The Pythagoreans classified numbers as Triangular (1, 3, 6, 10,...), Square (1, 4, 9, 16,...), Oblong (2, 6, 12, 20,...), etc., depending on the shapes into which they could be arranged. The Oblong numbers are those that can be arranged in a rectangle one unit wider than it is high; each is twice a Triangular number. The Oblongs have sides in the ratios 1:2, 2:3, 3:4, 4:5, 5:6,.... These give the intervals in decreasing order of consonance: Octave, Perfect Fifth, Perfect Fourth, Major Third (4:5), Minor Third (5:6), etc. The Tonic, Third and Fifth are the basic components of Triadic Harmony. D. The Circle of Fifths The basic process of the Triadic Genesis can be repeated to generate all the pitches of the chromatic scale: i. The Monad: Choose a pitch. ii. The Dyad: Double the pitch and so determine the Octave. iii. The Triad: Take the (Arithmetic) Mean of the Octave to get the Perfect Fifth. iv. Now take this new pitch as the Monad and repeat the above steps. This will produce that well-known "Circle of Fifths" that will eventually generate all the pitches of the chromatic scale and return to the starting point (after 12 cycles): C G D A E B Gb Db Ab Eb Bb F c (This will require slight adjustment to correspond to modern equal-temperment tuning, which replaces the Pythagorean exact ratio 3/2 = 1.5 with the approximation 2**(7/12) = 1.49831.) *** finis ***