Variations on a Theme from Athanasius Kircher’s Harmonia Mundi
‘What the dream suggests is this: an infinitude of analogues.’ Peter W. Travis
‘Oh baby, why don’t you just meet me in the middle?’ Sarah Aarons, Zedd, Grey, and The Monsters and the Strangerz
‘If everything were similar there would be no harmony, and, therefore, no beauty. Harmony is the unification of dissimilar and unequal things.’ Eriugena, Expositiones in Ierarchiam Coelestem
This table of correspondences showing the music of the spheres is from Athanasius Kircher’s Musurgia Universalis, vol II, book X, p.393 (1650).
In Neurobiology, Layered Texts, and Correlative Cosmologies, Farmer et al (2000) attribute it to Robert Fludd’s Utriusque cosmi maioris scilicet et minoris metaphysica, physica atque technica historia (1617-21), but though similar and related material is to be found there in the second volume (Tomus Secundus, e.g. pp.45, 90, 254, 259), it is quite different in form and substance, and lacking in many significant details included by Kircher.
Kircher’s table combines diverse traditions from many centuries: Pythagoreanism with its Egyptian and Babylonian sources, Empedoclean philosophy, neo-Platonism, early Christian angelology with its Old Testament roots, and medieval Christian scholasticism. Inspired by a vision shared with their contemporaries of a harmonious universe, Fludd and Kircher nevertheless found themselves in active opposition and noisy dissonance with those such as Marin Mersenne and Johannes Kepler who at the same time stood at the threshold of a more modern scientific outlook. 300 years later, physicist Wolfgang Pauli highlighted both gains and losses in this paradigm shift: in our detailed measurement of distinct units, we have undervalued the holistic, integrated view which sees connections between things seemingly unconnected, ourselves and our universe.
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The ten columns are labelled ‘enneachord’ (‘ennea’ simply meaning ‘nine’ for the nine rows in the table), and are numbered one through ten in Roman numerals, with each column listing nine elements of a single type that were together thought to characterise the harmony of the spheres.
To the left of the table are nine musical terms in Latin, representing a sequence of musical intervals, from the bottom of the table: tone, third, fourth, fifth, sixth, seventh, octave, ninth, and tenth.
In the first proper column (labelled mundus archetypus) come the nine choirs of angels, as defined by Pseudo-Dionysus and Thomas Aquinas, who drew on scattered biblical references to angels in the Old and New Testaments. From the bottom, in three groups of three, are: angels, archangels, and principalities; powers, virtues, and dominations; thrones, cherubim, and seraphim. Each group has a specific role: guiding, protecting and mediating between God and man; governing creation; and serving God.
For example, the angels corresponding to the middle row of this first column are the virtutes (virtues, also known as strongholds, fortresses and ‘shining ones’), whose role is to ensure the continuing operation of the physical laws, which reveal the miraculous perfection of God’s creation. From Gustav Davidson (The Celestial Virtues, 1970), we learn that an early Aramaic-Hebrew magical text, Sword of Moses, teaches that the ‘Virtues oversee the heavens and perform wonders.’ Examples of this oversight are maintaining the sun and the planets in their proper course in the heavens, and keeping the virtuous resolutely on the path of righteousness, acting in the right way, at the right time, in the right place, and for the right reasons, consistent with Aristotle’s ethical doctrine of the golden mean, taking the middle way between extremes.
The second column (labelled coelum empireum) is the only one to contain pairs of terms in a single column, and it illustrates the music of the spheres by connecting music and astronomy.
The symbols of the second column represent the sun, moon and planets, in order of their supposed distance from the earth. Listed, from the bottom, are the earth itself (‘Terra’), followed by the astrological symbols for the moon, Venus, Mercury, Sun, Mars, Jupiter, Saturn, and the fixed stars in the firmament. As for the order of Venus and Mercury in this sequence of planets, Flora Levin explains in her translation and commentary on The Manual of Harmonics of Nichomachus the Pythagorean (1994) that ‘the respective positions of Hermes [Mercury] and Aphrodite [Venus] seem, however, to have been unsettled in antiquity and this is reflected in Nichomachus’ ordering.’ (Levin, p.56). Both planets were correctly identified as moving in a way peculiarly constrained by the sun, from our perspective on earth looking at what are called the two ‘inner planets’ from a heliocentric perspective, with their ordering relative to the sun only depending on the ordering principle we elect to use (with the faster of the two nearer to earth, or nearer to the sun).
The Greek terms in the same column are names of degrees of the musical scale, or strings on the nine-string enneachord, ranging from proslambanomenos, the lowest sounding note, through two overlapping tetrachords: hypate, parhypate, lichanos, mese (the highest note of the low tetrachord, and the lowest of the high tetrachord), paramese, paranete, and nete. No musical term is given for the ninth degree, corresponding to the fixed stars: perhaps their fixity gives no corresponding sound. In ancient Greek music theory, these names do not define specific musical intervals, and do not address the matter of precise tuning. In fact, only three of these eight notes (proslambanomenos, hypate, and mese – in Kircher’s table corresponding to earth, moon and sun) were considered to have fixed tuning, while the exact tuning of other notes may vary according to the music to be played.
The conjuction of the musical and astronomical sequences in this particular order, with the moon taken as the closest, slowest, and lowest sounding of the heavenly bodies apart from the earth itself, through to Saturn as the furthest, fastest and highest sounding, is to be found in a fragment ascribed to Nichomachus (Levin, pp.191–192), and which is alleged by Nichomachus to antedate the reversed order given in the Manual itself. (Levin, p.55)
The quantitative arithmetic justification for the equivalences between planets and musical notes are not given in Kircher’s table itself, nor does he supply any reasoned link to the Latin intervals to the left of the table, though he treats these matters elsewhere in his treatise (e.g. p.378 of 𝘔𝘶𝘴𝘶𝘳𝘨𝘪𝘢 𝘜𝘯𝘪𝘷𝘦𝘳𝘴𝘢𝘭𝘪𝘴, vol II, but not in a manner directly compatible with the table). [1]
For the remainder of this exploration of Kircher’s table, I propose to continue to by walking through the middle term of each column, by way of example. This central diapente row takes on additional significance, not only as the centre of each column, but also as the definition of an active principle (sun), mediating between the fixed octaves (earth, firmanent) and guiding the other notes (planets, and all other things between heaven and earth). These ideas will recur in the following explorations.
Either of Nichomachus’ two versions of the ordering of the planets versus musical tones leaves the sun and mese at the centre of the second column. ‘Mese’ in Greek means simply ‘middle’ – the middle string of the enneachord, to be played by the middle finger. The sun also has a role as the middle orbit in an overall geocentric scheme consistent with the model of 1st c. CE Alexandrian Ptolemy which recognised the special status of the ‘inner planets,’ as later developed by Tycho Brahe. So, all considerations of periodicity and frequency of heavenly and musical movement aside, the sun and mese play an analogous role at the middle of their respective enneachords.
The third column of metals (mundus mineralis) specifies gold in the middle position. Because of its physical appearance, gold is an obvious analogue of the sun, an association found in many world cultures. The Latin word used here for gold, aurum, is derived from the Proto Indo European root *Hau-s-, also associated with the sun in the words Eos (a goddess of dawn), easter (the festival of solar rebirth), east (the direction of sunrise), and aurora. Gold is also distinctive for its malleability, and its resistance to moisture and corrosion even on its surface, preserving its attractive shiny appearance over time.
In the next column for stones (lapides), the correct translation of Kircher’s Latin term pyropus is less clear. Farmer et al (ibid.) translate it as fool’s gold, or iron pyrite, which makes an attractive connection as far as superficial appearances are concerned, but which I’ve not been able to substantiate anywhere. Other translations sometimes given for pyropus in Classical Latin include bronze or bronze/gold alloys, all metallic materials which are not in the category of minerals, and so are unlikely candidates here.
On the other hand, W.J. Oates in The Stoic and Epicurean Philosophers (1940) translates Lucretius, De Rerum Natura II: ‘ruddy with the hue of bright pyropus.’ The description of a kind of carbuncle stone of fiery red is supported by the Oxford English Dictionary, which gives near contemporary definitions of pyropus as a red-coloured gemstone or carbuncle (e.g. ‘1553 EDEN Treat. Newe Ind. (Arb.) 14 Piropi (whiche are a kind of Rubies or Carbuncles)’), this being followed by several dictionaries in the following centuries. Hans Helander’s (1985) notes on Swedenborg’s neo-Latin panegyric on Charles XII, Festivus applausus (1710), state: ‘Some classical words are used in a new sense, but quite in accordance with the entries in Jonas Petri Gothus’ dictionary (1640): pyropus probably does not mean “goldbronze” but “ruby”’ (p.32), which was evidently the only meaning given in this dictionary and others at the time (ibid., p.139). In The Diamond: A Study in Chinese and Hellenistic Folk-Lore (1915), Berthold Laufer states that several of his contemporaries, ‘without giving reference to any passage, are unanimous in the belief that the carbuncle is the chief night-shining jewel of the ancients. It would be interesting to learn what alleged passage in an ancient author these scholars had in mind. As far as I know, the carbuncle appears as a night-shining stone only in the mineralogical writings of the middle ages, for the first time presumably in the fundamental work De lapidibis pretiosis of Marbodus (1035–1123), the famous French Bishop of Rennes.’ He goes on to give later examples from the mythical Prester John (c. 1165), and Konrad von Megenberg (1309–1378) which extol ‘the carbuncle as the noblest of all stones, combining all their virtues. Its colour is fiery, and it is even more brilliant at night than in the daytime; during the day it is dark, but at night it shines so brightly that night almost becomes day. This belief still prevailed in the seventeenth century,’ as he exemplifies in a long extract from A. Boetius de Boot (1636).
Ultimately, it’s not a definitively correct translation of Kircher we’re aiming at, which is unfeasible since the author himself may have had multiple meanings in mind, but an insight into the idea behind the association, and what connecting features were meant to make the forms sympathetic. In the case of pyropus, it may have been the literal translation of pyropus as a stone with ‘fiery aspect,’ which could possibly be read as fool’s gold, but to my mind it’s the fiery red carbuncle generating its own light in the darkness which gives the strongest and deepest analogy with the sun, producing light in the darkness.
We can be sure Kircher was familiar with what we now know as a sunflower (Helianthus), because in 1641 he documented and illustrated an experiment which tried to use one mounted on a cork and fixed with a mirror as a clock (though in the illustration he calls it solisequa). Earlier writers using the term heliotropium, such as Dante, cannot have been referring to Helianthus, since the genus was brought from America by explorers in the 15th century. In The Heliotrope Tradition (1937), Erika von Erhardt-Siebold makes a comprehensive study of potential referents of the term Heliotropium from antiquity to the sixteenth century, and concludes : ‘The question which particular flower a certain writer meant by “his” heliotrope is, of course, perfectly senseless in all those cases – and they are by far the most numerous – where the author himself would never have been able to point to a definite object.’ A unifying feature is a supposed tendency (not actual in many cases) for a flower to turn, or for its appearance to be otherwise affected, through opening or closing its petals, according to the sun’s rotation. She concludes, of the period she covers after 1200, that of ‘the numerous suborders of the Compositae the genera and species of the suborder Lactuceae undoubtedly ranked first in furnishing heliotrope flowers.’ This is a plant tribe in the daisy family which includes thousands of species, but von Erhardt-Siebold singles out the dandelion, whose downy seedhead is known as a dandelion clock, and is used in traditional child’s play to tell the time. She derives the name dandelion through the familiar etymology of dent de lion in French, but rather than following this back to Dens Leonis in Latin, she proposes an earlier derivation through the term for heliotrope used by Dioscorides and Apulius, dialion, to the family of words associated with ‘divine/deity’ and ‘diurnal/day,’ citing Pliny’s statement that the heliotropes were sun-dials. While the shared roots of words asociated with divinity and day are well supported elsewhere (e.g. Gamkrelidze, T., Indo European and the Indo Europeans, 1995, vol.2, p.196), as is the etymology of the heliotropic daisy to the same root as day (e.g. American Heritage Dictionary Indo-European roots), I’ve not found further support for the derivation of dandelion from dialion.
In Ovid’s Metamorphosis IV, 268, the nymph Clytie wasted away for love of Apollo, and transformed into a sun-following heliotrope with its violet flower (clearly not the golden dandelion) forever turned towards the sun. Other Classical references quoted in Dioscorides are unanimously identified with Boraginaceae Heliotropium, according to von Erhardt-Siebold, Linnaeus having later borrowed his scientific name from this attribution. Again, it’s not the exact genus of plant Kircher was referring to which is of most relevance, but their sun-following behaviour, and in the Ovidian source of the myth of Apollo and Clytie, we have found another reference point to explore the analogies in this row of the table.
In the sixth column is the only instance in the row giving two distinct elements of the same type, the laurel tree and the lotus tree.
Apollo is associated with the laurel by metonymy in many ways: in Ovid’s Metamorphoses, Daphne (literally ‘laurel’ in Greek) transforms into a laurel to escape Apollo’s advances, and (in some versions) subsequently pities him and gives him a crown of laurel leaves, which are one of his distinguishing emblems, and symbol of the artful poetry and music which may arise from unrequited and sublimated love. Lotis was similarly metamorphosed into a lotus tree to escape the unwelcome attentions of the very un-Apollonian Priapus. The connection to Apollo comes in a later episode in which Dryope, having been raped by Apollo and now mothering his young child, plucks a flower from the same lotus tree, which is wounded, bleeds, and trembles for fear of pain. Dryope is in turn punished by transformation into a tree. Ovid describes the transformation and aftermath of both Daphne and Lotis in such a way that shows them to still retain aspects of their original nature, and to be at the threshold of existence as a tree without being entirely transformed.
Many mythical transformations result in trees, and several directly involve Apollo, such as Cypressus’ transformation into a cypress tree, which is included in Kirscher’s table in the row associated with Saturn. So why are the laurel and lotus trees presented in this row together?
The laurel is easier to understand: there is a stronger direct relationship with Apollo and his symbolism, and the laurel was thought to have been used in the oracular ritual at Delphi (though it’s recently been suggested it could have been oleander going under the name of laurel: Harissis, H.V., A Bittersweet Story: the true nature of the laurel of the Oracle of Delphi, 2015). But also, the transformation of Daphne was the first after the deluge in Ovid’s first book of Metamorphosis. ‘From the transformation of Daphne on, the poem will feature countless metamorphoses producing new beings – from plants and animals to geographical features and even constellations – further enriching the population, landscape, and history of an ever-changing world.’ (Zatta, C., Plants’ Interconnected Lives: From Ovid’s Myths to Presocratic Thought and Beyond, 2016). So Daphne’s transformation is especially significant as an initial examplar of how the principal of metamorphosis will subsequently develop to produce all other things (like the Pythagorean diapente produces other musical tones).
Furthermore, writing in the 5th century BCE, Empedocles wrote (fragments 26–28, in van der Ben, N., Empedocles’ Poem on natural philosophy, 2019) that the lion, laurel, dolphin and eagle are the fulfilment of their respective creature types. Later Platonists read this in the light of their own views on transmigration of souls, which van der Ben does not ascribe to Empedocles however, nor any ‘Circean or Ovidian-style metamorphosis.’ Nevertheless, three of these four creatures appear in this same row in Kircher’s table (the eagle being associated by Kircher instead with Jupiter), suggesting some kind of influence, but adopted with what in mind? These creatures are not presented as the mean or average of their respective types, but the fulfilment, the telos, the highest species of their group. To simultaneously present them as a middle-point only makes sense if they are also an inflection point between the different groups in an ordered hierarchy, as the Platonists had believed.
The lotus tree however is more of a challenge, not least because again we are in doubt which trees Kircher and earlier writers are referring to. For Ursula Sdunnus (The Story of Dryope: A Rare Subject from Ovid’s Metamorphoses, 1990): ‘the tree in question is probably either a nettle tree (celtis australis) or a date plum (diospyros lotus). Ovid certainly did not mean nympheas for which the name lotus is now more common. But since the “lotus” trees were already confused in antiquity it is impossible to say exactly which Ovid had in mind.’ The plucking of the branch is the only component uniquely specific to the story of Dryope, which is linked to this row through the theme of mediation, the particular emphasis on the liminal existence of the tree/nymph being analogous to the position of the sun between inner and outer planets, and mese as the midpoint in the enneachord. But since the theme of liminality is already evident in the story of Daphne, it remains unclear to me why it needed reemphasising with a second example.
Dolphins, in the seventh column, are directly associated with Apollo, one of whose epithets is Delphinios, and whose oracle at Delphi was established after Apollo appeared as a dolphin and coopted a party of merchants into his priesthood. Recent research (e.g. Pilleri, G., Knuckey, J., The distribution, navigation and orientation by the sun of Delphinus delphis L. in the western Mediterranean, 1968) attests the dolphin’s ability to direct itself with reference to the sun, a gift which may have been informally observed or supposed from ancient times, contributing to an association with the sun and Apollo in Classical Greece and related cultures. And anyone who’s watched a school of dolphins gracefully arcing over the surface of a calm sea, their dorsal fins tracing congruent curves through the air, may be reminded of the daily procession of the sun and planets along the ecliptic.
The theme of liminality also arises with reference to the dolphin. The same Proto Indo European root *k’oelbh gives us, in Greek, delphús ‘womb’, and delphinos ‘dolphin’ (Gamkrelidze, ibid, vol.2, p.716): the dolphin is a sea animal, though a mammal and not a fish. They were also said by the ancients to have human qualities: ‘Diviner than the dolphin is nothing yet created, for indeed they were aforetime men and lived in cities along with the mortals.’ (Oppian (Loeb translation), Halieutica I, 200 CE). One author documents (without a specific reference and very probably unreliably, also in attributing the dolphin to Venus rather than Apollo) a Greek myth in which a dolphin gives birth to the sun. (Hannay, J.B., Symbolism in Relation to Religion, 1913)
As we have seen, Empedocles holds out the eagle as the fulfilment of the avian creatures, and we have this particular fragment through a quotation by Oppian who mentions both the eagle and dolphin. But for once, Kircher departs from Empedocles and assigns this row to the cockerel, perhaps because of its strong association with solar timekeeping, its cry being the harbinger of dawn in the popular imagination. (In fact, the cockerel is more frequently associated in folk tradition with Mercury, to whose row in the table Kircher allocates the parrot – but that’s another story.) In the Nun’s Priest’s Tale, Chaucer presents the magnificent cockerel Chauntecleer, which puts clockmakers to shame with the accuracy of his timekeeping:
Wel sikerer was his crowyng in his logge
Than is a clokke or an abbey orlogge.
By nature he knew ech ascencioun
Of the equynoxial in thilke toun;
For whan degrees fiftene weren ascended,
Thanne crew he that it myghte nat been amended.
And if Kircher had read his Albertus Magnus diligently, he’d have known that ‘the movements of choler occur at intervals of three – at every third year, every third month, every third day, or every third hour. As a choleric animal, the cock is subject to the movements of choler at every third hour.’ (Steadman, J.M., Chauntecleer and Medieval Natural History, 1959) In turn, this connects back to the ‘threeness’ of the mese, the three musico-mathematical means, and again the diapente interval.
The lion, in the ninth column, of quadrapeds, is another interesting case. If Kircher was following Empedocles as he had elsewhere, with trees and aquatic creatures, then we have already seen the source of this association. There are other factors which may not have been directly known to Kircher which support the association of the lion with the sun, Apollo, and gold. They may have been known at least in part to Empedocles, and may explain his original assessment of the lion as the fulfilment of the quadrupeds. Apollo, sometimes called the most Greek of the Greek gods, was the confluence of many cultural streams. Herbert A. Cahn argues (Die Löwen des Apollon, 1950) that Apollo was originally identical with the Anatolian sun god, that he came from the East, and that from the beginning the lion was his companion. In Old Hittite tradition the wild animals of the Middle World include the lion with Sumerogram UR.MAH, ‘of gold.’ (Gramkelidze, ibid, vol.2, p.406). Calvert Watkins (How to Kill a Dragon: Aspects of Indo-European Poetics, 1995, p.148–9) defends various Hittite, Doric and Mycenean sources for Apollo, and while he doubts that ‘his later connection with the sun can be projected back to the second millenium’ he argues for a cultural continuum which links Apollo to earlier sun deities: Luvian Suwasuna, and Anatolian Istanu. Aelian (Loeb translation, On the characteristics of animals, 12.7) reports: ‘In Egypt they worship lions… And since he is of a very fiery nature, they say the lion is the house of the sun.’
Superficially, some of the earler associations in this row directly involved the colour gold in this last and tenth column (the sun, the metal gold, the dandelion, the lion, but also the specious translations of pyropus as fool’s gold, and heliotropium as sunflower/helianthus), but some others did not: the sun of the dawn and dusk is red; the carbuncle is red; the heliotropium is violet in Ovid; cockerels may or may not be golden, and can also sport a distinctive red comb like the sunrise; there is nothing inherently golden about an actual dolphin, or a laurel. The most notable connection of the golden colour with the other elements in this row, of course, is its association with gold itself, and thereby all that is valuable and reliable in that metal. After all, what is it that is ‘golden’ about the golden section, the golden mean, the golden age, and the golden rule? Nothing other than their being, like gold itself, the supposed epitome of their respective types.
The golden light of the sun is the active principle which gives rise to the other colours in daylight, for example for Lucretius (De Rerum Natura, II): ‘what colour can there be in total darkness? nay it changes in the light itself according as its brightness comes… sure enough you must believe that [colours] cannot be produced without it.’) In this way, golden sunlight takes its place in the colours alongside the generative and ordering intermediators of diapente, mese, the sun, laurel, lion and dolphin.
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Holding that all of language works on the same principle as metaphor is an idea on the ascendant in some linguistic circles (e.g. Ortony, A. (ed.), Metaphor and Thought, 2nd ed., 1993). The extent of the arbitrariness of vehicle and tenor in the operation of this principle is variable – there are better grounds for some substitutions than others – and what we’ve done here is explore both vehicle and tenor (signifier and signified), to try to understand the grounds on which their connection has been made.
Kircher’s table is the product of a mindset which seeks connections between the heavens and the earth, and between the earth and its denizens, including humankind, and demands that it be understood as a harmonious whole. As Henry Cornelius Agrippa had said a century before Kircher (De Occulta Philosophia, I:37, 1533): ‘The world is the image of god, man the image of the world, animals that of man, plant life that of animals, metals that of plant life, and stones that of metals.’ As we have seen, it is an old tradition that reaches back through the ancients, and into prehistory.
I like to imagine Kircher had a sense of humour, and that he had good fun assembling and adjusting this table slowly over time, as he continued his polymathic research. But to what extent is an apparently complete and detailed table such as this comparable to the great homologies so powerfully celebrated in the Rig Veda, Chinese Feng Shui, or African magical traditions, as proposed by Farmer et al? It is similarly comprised of traditional elements organically accrued by a culture from diverse sources over many generations, though perhaps its presentation as a finite table pretends to a finality and completeness which is not entirely warranted.
I discussed tables of correspondence with J.H. Prynne in 2009: ‘Ah yes,’ said he, with what I heard as heavy irony, ‘we like tables.’ I took it as an admonition at the time, that such tables can become problematic if they are taken too literally, if they are read in a way that does not admit playful multivalency and polysemy, and if, in making them, we are not patient enough to leave some gaps where we have no ideas, perhaps for now.
Much of the above has centred on the sun, and along with some doubtful equivalences with fool’s gold and sunflowers, there have been many other initially opaque but ultimately illuminating associations in other domains. We have sought that same commonality Edmund Husserl sought through his phenomenological Variations (Experience and Judgment, 1973, p.348–50) which tends towards the very essence of a thing, an essence which itself has proved for some to be fool’s gold, merely a dream of a false dawn. We can intuit an essence within the elements of the row, but it does not go by one name alone: it is the Apollonian ordering of all things into harmonious parts, containing both the Empedoclean telos of a type, and at the same time its Pythagorean middle term.
But where Husserl’s endgame is to intuit the essence of a fact, and reduce reality to idea by removing difference from its variations, our game has been to multiply the particular cases in which a fact can be found as a variation, and augment reality to something more than itself through connections based on similarity.
Husserl’s assertion that adequate perception of a melody must include more than just the note now being heard influenced Jan Mukařovský to propose the syntagmic wholeness of a sentence over its individual words. And like Sergej Karcevskij’s sign which is a synonym of all other words that have been or might be applied, like Vedic homologies, analogies in feng shui and Chan/Zen reexpressions, and African homeopathic magic, our own game of variations aims to be a (never-ending) melody in which the whole is simultaneously potential in a present that never arrives.
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[1] Note on the ‘harmony of the spheres’
The harmony of the spheres is an idea which persisted from the times of Pythagoras, through Plato, Boethius, Dante, and Chaucer to Kircher.
Dividing a string equally into 2, 4, 8 &c parts gives a sequence of octaves considered musically equivalent: e.g. a series of Cs in modern western terminology. Dividing it into 3, 9, 27 &c parts gives a new note in the scale each time: e.g. G, D, A and so on, which is known in modern terminology as the ‘circle of fifths’.
By dividing the string into powers of three, across the division into diapason/octaves formed by powers of two, the circle of fifths of Pythagorean tuning is set in motion.
The Greek ‘diapente’ musical interval (a ‘fifth’ in modern terminology) is obtained by dividing a vibrating string into three equal parts: it is the third step of the natural overtone series, between the two octaves which result from dividing the string into two and four parts. The interval of diapente/fifth also implicitly defines the ‘diatessaron’ (a ‘fourth’ in modern terminology), which is the part of the octave remaining outside the fifth, and which forms the basis of the tetrachord interval which can be further divided into various non-Pythagorean modes.
There are a triple of mathematical means in evidence here: for example, the geometric mean of 1 and 4, the double octave, which equals 2 and sounds the diapason/octave; the arithmetic mean of 1 and 2, the octave, which equals 3/2 and sounds the diapente/fifth; and the harmonic mean of 1 and 2, the octave, which equals 4/3 and sounds the diatessaron/fourth.
Boethius’ Consolatio 3, IX is a short poem at the exact centre of a long work which was very influential on Kircher’s predecessors and contemporaries. The following lines are themselves at the centre (lines 13–14) of the 28-line poem:
Tu triplicis mediam naturae cuncta moventem
conectens animam per consona membra resolves
In Time and the Crystal (1990), a study of Dante’s Rime Petrose, Robert M. Durling and Ronald L. Martine translate the lines as follows:
To be the mid-point of triple Nature, to move all things,
You attach the soul and diffuse it through adapted members
In a note, they explain: ‘This translation departs from the way the verse was understood in the Middle Ages: as most of the commentaries agree, “triplicis . . . naturae” was understood as a modifier to animam rather than to mediam […] referring to Timaeus 35a-36b [Plato’s famous exposition of musical origins]. Dante seems to have understood the verse as we do, as a reference to the position of the soul as the mean between the two extremes.’
Chaucer translated the lines (in c.1380–85) as ‘Thow knyttest togidere the mene soule of treble kynd thingis, and divydest it by membrys accordynge,’ and H.R. James (1897) has ‘A soul of threefold nature.’ (both from Phillips, P.E., The English “Consolation of Philosophy”: Translation and Reception, 2008)
I prefer the reading Durling and Martine associate with Dante’s, but (with James) I would not capitalise ‘nature,’ so that the the ‘mid-point of triple nature’ may be read as the three mathematical means which are explicitly featured in Boethius’ source in Plato’s Timaeus. Then these three musico-mathematical means may be taken as the maker/mover of all things. As in the Timaeus, what is described in these lines is a specific empirical musical operation, which is the vehicle for a metaphor of the creation of the world soul according to musical principles:
You, the triple-natured mid-point, moving all things,
Connecting the soul, into harmonious parts she will divide.
