“Self-limitation” in Pythagorean allegory
Ernest G. McClain
Richard Heath’s interactive website called HarmonicExplorer.org is a brilliant introduction to the cosmology of our ancestors, East and West, made accessible to children of all ages by way of our inherited harmonical mythology. The subtitle of one of Heath’s books, HOW STORIES CREATE THE WORLD, is my guiding light here.
HARMONIC EXPLORER (hereafter simply HE as acronym for Pythagoras, imperturbable Greek know-it-all who faked every experiment,) opens with a default setting of 8,640 that helps Socrates tell Plato’s story in the fourth century BCE. Trusting vision as most reliable of the senses, HE opens with the Chaos that Plato deliberately bequeathed to Greece science upon his death in 349 BCE. ‘Nature likes to hide,’ so the Pre-Socratics liked to remind each other, and it is their mystery-mongering that he plumbs to its depths. Plato buries earlier tales under his own ironically cryptic abstraction of logic presented with a new mythology of his own. HE effortlessly decodes the ancient Egyptian arithmetic that Plato posits as origin of all Greek science, and so frees our concern from technology for immediate consideration of meaning. The result is proposed as a reconstruction of Aristotle’s lost book on “Pythagoreanism,” the first ever written. Its suppression seems likely to have been deliberate, for the clues have survived, colored with deceit.
- ‘Trinitarian’ analysis in three perspectives
HE’s default setting of 8,640 is “upper bound” of a minimal numerology within the octave 2:1 imagined as 8640:4320 when the limit is halved. This range of 4,320 units has descended within the Platonic tradition [c.f. John Adams] to emerge in the 20th century as the ‘period of the phoenix,’ traditional “bird of paradise” in various cultures. The result is pictured here in three different ways as a Trinity that can be judged equally sacred or profane, as we please.
1.1) Factor analysis of partial products of “regular” primes 2, 3, and 5 (the first three) is keyed to “a tonal algebra” for 263251 meaning 2x2x2x2x2x2=64, times 3×3=9, times 5. The result, 8,640, in retrospect becomes a Platonic ‘chaos’ framed historically in smallest integers. This ‘bird’ of apparently 4,320 linear ‘distances’ contains only 30 that define useful ‘building bricks,’ and only 19 that matter at the moment for musical cosmology–elegantly compacted. If these ideas are new, let yourself be amazed; explanation follows; understanding comes last, and in its own ways to yourself.
1.2) Pattern is concretized in a ‘holy mountain’ of 30 ‘baked bricks’ numbered within Pythagorean ‘Ten-ness’ (integers 0-1-2-3-4-5-6-7-8-9) become Plato’s “form” numbers in defining the larger integers restricted to tonal significance, now visible on the bricks. They correlate with seven alphabetical tone names (A-B-C-D-E-F-G) in ways that identify 19 elements–their darkened bricks now lie within the first three rows as 6+7+6 that Plato reads as “brothers and sisters,” sprung to life within in the same octave-double, 2:1.
1.3) Resultant tonal meaning in “cyclic permutation” is graphed as “vector analysis” by radial lines in a “tone circle” that ensures perfect cyclic coincidence at every numerical doubling. Think of them as emanating from the ‘middle of the sun’ from which fallen hero ER describes the universe at the end of REPUBLIC (more accurately translated as POLITY), for the books, as considering the meaning of “Justice” that Pythagoreans considered to be “Four.” Thus ‘2’ as its ‘square root’ and smallest prime becomes both easiest to use and most powerful for understanding. (To the initiated, division by 2 instantly reveals the deepest secrets.)
[In harmonic analysis modern “exponents” conveniently behave as ancient “god powers.” Root meanings of “joint, proportion, concord,” fitting together as if ‘naturally’ is our aim in cooking and carpentry. HE honors Plato by studying ‘philosophy mixed with music’ as he contrives paradox upon paradox to awaken reflection during the darkest period of Athenian history—the early fourth century BCE, when Athens was shamed by the complicity of Plato’s own family in political disaster.
Default limit 8,640: toolbar technology for ‘Pythagorean’ conflict
2.1) Within the infinite reach of the integers (our natural “counting numbers”) any convenient limit becomes our best friend. Click the cursor on its window one once to highlight, and a drop down window then offers many alternatives. Choose another by clicking on it, then activate with a click on the check mark A to its right. Whenever confused (and it will happen often), simply overtype your own choice; or close the program and reopen with default 8,640.
2.2) Test the first ten toolbar windows that follow with a click on division (÷) or multiplication (x) by 2, 3, 5, 10 and 60 to witness simultaneous changes in all three displays. Alternate clicks to change and restore defaults to your pleasure. Keep a ‘playful attitude’ until you acquire control. Feeling counts.
[Useful trick: Hold down CTRL while rotating the mouse wheel slightly forward or back to enlarge or decrease type size, also affecting the toolbar; it is often most comfortable for eyes when the bottom row of bricks almost touches the circle, but larger limits may require adjustment or separate display.)
2.3) Toggle cents to transform the referent 12 hours into 1200 modern logarithmic cents–as if each equally tempered semitone is subdivided into a hundred micro intervals, each too small to be noticed by human ears. (A dog does better, and Pythagoras considered it a “philosopher” for greeting a friend or stranger appropriately.)
[Toggling wings us past 5000 years of ‘spiritual’ warfare (and billions of archaic ‘dust’ particles) into the rarefied mind-space of metaphysics where only false pride can be injured. Attention to ‘12 hours’ leads naturally to the zodiac, counting days by solar sunrises, but within periods determined by the wayward moon. HE is designed to accommodate Plato’s whims, and those of scribes in other cultures.]
2.4) Toggle “brick pile” to see it disappear and return, then toggle “notes” to see their names and circle do the same, and then toggle “circle” to see it disappear by itself. These controls enlarge or decrease both displays and printing. HE assumes that musical “wholetones” are double hours, 6 to a cosmic cycle; as if 3 “watches for the night,” and another 3 for the day, meaning that the chromatics scale of 12 semitones correlate symbolically with the 12 signs of zodiacs, each sign considered as extending to 30 degrees in a circle. Thus explanations employ the odd verbalism of “semitone hours” purely for numerical convenience.
2.5) But HARMONIC EXPLORER has a mind of its own—favoring products of 2, 3 and 5 in the Neolithic 4th millennium BCE, 5 or 6 thousand years ago. When we divide numbers with larger prime factors, HE answers as expected until their multiples are exhausted; from that point on it responds with nearest approximations among its own “regular” numbers (products of 2, 3, and 5).
Explanation: Biblical Jews chafed for over four hundred years ‘making bricks for Pharaoh’ under restrictions (that perhaps were Sumerian in origin) before ‘approximately 600,000’ rebel under the leadership of Moses. HE helps us share their pain. The factors of 26=64 in default 8,640 descend from glyphs representing the right eye of Horus, the hawk guarding the throne of Egypt, now reduced to Platonic “nursemaids” (indistinguishable from mothers, daughters, and sisters) as ‘empty vessels’ for “male seeds” (as in Hippocratic genetic theory). “Trinitarian” upper caste factors of 32=9 descend with Philo’s later Jewish blessing as “the most warlike of numbers” in the Bible and (when looking at 1x3x3 = 9 become examples of ‘Cretan bull-leaping’ in Homeric metaphor. All of us are created naturally as Plato’s ‘auxiliaries,’ meaning lower caste “fivers.” This default setting of 8,640 cryptically correlates all of worst number problems in classical Greek mathematics into a perverse example of “smallest integers” that I suggest we accept as Plato’s ‘phoenix,’ in the sense of the Egyptian Benu bird.
2.6) With the “pointing” fingers of both hands on the screen at pitch class “D” (at 12:00 o’clock ,“our ‘zero hour’ on American time tables), move them simultaneously outward around the perimeter to count nine pairs of ‘Platonic twins” mirrored on each side of an imaginary “plumb line” from heaven. Let’s pretend that line is Alice in Wonderland’s first of 3 magic mirrors by which ancient “mythography” can be read with Plato’s famed stereoptical double vision but set in motion as dialectics, meaning in dialogue, most successful when some Other mind is on the opposite end of your log. This is 3-way conversation about past, present and future, (Aristotle’s beginning, middle, and end, the best advice he could append).
3) Plato’s heptatonic World Soul abstracted as default 864
3.1) With default limit 8,640 still showing, click once on ÷10 to flee Greek confusion for 864 as Plato’s notion of heaven embodied in the ‘octave’ cycle of 8 Sirens mounted on rims of planets in the “spindle” of his World Soul, each singing a single tone. The 8th as ‘boundary marker’ is hidden within the lower bound of 432 as 864/2, halving the referent “D” that HE always “enthrones” for immediate comparison. Within the simpler self-symmetries of “octave double” of 864:432 a heptatonic 7–tone limit of F C G D A E B lies in tuning order among the bricks as paired from the middle. (Place the pointing fingers of both hands again on brick “D,” and move them simultaneously outwards on G:A as ‘wholetones’ or ‘major 2nds,’ then on to C:E as ‘major 3rds,’ then on to F:B as “minor 3rds,” always counted from D to simulate this bi-lateral symmetry associated specifically with Apollo. Athena is “Justice” not only as “4” but also associated with 7 (united with 4 in 4-3-2-1-2-3-4) when viewed from the middle of the sun of Platonic insight. We are leaving childhood behind to think of ourselves as the vertical embodiment of bi-lateral symmetry, wherever we find it.
3.2) Now focus eyes and fingers on the tone circle and move your paired fingers in opposite directions around its circumference to recognize the same tones realigned into scale order as if intended for sounding on the Greek 7-string lyre—rising to the right and falling to the left as far as voice and instruments can let us sing or play.
3.3) Next, with one finger only, start on any tone, rotating to the 8th in either direction, to meet the same pitch class again as both upper and lower boundary markers of a modal octave that consist of 5 wholetones and two semitones, always at B:C and E:F, distinctly closer together, never again in the middle of tetrachords. Thus heptatonic modal octaves never enjoy the “Edenic” bliss of an extended pentatonic system. The result is seven distinctly different modal patterns, differentiated by careful attention to semitones. But notice also the growing semantic confusion as musicians employ the same words for quite different meanings that only context clarifies. (This can sound like gibberish for the uninitiated.) Music is practiced as an art of the particular, the concrete, often with great concern for detail. But the Muses had long been loving sisters, rapidly become estranged in new Greek science.
3.4) Our sophisticated naming system employs 7 letters modified upwards or downwards by a semitone with sharps (S) and flats (F) to become 12, plus 10 that accept double sharps (##) and double flats (FF), printed and in both upper and lower case letters, and giving rise to further trivial distinctions known as “commas.” Such algebraic nicety makes it easier to follow common European practice for the last thousand years by singing only the solfeggio symbols do-re-mi-fa-sol-la-si-do with the rising scale from C as Plato’s ‘true Hellenic mode, the Greek Dorian—or the falling scale from E as mi-re-do-si-la-sol- fa-mi that need the same arithmetic for contrary locations of their semitones. Greek ratio numbers and proper pitch names belong only to theory, not to musical performance. We are free either to heal or split our own souls over this area. Pythagorean issues must be left to those who find them interesting.
Comment: This falling E mode was considered more authentic for Plato’s Greek Dorian until Francis Cornford pointed out that our own rising C mode was assumed as implied by Plato’s paired reciprocals. He exercises future “guardians” of his model cities by ‘wrestling naked in the gymnasia’ in ‘paired teams’ of 2 to 10 each. Here we see two ‘3-man teams’ around their ‘referee.’ Any seven consecutive elements in any row of such a ‘matrix’ (i.e., mother) enjoys equivalent worth in some context.
3.5) We have arrived at the Platonic entrance to a welter of heptatonic 7-tone confusion that today is severely strained by modern 12-tone chromaticism. Each tone of the 12 is a possible “tonic” reference for all seven of the Greek diatonic modes (among very many other structures), translatable through 12 different keys producing—on paper—7×12=84 “mode keys” that are recognizable only among about one-percent of the population that enjoys “absolute pitch” memory. English habits recognize that the rest of us possess, at best, only a sense of relative pitch, thus favoring seven solfeggio syllables for all purposes. Pitch class “D” sung as ‘do’ (from Latin dominus, Lord) thus makes a delightful bridge to Plato’s ancient habits that, in turn, function as a bridge to a musicology emerged in the late Neolithic before the invention of writing.
3.6) With default 864 displayed, click once each on ÷2 and ÷3 (using their partial results) to divide by 6 [now available] into 144 for further study of these first two primes.
4) Default 144: Plato’s “human soul” descends from stars (not planets)
4.1) With division by 6 we reach Egypt with default 144 as sexually ambiguous “humanoid fivers” (displayed visually as 2+3) with greater distances of “minor thirds” A:C and E:G imposed between ‘feet’ and ‘hands.’ We appear articulated happily “at the waist” like an insect, near the diaphragm where “spirit” as the “wind of God” originates.
4.2) Only 5 bricks (darkened in the foundational base) among 12 “regular” candidates can be given tone names as twinned Platonically because they constitute the lower border (lacking opposites below). All 12 are required later for calendar and zodiac. What these 5 mean tonally is experienced easily by singing do re mi sol la (rising from left to right, or falling in reverse as mi re do la sol. These elements are best absorbed unconsciously very early in children’s songs and dances. Sing them while embodying them personally to link your own tonal feeling with the history of ideas: here is Platonic philosophy mixed with music. A mysterious sense of tonality likely will guide your pitch if you’ve been properly loved—for we are long conditioned harmonically by the natural resonances in our own voice and of those closest to us. Wholetones do-re-mi rise to the right in patterns of 3s, while sol-la continues in pairs, and all fall to the left in reverse as la-sol-mi-re-do, repeating at the octave 2:1 on any sixth tone (a contradictory verbal locution that proves convenient). On keyboards HE’s enthroned “D” correlates with the middle of three black digitals now named either A-FLAT or G-SHARP, but this confusion is avoided in Plato’s models by his exclusion of ‘enharmonic genera,’ eventually very widely exploited. (In the arts, whatever proves “worst” proves useful for some contrary purpose.)
4.3) Glide the cursor across these five favored bricks (white labels help them stand out) while noticing that pitch names are encircled in the cycle, alerting us that numbers no longer signify abstract “points of no dimension” as Platonic ‘boundary markers,’ but have become “embodied” pitches for ears that require spatial extension over some radial portion of the cycle, assumed to be about a quartertone. As a result, an ‘octave’ theoretically of six wholetones (difficult to sing accurately) and often referred to more simply as tones, invites further semantic confusion) while permitting practical halving into 12 semitones that never were standardized until Equal Temperament was recently imposed (for commercial reasons). Plato’s social theory goes no further than necessary to document Socrates’ concern with “What twelve is.” Here we see 3 wholetones and 2 semitones filling the space reserved for twelve intervals whose perfect equality matters only to astronomers and calendar makers, thinking of a lunar zodiac rather than ‘good enough” pitches. Thus Plato’s “numbers in motion,” by themselves and at our convenience (an idea still roundly ridiculed) serve very different sister Muses.
5) Platonic technology—from the one to the many
5.1) Plato’s arithmetic is disciplined teleologically by the end he has in mind, described in S. 546 as “3 multiplied by 4 and 5, and raised to the fourth power” as 3x4x5=601,2,3,4 =12,960,000– decomposed (when Socrates’ pretentious jesting is penetrated) into the square of 3,600 and/or a rectangle of 2,700×4,800. Since understanding comes last to a learner, let’s “saltate” (leap) to the end as he does–to the middle of the sun itself–and watch its ‘radiance’ grow as if “seeded” here by 34=81 in default 144 as fifth tone in the pentatonic subset. A click on ÷3 eliminates ‘arms’ and reduces the default to 48, with 36 and 27 showing to its right as a preview of their later roles in his analysis. Another click on ÷2 eliminates ‘legs’ and reduces default bricks to “the One itself” as the modern zero power all numbers, justifying the verbal insistence of some ancient authors that “creation proceeded from nothing,” although philosophers disagreed.
5.2) Now click 4 times on X60 to watch Plato’s explosive overview of harmonic theory, radiating as if from the middle of the sun. Our referent D now is 5th among 9 symmetries in the fifth row—an ennead traceable to 34=81 among 144 being reciprocated. This basic Mesopotamian tuning theory was published only in 1976 by Kilmer, Crocker and Brown as inferred from cuneiform documents from the early 2nd millennium BCE, 4000 years ago—when adapted to its 9-string lyre, with strings 1 and 8 both tuned in octaves, and with the 2nd and 9th tuned similarly. Pythagorean economy knew only the 7-string version, with pentatonic C G D A E extended by F and B; it is now doubly extended by Bb on the left and F# on the right. (Pythagorean Ten-ness is served by a lonely C#, still unnamed on the far right. “Citizens” older than 10 are excluded from foundational theory.)
5.3) Click twice on ÷60 to see the “diminished light” in 603 = 216,000
and 602 = 3,600, and then once more to see the reduced model of 601 itself as “big One” in the new arithmetic that Sumer is credited with inventing in the fourth millennium BCE near the end of the Neolithic period, before verbal writing appeared.
This version of pentatonic symmetry (with drooping ‘arms’) may look dejected, but head and feet are ideally located to expose the first paired symmetries. G and A are Plato’s “guardians of the highest property class” (meaning as musical 5th and/or 4ths in opposite directions from referent D, always enthroned). They now enjoy help from b and f (in lower case type face) as ‘auxiliary guardians’ of the second class (or caste). The ratio 40:50 integrates 4:5 and 45:36 integrates its reciprocal as 5:4. New symmetries of 5:6 as 50:60 also pair inversely with 36:30 (lower bound as half of 60). We are gazing on a revolution in arithmetic that happened before any stories were written, for base 60 cuneiform notation henceforth permitted division by 2, 3, and 5 to infinity to be simulated by merely rotating the matrix 180 degrees, half-a-turn, thus validating reading from both 12:00 and 6:00 o’clock (i.e., from 1200 and 600 cents.) When 50 is set aside (for chromatic extension), the remaining seven constitute a “Just” alternative to Spiral 5ths heptatonic tuning, falling D-c-bF-A, G-f-eF-D when the brick pile is upright, but rising D-e-fS-G, A-b-cS-D when it is inverted. The fixed limits of these pairs of similar “tetrachords” remain A-D-G as upper caste “guardians” among the bricks, but are realigned as G-D-A with the scale to accommodate 2 pairs of “auxiliaries” as “moveable sounds.” We can only wonder how closely Plato thought tones and planets behaved.
5.4) This revolution in “musicology” that happened long before Plato warrants expanding typeface and bricks with CTRL and the mouse roller, and then rotating the printed copy in our own hands. Cultures that preferred working in base 10 for its simplicity never needed to rotate “brick walls.” It was always assumed that, “As Above, So Below.” Our modern writing habits encourage working upward and to the right from the cornerstone. Scribes often could imagine the superimposed patterns without computing new reciprocals. But the underlying pattern reduces to that for only 3×5, so that for many purposes no doubling proves necessary. Plato provides for a tonal calendar that integrates reciprocals under a new limit. Lets exit here with default 60 to watch him integrate Time.
6) Plato’s 2-year Tonal Calendar as Heath’s “Lunar Zodiac”
6.1) With default 60 showing, click the cursor on 45 to see a surprising result as 45 rescues the “cornerstone 32” as a skewed pendulum from the throne. This proves essential to Semitic musicology and to the Bible’s, but not for Plato, who refuses to recognize approximation, however valuable. Heath has cleverly contrived this one exception to naming only Platonic twins; the extension to lonely small a-flat in “Just” tunings is given a dashed red vector not quite aligned with a ‘plumb line’ from the throne; this single a-symmetry achieves a remarkable digital economy. He is not as opposed to sensation as he sometimes appears to be.
6.2) With default 45 displayed, click slowly four times on X2 to see familiar “degrees” at right angles of 45-90-180-360, embraced within its double at 2×360=720 “days plus nights.” Although this concept of “degrees” is formalized only in the 2nd century of the Common Era (CE) by Claudius Ptolemy (assuming that 720 half-degrees map nearly the same area as a circle with a figure having that many equal sides, it is apparent that this coincidence with ancient practice became habitual with some scribes far earlier. Each hour on HE’s circle for 720 can be imagined as divided into 720/24=30 parts within the 2:1 octave as 60:30 with accurate ratio divisions at 30-32-36-40-45-48-54-60. Thus astronomical measures projected on this tone-circle in the 2:1 octave read as 720:360 would have approximated accuracy reasonably well while naked eye observation was being improve. Oddly, we know that for centuries after Plato’s death, Alexandrian Platonists tried to correct his tuning to fit better celestial observations.
But we can only guess how tone circles might have been viewed before producing “god numbers” for imperial tyrants.
6.3) Click now twice more on default 720 to reach default 2880 where small a-flat gains its twin as small g-sharp. This increase in numerosity by four times is obviated by rotation of the table. These twins function as Horus and Seth in Egyptian mythology, vying for father Pharaoh’s throne, and inspiring many tales that Plato tells in his own way. At 3 places in the circle (2:00, 6:00, and 10:00 o’clock) Plato places the “three fates” as “daughters of Necessity” (now enthroned at 12:00). These 3 poor girls, severely disciplined to spin out our solar fates, must constantly adjust his circles to keep them moving at a steady pace. Two of them (presumably at 10:00 and 2:00) reach out occasionally—one with the right hand, the other with the left–to adjust the speed of Plato’s “Spindle of Necessity,” carrying his 8 Sirens, but the third, presumably at 6:00, occasionally must use both hands. Under these slaving conditions we can appreciate why Plato postulates “Chance” as also a deity.
6.4) Now with default 2880 showing, click once on x3 to see default 8,640 reappear with both “Horus and Seth” surrounding 600 cents at 6:00 o’clock. Our pentatonic Egyptian model is displayed with purple radials, and the seven tones of the World Soul are displayed in upper case capitals as Platonic guardians, vs his auxiliaries in lower case.
6.4) One click on ÷2 vanquishes “bad twin Seth” as g-sharp to display (in my imagination) the Platonic “bird of Paradise” as 4,320 reported by James Adam, streamlined elegantly for action as the Egyptian Bennu. (Visit Wikipedia for a view). Then restore Seth to the default by a click on X2 to leave 8,640 still showing.
7) Platonic origins in the third millennium BCE: 8,640,000,000 as the universal flood
7.1) From default 8,640 multiply by millions with six successive clicks on X10 to extend the “tail” of the phoenix to represent the beastly Imdugud, of ancient Mesopotamia, winged with a “spiral of 5th and 4ths” reaching to the 21st feather as 320 = 3,486,784,401, and with a “peak” at 514=6,103,515,625. [Use Ctrl and mouse roller to adjust display to view limits.]
This vulgar (view also on Wikipedia) broke a compact with the serpent to “feed each other’s children” by traitorously swallowing those in the serpent’s nest. Now toggle cents to see 598 at the peak, and 602 as seventh in the base, meaning within 2/100th of a semitone when viewed from the throne as fourth in the eighth row. They are very near coincidences to the square root of 2, a point of central interest in Platonic mathematics, and the subject of his MENO where the untutored ‘slave boy’ (under Socrates questioning) proves wiser than his master. This picture had inspired not only Noah’s flood but hundreds of other stories throughout the ancient world. We shall return to it often.
7.2) Toggle the brick pile out of your way to view ‘Horus and Seth’ now doubly represented near 6:00, for the “flood twins,” far more accurate, have no practical role except to inspire wonder and reflection. Here, by the grace of Richard Heath’s skill as a web developer, mathematician, gracious author, and geometer, we see for ourselves the ancient foundations of Plato’s science in ways he probably never witnessed himself, and was trying to re-invent from traveller’s tales.
Here I must stop to take a breath.
And after 6 pages of being led through HE with an “Egyptian nose-rope” the reader probably will welcome a recess also.