In the book of Revelation a choir of 144,000 "male virgins" ("men who have not known women") sing "a new song before the throne" at both the first and second resurrections of the Savior. The arithmetic is developed by "David" himself as 4.6.4 in Hebrew alphabetical notation, meaning the RATIO 2:3 taken in opposite directions from the middle.
Now stand erect with your two feet spread comfortably to 5 and 7 o'clock and YOU "embody" David's ratios as "clock time" or "ratio values" in a cyclic octave 2:1 in which further arithmetical DOUBLING changes nothing of interest to philosophy. Now raise your two arms symmetrically to 10 and 2 o'clock (and stand facing a clock if you need help) and you become a "stick figure" of 5 elements (2 hands plus 2 feet plus head above) in imitation of the Egyptian glyph for a star. Hands and head demark the position of C, D, and E at the interval of a "wholetone" between each, while feet mark G and A as another wholetone in the same direction (however you count), but with "minor thirds" at A:C and E:G. Your five "body markers" thus define five "distances" within David's "cyclic double" of 6:3 that we now call an "octave" (but the Greeks called a "dia-pason" meaning "through all the tones," thinking in terms of seven as normative, of which 2 are still missing here).
Forget the Greek, in America most of us speak English or at least try to, more or less. But historians of mathematics like Neugebauer and Sachs (who restored much of the ancient material for us) regard your feet as "bad approximations" to the "square root of 2" that keeps your head "enthroned" towards 12:00 o'clock as the "zero power" of ALL numbers. And that is why NO number suffices to represent Deity EXCEPT "1" as the ZERO power of any that numbers can define. "One-o'clock" thus does NOT coincide with the throne but rather with one-twelfth of its "clock cycle" taken for granted as "twelve" (but sometimes also twenty-four). Now musicologists never forget this ratio of 1.059432.... (a "continuing fraction") for it now defines "the twelfth root of 2" so that twelve in successive exactly arrives at "2." This ratio is nearly six percent, so we are welcome to think that "six percent compound interest doubles ANY value" in about 12 years. [During the Great Depression that began in 1929 some banks, instead of foreclosing on home buyers deprived of jobs, allowed trusted borrowers to keep them by paying ONLY this 6 percent annual interest, thus automatically extending mortgage payments, and this is how my parents managed to "get by." The ratio eventually became indelible in my mind; it signaled "survival." That is no longer true.]
In ancient times, whenever that may have been, musicians never suspected any "cyclic flaw" in musical fifths of 2:3 UNTIL they noticed a near convergence on the 13th tone that signaled a very slight cyclic excess eventually known as the "comma." You can check this for yourself by "taking 3" 12 times in succession to pinpoint 531,441, and then comparing with "2" taken 19 times in succession to reach 524,288. Never mine those six digit monstrosities; nobody's ears notice what is happening in the last four digits. The Greeks accepted 53:52 as the measure of the comma, and realized that 53 commas thus fall short of the octave as 54 = 6x9 commas, and that is why they conceived the "wholetone" (six per octave) as approximately "nine commas" each. Our "theory" develops in retrospect from experience. But this kind of insight made the god of mathematics and music the ONLY deity who could ANTICIPATE the future! And that is the role of Sumerian ENKI (later EA), god 40, never the head of the pantheon, but crucial to its well being. Gods 60, 50, 30, 20, 15, and 10 have their own responsibilities, but ENKI holds the keys to the future, and becomes the "canal controller" in Mesopotamian mythology (anticipating Plato's "canal system" by thousands of years). And in early Sumer (and apparently Egypt and India, as later in China) the pentatonic (5-tone) model prevailed.
Here is how it must have been computed, with "Egyptian" technology-- preserved ONLY in later Chinese practice WHICH HAS ALREADY MODERNIZED IT by 700 BC! I display the Chinese algorithm here but with the warning that probably only the first HALF of it is ancient. (The second half is unnecessarily subtle, so I substitute my own "Egyptian" simplification.) My own methods are thus a compendium of Greek, Chinese, Egyptian, and Hindu materials studied with the canny insight of Neugebauer and Sachs into a procedure that is the very opposite of how we normally think today. I do not know how Bible authors carried out their calculations EXCEPT by the evidence in the Bible itself, and I read its testimony as restricted to the very simplest procedures possible. I reconstruct Bible arithmetic in ways that may be too ponderous to do justice to authorial skills, but I prefer to err in that direction in order to escape much worse arithmetical anachronism. Someday Bible scholarship will straighten out this problem for us, but it cannot yet do so, and so I beg my readers to stay with my very primitive procedure until they something better is confirmed. This is the arithmetic, in its crudest form, that helped to build the pyramids.
To define 5 new tones from ANY suitable starting place, "Take 3 four times," as in 3x3x3x3=81, thus touching on 3, 9, and 27 along the way. Chinese wisdom now continues "boustrophedon" (as the ox plows, reversing course at the end of a furrow), and the final solution reads: "add or subtract one-third." Now that ambiguity may seem pretty silly until you try it. Adding one/third to 81 produces the Buddhist rosary of 108 beads. Now subtracting one-third from 108 produces 72, the number of "translators" of the Septuagint. And this is the "lower bound" of the octave 2:1 defined by 144:72 (notice, containing 72 subdivisions) when it becomes the "leader" as "largest integer," and thus "sealing" the other four within it. Add one-third to 72 to find 96, also capable of "sealing" an alternative octave 96:48. Now subtract one-third from 96 to find 64 and we are on "the throne of Egypt" protected by the eyes of Horus, the hawk, in which every component of the eye symbolizes one of the "octave fractions" 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64.
The number 64 is the first (smallest) to be both a "square" and a "cube" (8^2 = 4^3). One of the loveliest "sleepers" in the Bible is Sarah's death at 127 meaning at 1+2+4+8+16+32+64=127. Here is a sly Jewish jest about why "she" is the most beautiful woman who ever lived, obviously desired by Pharaoh as a member of his harem. (Successive powers of 2 always sum to one less than the next power.)
But ANY power of 2 (think 2^0=1; 2^1=2; 2^2=4; 2^3=8; 2^4=16, etc.,) leaves "pitch class" (i.e., musical "tone-name") unchanged while EVERY successive power of 3 defines a NEW pitch class (in the sequence I write as C G D A E to keep notation simple, although ANY tone of our 12 functions as "C" for somebody somewhere some of the time). (We use these names as "fixed" when necessary, and as "variable" when convenient. And musical practice negates nothing in the arithmetic except somebody's inappropriate assumptions.) This is a wondrously "fluid arithmetic" for a "fluid art" that defies all "logical positivism" to its face. TO HELL WITH ALL TYRANNY. YHWH is not a "Middle Eastern" tyrant, despite "Calvinism" and good a bit of "Popery" (too often "poopery" requiring apology after heroes are long in their graves), and widely practiced by the most ignorant and untutored Protestants who think ONLY themselves "liberated"--as if God played favorites in their own very special case. (America is full of conceited "know nothings" but I rather doubt our originality.)
The Chinese algorithm has five solutions within the octave 2:1. Working "boustrophedon" from 81 nobody can go wrong in defining 5 pitch classes (because integers will be exhausted on the fifth (which is really the first as original reference). I'm going to tabulate the five possibilities here showing each of the five "sealing" integers; you may fill in the others at your pleasure, for I leave them blank. The point is always to gather all five within ONE "numerical double."
1 3 9 27 81 (i.e., "take 3 four times")
_ _ _ 108 81 a possible limit on 108
_ _ 144 _ 81 a symmetric limit on 144
_ 96 _ _ 81 a possible limit on 96
128 _ _ _ 81 a preferred limit at 2^7 in "philosophical pitch" only.
Notice that awareness that multiplication by 2 (i.e.. by simple doubling) can solve ALL problems here, suggesting to me that the Chinese algorithm may well represent a second and higher level of insight, rather than an "initial" discovery.
Now the computation for any set of tones in Spiral Fifths tuning (by far the best ancient approximation to our modern Equal Temperament, at least through eleven tones) works similarly. Only if we understand this do we realize how Abram fathers Ishmael (on Sarai's Egyptian maid Hagar, and at her suggestion) at the "age of 86" (meaning 8x8x8x8x8x8 = 64^3 or 4^9 or 2^18=262,144 "days") to cut off "Spiral Fifths" at 3^11=177,147 after the twelfth tone (and before the 13th can intrude as a "comma"). Here is the Biblical validation of Arab--and now Muslim--claim to precedence in Jewish "tribal history." (I believe there is more to be gained from conceding this than by continuing to deny it!)
Notice the "tails" in all of these six digit numbers. They become "leaders" in YHWH's "reduced values." (The Jews are pure "numerologists," but with an unimpeachable "Platonic" discipline older than Plato!)
The tail in 177,147 becomes Jacob's lifetime of 147 years within which 12 sons are born in "Just" tuning (not shown here).
The tail in 262,144 becomes the "leader" of the Revelation male chorus.
The tail in 524,288 becomes David's "288 sons of Asaph" in charge of Temple musicology.
The tail in 531,441 becomes the 1.4.40 or 1440 in Adam's name containing the "seeds" of all future generations in the DOUBLE OCTAVE 1440:720::720:360 as the Jews mock the pretentious arithmetic of both Mesopotamia and of Greek Pythagoreanism. (And the unit "tail" in 1441 is dropped to become a "free floating correction factor" needed always somewhere in harmonic theory because the four "rivers" of products of 2, 3, 5, and 7 can never avoid missing each other by 1 unit unless assembled with great care under "good shepherds." I am sorry that I cannot clarify this better for Duane's new commentary on Genesis 1 and 2 where principles are first laid down in the way a "logical" exposition is expected to begin.
I close here now by looking at the names of Abram and Sarai, changed later to Abraham and Sarah very cleverly indeed. Abram as summed to 243 defines six tones all by himself with either arm (he is ambidextrous about left and right) and so either arm enjoys five reciprocal twins in the other, Thus Abram defines eleven powers of 3 (to 3^10=59,049)--and don't miss the 49 here in the tail, for as 7x7 it is definitive in the Jewish Jubilee calendar cycle. Sarai's sum of 510 calls attention to the smallest "human" octave of 10:5 (with five subdivisions, and to the fact that "female" being only "half of a male" must be doubled in value). In preparation for the birth of a "legitimate" heir at the ages of 100 and 90 YHWH changes their names by adding an "h" worth 5 points to Abram so that 243+5=248 becomes a pure octave cycle itself 2
8 , thus as "God surrogate" he acquires a twelfth tone as we see in a moment, while Sarai become Sarah loses five points to become only 505. But SHE now describes Abraham's new matrix! That is, with six tones in each arm as they emanate "from the center"--the "zero power" of all numbers. Here is the pattern before I explain this apparently anachronistic mathematical interpretation. Abraham is now in a privileged position in the middle. Sarah "maps" the pattern.
Eb Bb F C G D A E B F# C#
6 5 4 3 2 1 2 3 4 5 6
Sarah as 5 as 0 Sarah again as 5
Now it is obvious that I am accepting Kabbalist "gematria" as a factor in Jewish narrative but I restrict it exclusively to names of God and to a very few of his favorites, and mainly names that He assigns or changes, and I do it ONLY where Bible arithmetic already justifies the assignments purely on arithmetical grounds. YHWH Himself as 10.5.6.5 can be read as 10^5x6^5=60^5=777,600,000 to define all eleven of Abraham's tones, but YHWH has a "sleeper" (a twelfth tone) hidden in the last zero of the tail thay defines a G# that transforms into A-flat on the left during reciprocation, so that what appears as 12th and last becomes lst among thirteen from another and "larger" perspective. This is devilishly clever and you must do the arithmetic to appreciate the subtlety. Nobody in history seems to have bothered decoding 777,600,000 with Platonic methods to discover this unity between YHWH and Apollo as "far-shooter." [Homer treats Apollo hilariously as a "little stinker" on Day 1 in his cradle--but I must suppress Homer's brilliance here to let YHWH's shine alone.] Nobody knows yet why there is so much agreement between Plato and the Bible--an agreement that stunned early Christians when they became aware of it. We can't be certain yet that either Athens or Israel understood what was happening in the other. They have a common heritage that forbids dogmatic interpretation until linguists study the evidence more carefully--with a musical and arithmetical insight that is long lost.
It should be obvious that IF I am correct, then there is likely to be a great deal of other gematria hidden from me in the Hebrew texts. But I refuse personally to become involved with ANY numerical speculation that I cannot already justify "musicologically" within "Davidic" arithmetic. These personal habits make me suspiciously lunatic both to mathematicians who normally avoid mythology like the plague it indeed can be, and to Bible scholars and other linguists who could not prove more totally innocent of ANY understanding at all of musical arithmetic and "harmonical mythology." I and I know no musicologist (except Siegmund Levarie) who will risk his reputation by making judgments about Hebrew texts. They prove HIGHLY problematical. This situation will continue until other musicologists pay attention to close arguments about Bible texts, and I can find none on the planet who really care--those who are both competent and still alive are all too busy with their own work (and there are some very fine ones indeed today).
And so I must continue "in limbo," really trusted by nobody. But that is a very common phenomena indeed in scholarship--and not really a bad idea at all. NOBODY can be trusted to fully test the evidence AGAINST his own theories. (Hence my own contempt for Popper's crippling dogmatism on this subject.) I had a very lucky academic career indeed, mainly due to Jewish friends. Some of my own most brilliant younger friends have not been able to find positions at all, and some are discouraged by the attrition of intellectual standards since they were students. Life is full of chance. I have nothing to complain about (though I still grouse a bit every day).
I continue to do the best I can to explain what I believe was a very simple but very subtle and brilliant grounding in "harmonical allegory" covering most of the ancient world. There are plenty of young people with enough brains to find and correct my errors, including those I can't expect myself to see. I can imagine a glorious future for a brave new world--unless some fool blows up the one we have. And that danger looms in the present. Ernest