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Against Archytas:  How the West Lost Alchemy or Paranormal Complimentary Opposite Harmonics

Happy holidays. I’m presenting another article by drew hempel. I’ve asked drew to write in more detail on the subject of the connection between math and music and how it connects to the paranormal. I felt that his article
The Secret of Psychic Music Healing was assuming too much of the reader and so asked him to write a more explanatory article, which I now present.

Against Archytas:  How the West Lost Alchemy or Paranormal Complimentary Opposite Harmonics

by drew hempel, MA

(anti-copyright, free distribution)

The early Greek mathematics used the 60-based number system of Babylon from which Archytas, a collaborator with Plato, received the harmonic tetrachord or the continued proportion 6:8::9:12.  This tetrachord creates a geometric mean between the octave, perfect fourth and perfect fifth music intervals, or 1:2:3:4, through “divide and average” logarithmic-based mathematics.  So 6:8 and 9:12 are in the continued proportion 3:4, the perfect fourth music interval, while 6:9 and 8:12 are 2:3, the perfect fifth music interval, and 6:12 is 1:2, the octave.  The geometric mean is A:B::B:C or B squared = AC or the square root of AC = B. What Archytas added to this Babylon “divide and average” harmonic mathematics was the concept of the Greek “incommensurable” – the algebraic axiomatic proof of “alogon” or a precise irrational number, the square root of two.  This process ushered in what’s called “The Greek Miracle” that continues to be the structure of science:  symmetry-based mathematics.

Instead of the above system, the alchemical Pythagorean Tetrad relies on complimentary opposite harmonics so that an equilateral triangle of geometric points equals the continued proportion 1:2:3:4 as the octave, perfect fifth and perfect fourth music intervals.   In “orthodox” Pythagorean harmonics this was known as the “subcontrary mean” whereby the complimentary opposites of the Tetrad were maintained in violation of “divide and average” mathematics.  So for the Tetrad A:B is 2:3 and B:A is 3:4 against the commutative property, A x B = B x A.   In music theory this complimentary opposite inversion of the perfect fifth and perfect fourth is taught as 2:3 is C to G while 3:4 is G to C.  This process of complimentary opposites is listened to, as the perfect fifth, perfect fourth harmonics, which create all the notes.  Most importantly the complimentary opposite harmonics transduces sound throughout the whole energy spectrum, as I’ve described in previous articles..   

Philolaus, one of the early Pythagorean writers, detailed that this “subcontrary mean” or complimentary opposite harmonic caused any attempt at subdividing the scale into symmetry as a failure.  In contrast Archytas changed the “subcontrary” complimentary opposite mean into the “harmonic mean” using “divide and average” mathematics.  The outcome has precisely opposite the meaning of “harmony” which for Pythagoreans referred to the paranormal source of sound as the Goddess Harmonia or what I call female formless awareness.  For Philolaus the perfect fifth as 2:3 could be inverted to 3:2 and then extended another fifth to 9:4 and then divided back into the octave, below 2, for the major second interval of 9:8 or C to D.  Yet 9:8 cubed or three major second music intervals equaled the 3:2 perfect fifth music interval, plus a tiny ratio called “the comma of Pythagoras.”  

This “comma of Pythagoras” is the difference between the “divide and average” octave system adopted by Archytas and the complimentary opposite fifths inverting into fourths, used by the orthodox Pythagoreans.  The “comma of Pythagoras” is the key to harmonic alchemy whereby 2:3, the perfect fifth, is yang in Taoism and 3:4, the perfect fourth, is yin.  As Gurdjieff desribes the “shock” of the diatonic scale, whereby the “inverse ratios” do not line up with the octaves, is intensified as the octaves expand.  In contrast Archytas argued that 9/8 cubed or three major second intervals equals the square root of two as the Greek Miracle, the axiomatic algebra of the precise incommensurable irrational number.  What Archtyas essentially did, as I’ll describe, is equate the perfect 5th or 2:3 with the perfect 4th or 3:4 as equally-divided or symmetric ratios through a “divide and average” mathematics.

Archytas took the Babylonian geometric mean of 6:8::9:12 used for harmonics and then applied the Pythagorean Tetrad 1:2:3:4 so that the 2:3 ratio of complimetary opposite frequency was converted to 3:2 as a materialistic vibrating string length.  This became known as the Law of Pythagoras even though it goes against the true meaning of the complimentary opposites when this “inverse ratio” is combined with the “divide and average” commutative property. Gurdjieff, for example, still relies on the “inverse ratio” of density or string length versus frequency or consciousness.  But Gurdjieff does not use Archytas’ “divide and average” symmetric-based mathematics, instead Gurdjieff relies on the Law of Three aka the Tetrad, or octave-fifth-fourth,  to resonate through the comma of Pythagoras as the “shocks” of alchemy.  In Taoism this system of alchemical shocks is taught as the 12 harmonic nodes along the outside of the body, enabling healing and paranormal energy, an exercise called “the small universe.”

The equation used by Archtyas, from Babylon, was arthimetic mean times harmonic mean = geometric mean squared.  For the Pythagorean Tetrad this means if A = 1 and C = 2, the octave, then B = 3/2 for the arithmetic mean (A + C divided by 2) and B = 4/3 for the harmonic mean or 2(AC) divided by A + C. Meanwhile B = the square root of two for the geometric mean or the arithmetic mean times the harmonic mean equals the geometric mean squared (3/2×4/3 = 2).

The error that Archytas makes is ignoring the “comma of Pythagoras” arising from complimentary opposites harmonics so that the square root of two is equated with 3/2 through the “divide and average” converging sequence.  The square root of two is greater then one and less than two which can be solved through the “divide and average” geometric-based continued fraction series of (1 + a)squared = 1 + 2a + a squared.  2a + a squared = 1.  Or a(2 + a)=1.  Therefore a = 1 divided by 2 + a.  Then just keep replacing “a” with “1 divided by 2 + a” for an infinite series that converges as the square root of two.  The first iteration is 1 + 1/2 or the approximate series solution, 3/2 = the square root of two.

It needs to be emphasized that the square root of two is actually a “transcendental” number although this is only acknowledged in number theory and not in standard math.  There is no positive proof demonstrating that the hypotenuse of the 1-1 triangle equals the algebraic symbol the square root of two since there’s no algebric equation demonstrating the square root of two, only an infinite algebraic series that is not a closed set.  Most people just learn the geometric proof for the square root of two, demonstrating a “proof by contradiction,” which simply states that the ratios are not rational, but it does not consider other options for the ratios, especially the case of diverging complimentary opposite harmonics.

In contrast music theory, based on logical inference, uses listening to complimentary opposites, 1:2:3:4, as proof that the foundation of reality is female consciousness with the male number 1 as an infinite resonance creating an octave, 1/2, that expands in frequency through the perfect fifth or 2:3 or yang, inverting to the perfect fourth or 3:4 as yin.  In western music theory this is taught as the “circle of fifths” so that the perfect fifth or 2:3 starts with C to G then continues to finish the octave C through the 12 notes of the scale.  C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C.  This return back to “C” again ignores the empirical fact that the perfect fifth overtone inverts into the perfect fourth interval – through complimentary opposites, a dynamic that can be heard in music.  So if you have a string marked  into 1/2, 1, 3/2 and 2, a node at 3/2 actually is 3/4 of the string.  Or you could start the string with zero, and again a node at 2/3 is 3/4 of the string.  Similarly a node at 3/4 is 2/3 of the string.  What’s important is the yin-yang dynamic of this complimentary opposite harmonic thereby creating alchemy.

In other words there’s a complimentary opposite between the nodes and the numbers which can be heard as harmonious.  For this reason that musical “inverse induction” extension of the perfect fifths does not line up with the the extension of the octaves and this difference is called the “comma of Pythagoras.” In Nature there is no “circle of fifths,” but rather an infinite spiral of energy transduction that starts and ends with the male number one resonating as a complimentary opposite into female formless awareness and then resonating out as the perfect fifth harmonic which inverts as the perfect fourth to pull back to one and then, through its complimentary opposite – female formless awareness.  This process continues as resonance overtones, just as Dennis Gabor’s Quantum Time-Frequency Uncertainty Principle describes:  As the time gets less the frequency spreads across the whole energy spectrum.  It’s no accident that in quantum mechanics the commutative property is also violated with momentum times position not equaling position times momentum – only in science this is converted back into symmetric-based math using logarithms.  Quantum physicist Henry P. Stapp makes this paradox of Number and Order central to the mystery of consciousness in quantum mechanics.

The Golden Ratio also does not converge geometrically, unless, like the perfect fifth music interval, 2:3, the order of the Fibonacci number series is reversed so that there is symmetry between zero and one, using the “divide and average” commutative property.  In the converging series expansion the continued proportion A:B::B:A + B of the Fibonacci series, 1, 2, 3, 5, 8, 13 is reversed to A:B::B:A-B so that A(A – B) gives a positive solution for geometric convergence.  What this means, again, in terms of music is that the frequency ratios of 4:5, the major third, and 5:8, the minor sixth have to be reversed to materialistic string length as 5:4, the cube root of two, and 8:5, an approximation for the golden ratio and also the cube root of four.  Kepler used this Golden Ratio as the 3:5 music harmonic to combine it with 5:4, the major third, in order to justify his elliptical orbit analysis.  Similarly Newton’s inverse square law of gravity was derived from his application of Archytas’ symmetry-based logic so that it takes four times the weight on the end of a string to increase the tension to twice the string’s frequency.  Not until the 1960s did the Philosophical Transactions of the Royal Society of London publish this newly discovered Pythagorean-based source for Newton’s gravity.

I discovered that Archytas’ proof for precisely doubling the cube, his most famous equation, relies on this secret harmonic conversion of ratios into irrationals.  Up until now, only a geometric magnitude proof is given for doubling the cube so that A:X::Y:2A refers to X being the side length of the cube that needs to be doubled while the volume as 2A needs to be cubed with the geometric mean equation as 1:2::4:8.  The original side length equation is 1:cube root of two::cube root of four:2.  In fact these geometric mean ratios are from music harmonics as the major third and its inverse, the minor sixth or Golden Ratio, giving 1:5/4::8/5:2.  None of the mathematicians have been able to make this connection to Archytas’ conversion of the 2:3, perfect fifth music interval, for this “doubling the cube” proof that creates the Greek Miracle – an axiom for an irrational number.  David Fowler’s Mathematics of Plato’s Academy book certainly acknowledges the mystery of the music ratios origin for Archtyas’ doubling the cube proof.  Other mathematicians, like Professor Luigi Borzacchini, with whom I’ve corresponded, have certainly pondered the math-music incommensurability issue.

And so again the continued proportion proof for the square root of two in its first iteration gives the value 1 + 1/2 or 3/2 = the square root of two which is what Archytas relied on.  This same proof is tested by squaring both sides so that 9/4 = 2 (a first iteration in the series) or algebraically y squared + or – 1 divided by x squared = 2.  In the geometric mean proof, also used for the Golden Ratio, AC = B squared or N = x(x-1).  This also converts back to the Babylonian geometric mean equation so that 6:8::9:12.  Because 9/4 = 2, as the first iteration of the square root of two series, this is actually based on the Tetrad harmonics of 3/2 squared which must be divided back into the octave, less than two, as 9/8, the major second interval.  Or again Archytas combined the complimentary opposite harmonics of 1:2:3:4 into the geometric mean 6:8::9:12 for an algebraic solution to the square root of two with the iteration of 3:2.  So the geometric mean became paradoxically 1:8::9:2 or 1:2 cubed::3 squared:2 but the order is reversed to 9/8 since it’s a materialist string length, and not just frequency ratios.  In terms of the geometric mean for doubling the cube we can see that 5/4 is just 9/8 as two major second intervals or 10/8 reduced to 5/4, now justified as the cube root of two as the major third while the minor sixth, as 8:5, is just the half and then inverted.   

Again Archytas’ use of “just tuning” diatonic ratios, also used by Gurdjieff, such as 9:8, 3:5, and 5:4 were solely due to Archtyas’ conversion of 2:3:4 as complimentary opposites into the “divide and average” symmetry or the arithmetic mean.  So 2:3 became 3:2 from A + C divided by 2 = B with A = 1 and C = 2 as the octave.  This conversion of the complimentary opposite Tetrad perfect fifth of 2:3 into 3:2, as an arithmetic mean, then could be doubled, 9/4, and inverted back into the octave as 9/8, the major second interval and then converted to the geometric mean as three major second intervals or the square root of two also known as the “Devil’s Interval,” the tritone – C to F#.  

An easier way to understand this is described on this physics website:

“Solution … 
The ratio of the diagonal of a square to a side is √2:1. (Galileo stated the order of the ratio the other way around, but this is a minor detail.) Each half step (a semitone) up the equal tempered scale multiplies the previous note by the twelfth root of two, two half steps (a whole tone) multiplies the note by the twelfth root of two squared, three half steps by the twelfth root of two cubed, and so on …

1 semitone   minor second 12√2 12√2 
2 semitones  = 1 tone (whole tone)  major second 12√2 12√2 6√2 
3 semitones   minor third 12√2 12√2 12√2 4√2 
4 semitones  = 2 tones (ditone) major third 12√2 12√2 12√2 12√2 3√2 
5 semitones   perfect fourth 12√2 12√2 12√2 12√2 12√2 2.4√2 
6 semitones  = 3 tones (tritone) augmented fourth  12√2 12√2 12√2 12√2 12√2 12√2 2√2

Six semitones is equal to the twelfth root of two to the sixth power, which is equal to the square root of two. This interval is called a tritone, an augmented fourth, or a diminished fifth; for example, C and F♯ (G♭) or F and B. Had I given you a more complete quote from Galileo you would have already known this.”

In orthodox Pythagoreanism this use of an attempt to equal-temper the scale into 12 fifths, using the 9/8 ratio as the geometric mean, would not have been allowed since it enables setting up a quadratic equation based on the commutative property and the Pythagorean Theorem.  A square x B squared = C squared was always averaged in other mathematic systems but now it could be combined with zero to create a geometric convergence as an irrational number – the Greek Miracle based on a deep disharmony.  The complimentary opposite harmonics which naturally resonate as yin-yang dynamics have now been destroyed.  Even supersymmetry is just an extension of what Professor Oliver Reiser called “the music logarithmic spiral” so that mass squared is inversely proportional to energy frequency distance, as detailed by physics professor Gordon Kane.

There is no pure science.  Archytas’ “doubling the of cube” miracle was used for catapult technololgy while Newton’s “inverse square law” from Archytas’ geometric mean analysis has been the key for all projectile military technology just as Galileo’s Pythaogorean math was used for cannons.

What enabled Archytas to ignore the “comma of Pythagoras” which in alchemy creates an infinite resonance of energy through complimentary opposites was the vast difference of error between the Golden Ratio, the slowest converging irrational number and the square root of two series.  So that the inversion of 5:8 into 8:5 for both string length use (instead of frequency) and for use of the quadratic zero, geometric convergence, does not greatly affect the harmonic series accuracy.  It’s inverse is the conversion of 5/4 or 10/8 into the cube root of two.  This is the true secret of the Greek Miracle – combining the Brahmin cipher system of the commutative property, using zero, with Babylon’s equation of the arithmetic mean times harmonic mean equaling the geometric mean squared.  Eudoxus simply extended Archtyas’ use of the harmonic mean so that it could be applied as the Golden Ratio mean.  As the book Excursions into Number Theory notes:  “For instance, 99/70, the sixth convergent of square root of two, differs from square root of two by .000072; but 13/8, the sixth convergence of the Golden Ratio, differs from the Golden Ratio by .0070, showing an error nearly 100 times as large.” (p. 134)

In conclusion – mathematicians have never been able to figure out why the ratio 2/3 was sacred in Egypt.  Now you know the secret – or at least to what extent complimentary opposite alchemical harmonics have been covered up.





  1. Well the most important to learn is HOW TO LEARN. haha.
    Anyway — here’s a new source — see the page 14, the first paragraph, after the chart, for discussing the tritone as the comma of Pythagoras plus three 9/8 intervals. This is either a Perfect 4th plus the comma of Pythagoras or a Perfect 5th plus the comma of Pythagoras — to equal the tritone.,+philolaus,+3/2,+comma+of+pythagoras&hl=en&ct=clnk&cd=15&gl=us&client=firefox-a

    The same issue applies — application of the commutative principle.

    Ok here’s the main sources I used, plus relying on my previous research, chapter four, found at

    Here check this out. A chapter of someone’s 1,000 plus page tome on scales and tuning — is focused on this issue of the ORDER of ratios as string length vs. frequency.

    Does he understand the secret meaning of this paradox? haha. Don’t think so but at least he addresses this issue.

    Here’s another REALLY excellent article on Archytas’ origin of the harmonic mean — and it’s reliance on “divide and average” mathematics.

    And the final clincher — this new book on Archytas details on Archytas and EUDOXUS (note last night’s PRECOG reference) changed

    Philolaus’ use of “subcontary” as COMPLIMENTARY OPPOSITE into

    Harmonic — as “divide and average”,M1

  2. Bryan: The article clearly states I’m talking about a CONVERGING series — with the second divide and average value for the square root of two as 3/2 (1 plus 1/2). The crucial point is that a “divide and average” series relies on symmetry, or the commutative principle, whereas the music harmonic arithmetic series relies on complimentary opposites. So Archytas argued that 9/8 cubed (3/2 squared as 9/4, divided by 2, equalling 9/8 as the major second interval), then is the tritone interval or the square root of two. In contrast Philolaus states that 9/8 cubed is 3/2 plus the comma of Pythagoras. The difference between the two is that the latter, the “comma of Pythagoras,” relies on an arithmetic series that violates the commutative principle. Squaring a number, 3/2, does not rely on the commutative principle which uses a one-to-one correspondence between letter, as geometry, and number. Both the proof for the square root of two AND Archtyas’ use of arithmetic mean x harmonic mean equals geometric mean squared rely on the commutative principle. What complementary opposite harmonics shows is that the proof for the square root of two is WRONG, because the arithemetic series, through complementary harmonics, violates the commutative principle. In other words the squaring of the perfect fifths as 3/2 does not “divide and average” into the square root of two, as the squaring of the octaves, and this EMPIRICAL difference creates the comma of Pythagoras of complementary opposite harmonics. There’s a fundamental difference between the harmonics of 1 plus 1/2 as an arithmetic series, or pure number, in contrast to number (1) as a geometric side (square root of two) of an area. This difference, only demonstrated in complementary opposite harmonics, disproves the square root of two which is the foundation for all of mathematics. Archytas relied on squaring as geometry, with arithmetic mean x harmonic mean equalling geometric mean squared, so that the octave is now not an arithmetic double as is the case with the harmonic series 1:2:4:8:16, but a geometric square (3/2 x 4/3 equals 2, geometric mean squared, as the octave of 1, with the square root of two as the tritone interval (9/8 cubed). The lie here is that Archytas is equating the 3/2 squared (9/4 divided by 2 and then cubed) as the same as the octave squared whereas the first is an arithmetic series whereas the second is a geometric series (since the octave doubles in the harmonic series, not squares). Archtyas’ system works only because the octave is SQUARED to 4, using weight as tension to create an inverse square positive coupling of the octave. So under Archytas system when the weight as frequency is 4 then the square root of geometric mean is 2, the positive coupling of octave and string length. This positive coupling of string length and frequency, using weight, is the source for Descartes, Galileo and Newton’s measurement of frequency as geometric momentum of weight, whereby twice the distance or string length also equals twice the speed. So that, as Archytas first argued, the string length of two can also be the octave frequency of two. In other words time is no longer measured as frequency of number, but as geometric mean distance using weight for speed. Again the previous inverse ratios between string length and frequency, aka the Law of Pythagoras, have the octave as 1, when the string length is 2, or 1/2 with the string length is 1, not the octave as 2 (with the half when the string length starts with 1 and goes to 2, as the tritone. Under the harmonic series, the half string length is the octave frequency, again thereby having Archytas argue that 3/2, between 1 and 2, is now the tritone or the square root of two, only because of geometric-based time using weight tension). Complementary opposite harmonics, creating the comma of Pythagoras, relies on squaring as number, in violation of the commutative principle, upon which all of math is based. Again the Perfect 5th music interval of the Pythagorean Tetrad, 1:2:3:4 is Yang as 2:3 or C to G, while the Perfect 4th is Yin as 3:4, of G to C. G x C does not equal C x G, thereby creating the comma of Pythagoras so that 1:2 does not line up with 2:3 or it’s inverse, 3:4, which then continues to resonate as complementary opposites yin and yang, through the whole energy spectrum. The sound resonates into ultrasound, thereby ionizing the electrochemicals of the body, which then are further ionized into electromagnetic fields and light the bends spacetime, returning to it’s source of time, as frequency: female formless awareness.

  3. Yeah there’s a lot out there on harmonics and healing and shamanism, etc.

    I’ve corresponded with professor Joscelyn Godwin about my research on yin and yang as complimentary opposites. It amazes me that this angle is still left untouched by all these occult harmonic, sacred sound analysts! This site has a lot on this issue — geometric mean, ratios, etc. but completely ignores this violation of the commutative principle as the secret of the law of three.

    There is one way in which Godwins’ analysis is justified: The Pythagorean Lambdoma or the Law of Pythagoras frequency as inverse of string length 1/2 to 2/1, 2/3 to 3/2, etc., as a graph, is the SAME as Cantor’s “domain of rational numbers.” Cantor argued that the domain of real numbers is a greater infinity even though it can’t be graphed, since there’s a greater number of irrational roots than there are rational roots of squares. This paradox was first presented by Galileo who relied on Archytas’ harmonics for his science.

    This argument would be the acceptable counter-explanation to my analysis on harmonics except that, as I’ve pointed out, there is a fundmental difference between geometry and number, whereby the first irrational root is not justified as a number. This fundamental difference also exposes the asymmetry built into the rational number domain (because of the comma of pythagoras), thereby disproving Cantor and Galileo, etc.

  4. OK let’s review the basic math again. Simon Stevin used the geometric mean equation for equal tempered tuning as such: A/B=C/X. He assumes that the octave is twice that of the fundmental tone and then states that he can split the scale using cube root of two. 1:5/4=8/5:2.

    But, under the harmonic series, the octave does not use the commutative principle, which Simon Stevin relies on. So when Archytas used the same geometric mean equation he did not have 4 terms (A:B:C:X) to solve. To reduce Simon Stevin’s octave to the square root of two is just 1 x 2 = square root of two x square root of two. It’s the same as the algebraic equation given for the square root two — but neither equations show the connection to solving for the number value. To achieve a number value for the square root of two equation, Archytas relied on the Babylonian geometric mean equation with only 3 terms needed (not needing 4), thereby demonstrating how math comes from music ratios. So arithmetic mean times harmonic mean equals geometric mean squared. To plug that back into Simon Stevin’s equation 1 divided by 4/3 = 3/2 divided by 2.

    So this is the equivalent of stating that 4/3 and 3/2 ARE the square root of two, which may be justified as the “divide and average” continued fraction solution but it’s NOT justified as the harmonic series which violates the commutative principle. Since the octave as string length, 1/2, does not equal the octave as square root of 2, Archytas’ proof violates the arithmetic doubling of the octave as the harmonic series. Archytas relies on using string length as 3/2 and 4/3 (instead of frequency less than 2, or 2/3 and 3/4, because he has to double the octave to justify it’s square root value, with the double octave, 4, having the octave value, 2, through string length tension. The same “squaring of the octave” continues using the arithmetic mean x harmonic mean = geometric mean squared equation. In other words Archtyas ONLY justifies a number value solution for the square root of two “divide and average” geometric mean by VIOLATING the complimentary opposite harmonics of the perfect fifth, 2/3, and the perfect fourth, 3/4, of the Pythagorean Tetrad — or the yin and yang of Taoist alchemy.

    Again this means that time has been reduced to space as length (now measured with weight as momentum) and thereby ushering in modern science when rediscovered by Galileo, Descartes and Newton. The square root of two proof by contradiction ALSO relies on the commutative principle, just as Archytas and Simon Stevin used to solve for equal-tempered tuning. But again logically the proof by contradiction refers to geometry (area and size) NOT to length as number. It would appeared justified to equate the two (especially since you can square the “divide and average” number so that the infinite irrational converges to the number two as area) but now that it’s been proven that such equation DIRECTLY VIOLATES the complimentary opposite harmonics of number as string length, we can see exactly why science is inherently destructive (not for moral reasons but for illogical reasons).

  5. OK Bryan Olson, you didn’t understand Bertrand Russell so well and this may be embarrassing for you.

    In consolation, it is true that there is a “positive solution” for the square root of two, given in the
    book “Number: The Language of Science” by Dantzig
    (praised by Einstein as the best book on number

    The positive solution for the square root of two
    is only found by SQUARING the continued fraction
    series of 1.414 which, while an infinite irrational,
    does converge to 2, again when squared.

    This is the reason that the square root of two
    is not considered “transcendental” by mainstream
    science but, as Bertrand Russell argued, the continued
    fraction to solve the square root of two is not a
    closed converging geometric set, rather it is a
    THINGS or just pure geometry.

    This difference between calculus as geometry and
    algebra as number theory is also the focus of the 1999 Philosophy journal article, “Did the Greeks Discover
    the Irrational?” by Professors Hugly and Sayward.
    The proof by contradiction that combines algebra with geometry is a negative proof referring to AREA and
    SIDE (pure geometry notions) not the LENGTH as a
    algebraic NUMBER. Again this is why the square root
    of two is transcendental, because the algebraic
    proof is not logically valid.

    What this means is rather esoteric yet crucially
    important for the issue of alchemy through
    complimentary opposite harmonics.
    As I detailed
    Stephen Hawkings notes that in India there was no
    concept of the GEOMETRY being “incommensurable” or
    irregular. Rather the Indians (Babylonians, Chinese,
    etc.) understood that the divide and average
    continued fraction was infinite as a time-based
    iteration but the SACRED geometry was finite and
    regular (not incommensurable).

    Western science converts time as an infinite process
    to space as an infinite process but the space is
    IRREGULAR (asymmetric). This means that mathematics
    relies on symmetric-based algebraic equations but
    the result is a convergence on space through
    destructive technology.

    In contrast, in alchemy, time is not defined through
    distance but rather as something that is listened
    to, through complimentary opposite harmonics. Again
    this is why the major second, as 9/8 cubed, equals
    the Perfect Fifth (YANG) or 2:3 PLUS THE COMMA OF
    PYTHAGORAS. Archytas converted the doubling of
    frequency through octaves into a SQUARING of frequency
    so that there is now a “positive coupling” of
    string length (using weight) for time, now as
    distance. Now half the octave is the square root
    of two, based on 9/8 cubed, in disregard of the
    difference between the fifth-fourth inversions and the
    octave doubling. In algebra, using the harmonic series,
    half of the octave as string length, 2, is 1/2 — NOT THE SQUARE ROOT OF TWO. What Archytas did is convert
    time as an infinite resonance which reverses itself
    through consciousness and instead defined space
    as an infinite process that is “incommensurable”
    or irrational. Time became a linear finite process
    that converges to zero or in terms of technology,
    the apocalypse.

    • I’ve no idea what Drew is trying to pull now. Why should I be embarrassed about checking his reference and finding that he was wrong?

      Drew was unable to locate his own reference (the book was checked out), but he looked elsewhere and found I was right: as he wrote, “Bertrand Russell accepted the validity of irrational numbers.” Yes, I had though so. Quite different from his previous assertion, the one I asked him to cite, “Russell calls the square root of two a ‘convenient fiction.’”

      Drew writes:
      “This is the reason that the square root of two is not considered “transcendental” by mainstream science”.

      The reason the square root of two is not considered a transcendental number is that it isn’t — contrary to what Drew had emphatically claimed. How many times do we have to over a matter already so easily proven? We’ve seen the definition. We’ve seen the proof.

      Drew’s articles are a mess. There are some correct assertions, parroted from reasonable references, but Drew has no real understanding of what they are talking about, or even which propositions are factual. Google “complementary opposite harmonics” and one finds Drew just making stuff up. Each word is comprehensible, and Chinese Yin/Yang describes complementary opposites, but Drew has gone off into his own fantasy land.

  6. The Principles of Mathematics by Bertrand Russell is
    currently out on loan at the University of Minnesota
    but it is true that Bertrand Russell accepted the
    validity of irrational numbers, as he finally stated
    in a footnote, as detailed in this source:

    Review of Bertrand Russell, Towards the “Principles of Mathematics”, 1900-02, edited by Gregory H. Moore, and Bertrand Russell, Foundations of Logic, 1903-05 edited by Alasdair Urquhart with the assistance of Albert C. Lewis
    Irving Anellis

    Source: Rev. Mod. Log. Volume 8, Number 3-4 (2000), 57-93.

    To state that I’m half right and then not give any
    quotation is silly and I still stand by the fact that
    Russell calls the square root of two a “convenient
    fiction.” Which half of that quote is wrong Bryan
    Olson? haha.

    What can’t be denied is that I’ve given several
    detailed sources in this comment section debating
    the “transcendental” value of the square root of two –
    specifically that the algebraic equation is a
    proof by contradiction that does not establish
    a POSITIVE value for a precise closed set symbol –
    the square root of two.

    This is best detailed in Professors Hugly and Sayward
    1999 essay, “Did the Greeks Discover the Irrational?”
    a work I just reread. Hugly and Sayward specifically
    rely on BERTRAND RUSSELL as the source with which they
    argue. The issue is also clarified by Stephen Hawkingss
    — the square root of two is a GEOMETRIC symbol
    replacing the symmetrical geometry and an infinite
    number series with an asymmetric geometric and a
    closed number series.

    • Drew, who said you were half right? I said that where I know something about the subject, I can tell that you do not.

      I expect most readers do not know what a transcendental number is, and do not need to. What I find stupid is writing “It needs to be emphasized that the square root of two is actually a “transcendental” number”. No, it is not, and everyone that does understand the term knows it.

      Drew wrote:
      What can’t be denied is that I’ve given several detailed sources in this comment section debating the “transcendental” value of the square root of two [...]

      No Drew, you are fooling yourself. You do not understand the material; there is no such debate. Your sources did not say the square root of two is a transcendental number. You do not know what they are talking about, so when you try to work from them, you do not know what you are talking about.

  7. OK Bryan, as I’m at the engineering library at the U of MN, I did a Science Citation Index search on Betrand Russell’s Principles of Mathematics. The following essay states that Russell DID NOT AGREE WITH CANTOR’S INFINITE NUMBERS — so Bryan Olson, your reading of Bertrand Russell is incorrect.

    The collected papers of Bertrand Russell, vol 3, Toward the ‘Principles of Mathematics’
    Source: International Journal of Philosophical Studies [0967-2559] Levine yr:1998 vol:6 iss:1 pg:87 -127

    “Further, rather than allowing that infinite collections may have a number, albeit one which is not assignable, Russell denies that they have a number at all, and thereby rejects Cantor’s view that there are infinite numbers.”

    • For some reason Drew Hemple quotes Levine writing:
      “Russell denies that they have a number at all, and thereby rejects Cantor’s view that there are infinite numbers.”

      Infinite numbers? Drew, do you think the square root of two, or any other irrational number, is infinite?

  8. Bryan you’d have to actually type the quote for me or anyone else to know what you’re referring to. Again willful ignorance is not a very effective means of communication! haha. As far as the definition of transcendental number — obviously if the definition for the square root of two is a “convenient fiction” then, as the first “algebraic” real number — other real numbers, like transcendentals, would not have the same definition (not being defined in an algebraic equation). That’s just logic — not semantics.

    I’ll look up the Principles of Mathematics quote just so others won’t be confused by your willful ignorance but as I remember it, Russell is discussing Dedekind AFTER Cantor — stating that both were not able to solve the paradox of the square root of two. This is also discussed in Carl Boyer’s A History of Calculus as well as in several other math books. It’s very well known that Cantor did not prove the real numbers as a closed SET (or geometric series ending in the square root of two) but only as an open series.

    Bryan what you are trying to argue completely ignores the real issue — the connection with complimentary opposite harmonics. So it’s a nice attempt to repress and deny the issue — a technique of transference — but, as I’ve already pointed out, the type of semantic word play you’re engaging in is not important. Mathematics and logic both ignore the music connection, just as you’re doing and THAT’S the issue at hand. There’s dozens of books debating the true meaning of the square root of two but none of them connect the issue to complimentary opposite harmonics which solves the conundrum, as I’ve detailed.

    • I checked your reference, Drew, and I cited the section which is more than you gave me to go on.

      The definition of a transcendental number that you yourself offered is close: “A transcendental number is a number that can not be solved in an algebraic equation.” Actually it is a real or complex number that is not the solution to any non-zero single-variable algebraic equation.

      Here’s what an algebraic equation is:

      Here’s an algebraic equation for which both square roots of two are solutions:

      x^2 – 2 = 0

      No long-winded ramblings needed.

      Yes, I ignore what you call the “real issue”, and I generally recommend ignoring made-up nonsense.

  9. Were you trying to quote from Russell’s /The Principles of Mathematics/ section 267? Your wording is close, but Russell is describing a specific defect he sees in Dedekind’s theory of irrationals. Russell prefers Cantor’s, which does not have the same problem.

    And why are quoting another source about the square root of two being irrational? You claimed that it is a transcendental number. It is not, and that is not a matter of opinion. Look it up:

    I prefer ignorance to believing what is wrong. Most of Drew’s writing is yet worse, not even wrong.

  10. OK I was just reading Stephen Hawkings’ new book “And God Made the Integars” which has a fascinating background discussion on the square root of two. Hawkings emphasizes that the Vedic mathematicians did not prove incommensurability — aka the Greek Miracle — because in Indian mathematics it’s assumed that the square root geometric series are still regular divisions (halves and thirds of each other, even though it’s an infinite division in algebra).

    Whereas the Greeks REVERSED this “harmonic” right-brain visual geometry math of the Indians, stating that the algebraic series was not infinite, but rather a precise incommensurable or asymmetric geometric division called the new number, the square root of two. So the Indians assumed the geometry WAS commensurable even though they knew the number series did not converge geometrically, while the Greeks did the opposite — assume the number series DID converge as a new algebraic symbol, the square root of two or alogon, because the geometry was INCOMMENSURABLE.

    As I’ve stated before to prove the algebraic solution for the Golden Ratio series, the numbers had to be REVERSED from 5:8 as the harmonic series ratio (minor sixth) to 8:5 as an irrational fraction. This reversal was necessary to use zero so that the quadratic root gave a positive solution. Kepler, for example, was against this Golden Ratio reversal for an algebraic solution.

    The same is true for the square root of two — the harmonic division of geometry, as an infinite number series 1:2:3 (or 1 plus 1/2 = 3/2) is converted to a quadratic algebraic equation to cut off the infinite number series by REVERSING the harmonic ratios to 3:2 (arithmetic mean) so that a geometric INCOMMENSURABLE mean could be found – as the harmonic mean x arithmetic mean = geometric mean squared.

    So just as Archytas’ “doubling the cube” has a real number solution (not just the geometric proof given by Euclid) that is discovered in music harmonics 1:5/4::8/5:2 (double the volume is the cube root of two or major third, 5/4, as double the major second music interval, 9/8) so too does the square root of two have a real number solution (perfect fifth or 3/2 as the tritone convergence) that reverses the infinite number series into a “precise” number now applied to irregular geometry.

  11. Bryan your practice of “willful ignorance” is a common rhetorical device in debate but it means you accept being the loser at the onset, surely a disadvantage on your part. I recommend you practice the habit of willfully pursuing knowledge and then you won’t come across as a simpleton. Betrand Russell’s book is in the public domain, available at any decent library. You’ll find the quote in the section titled, “Limits and Irrationals.” I’ll be happy to look it up for you as the online free book is still being set up but keep in mind I’M NOT SELLING ANYTHING so if you want to continue your “mall science” attitude it might be better to join some cult like the Skeptics.

    Now then I did google “Principles of Mathematics, betrand russell, the square root of two is a convenient fiction” and found this recent discussion of the issue which again perfectly describes the same transcendental understanding of the square root of two. (see below) Finally, again, Bryan, I urge you to engage with the issue at hand — that complimentary opposite harmonics DOES solve the logical paradox of the square root of two as a number (not a symbol) which can not be solved in algebraic form. The truth has been concealed by fraud but, to quote Freud, the repressed will return, through natural resonance.

    “Saturday, September 15, 2007
    Kline on Cantor on the Square Root of 2
    Morris Kline, Mathematics: The Loss of Certainty, Oxford 1980, p. 200:

    “. . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the argument that the irrational numbers, such as the square root of 2, when expressed as decimals involved actually infinite sets because any finite decimal could only be an approximation.

    Here may be one answer to V. Vohanka’s question that got me going on this series of posts. He asked whether one could prove the existence of actually infinite sets. Note, however, that Kline’s and Vohanka’s talk of actually infinite sets is pleonastic since an infinite set cannot be anything other than actually infinite as I explained here. Pleonasm, however, is but a peccadillo.

    Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.14159. . . . And yet the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite?

    This is not an argument, of course, but a gesture in the direction of a possible argument.”

  12. Drew, the issue is not what the Babylonians knew. You stated ‘It needs to be emphasized that the square root of two is actually a “transcendental” number’. I just showed that the square root of two is not a transcendental number.

    I am curious as to the context in which Bertrand Russel called the square root of two “a convenient fiction”. Can you provide the exact citation?

  13. The below online excerpt from “Geometry for Change” aptly describes the difference between the quadratic algebraic equation for the square root of two (which does not give the actual number value) and the algebraic continued fraction solution converted to geometry by the Greeks — with it’s dependence on Archytas’ concept of geometric mean:

    “The discovery that the square is doubled (or halved) by a different principle than a line, is indicated by Pythagoras’ determination of the incommensurability between the side of a square whose area is one and the side of a square whose area is two. This relationship determines a new type of magnitude, that which like all numbers, is not susceptible to formal definition, outside the physical relationship from which it originates. In other words, the square root of 2 is not the number 1.14142135…, but a magnitude that exists only within the physical relationship of two squares whose areas are in the proportion of 1:2.

    As Plato reports in the Theatetus, this magnitude is only a special case of a whole class of magnitudes, that can be characterized as the relationship of one geometric mean between two extremes.”

  14. Bryan — the solution you refer to already existed in Babylon, China, India, etc., as an infinite approximation or series. That’s not Euclid’s proof for the square root of two — which is from Archytas. The “Greek Miracle” was to take that “divide and average” algebraic series for the square root of two and then apply it to geometry as the PRECISE SYMBOL, “the square root of two,” — no longer an approximation. It was called “alogon” but was similar to the apeiron or “negative infinity” used by Plato to introduce zero. The article I refer to argues, as is obvious, that the algebraic solution is a proof by contradiction, when applied to geometry, and as an algebraic series, the equation you use above, the solution is infinite and not geometric. The algebraic equation you go does not converge to the square root of two as a positive solution and even Cantor could not prove that the real numbers converge to a set, but only exist as a series. Therefore the PRECISE symbol for infinity, the square root of two, is logically flawed and, again, “a convenient fiction” (Betrand Russell).

    This logical issue is contemplated in math (for example in Carl Boyer’s book “A History of Calculus”) and ignored because math “works” — but the question is for whom does math work? It’s based on a rotten root and so as the math expands, as mass squared inversely proportional to energy frequency distance, so too does the ecological crisis expand.

    My point is to reveal the secret complimentary opposites foundation of where the square root of two originates — in music harmonic ratios. So just as the Golden Ratio requires REVERSING the order of the Fibonacci Series to enable a geometric quadratic solution, using zero, so too does the square root of two algebraic solution ignore the REVERSAL of the harmonic series ratios as frequency, into time as geometric distance.

  15. Just to keep things fun — here’s someone arguing exactly the OPPOSITE of my argument. haha.

    This is nice since it helps clarify my point. Again just because the 9th overtone, as an arithmetic addition is D while the 8th overtone is C, the octave, this doesn’t mean that 3/2 squared should be 9/8 as 2 is the geometric mean squared.

  16. Bryan — you have incorrectly assumed that my email is private. (I’m sure the NSA would disagree!) Just because I referred to an email doesn’t mean it’s private and I already referred you to the discussion about this email at Not only did I post my emails from math professor Joe Mazur there but someone reading them looked up his public email address (at Marlboro College) and then emailed him for a response. Professor Mazur’s response was then posted at and as I stated he confirmed his “very valuable” statement about my research.

    In fact I emailed math professor Joe Mazur about this article and asked for his response again.

  17. OK so let’s start with the harmonic series as the Pythagorean Tetrad, 1:2:3:4. Now let’s plug in Simon Stevin’s equation A:B::C:X. Clearly the harmonic series Tetrad, as it develops in Nature, violates the commutative principle. What can be understood by this is that the source for Archytas’ geometric mean, the harmonic tetrachord 6:8::9:12 REVERSES the order of the Pythagorean Tetrad harmonic series so that 3:4 comes first instead of 1:2.

  18. Hi Bryan: I’m glad you show continued interest in this most perplexing and important issue.

    A transcendental number is a number that can not be solved in an algebraic equation. The essay “Did the Greeks Discover the Irrational?” (1999) by Professors Hugly and Sayward argues that the square root of two is NOT solved as an algebraic equation — specifically that while it’s proven there’s no rational number, this is not an algebraic proof for a positive number called “the square root of two.” In other words the answer to their title “Did the Greeks Discover the Irrational?” is NO and no one else has as well. This is why Bertrand Russell called the square root of two a “convenient fiction.” Or to put it another way it’s proven, by contradiction, that there’s no ratio of phonetic letters, based on the commutative principle, for the hypotenuse of a 1-1 triangle. That is also the definition of a transcendental number — that there is no algebraic equation demonstrating the existence of the square root of two. I already gave an earlier source discussing the transcendental value of the square root of two — the book “Enquiries In Number Theory.”

    The real clincher for this complimentary opposites argument is Simon Stevin’s 17th C. conversion of Archytas’ diatonic scale into equal-tempered tuning. Stevin relied on the octave defined as a starting value of 5000 with it’s “double” as 10,000. Stevin then argues that half of the octave is the square root, or the tritone and so a third of the octave, or two major second intervals, the major third is therefore a cube root of two.

    This fully accepted modern basis for equal-tempered tuning — that which you consider to be the truth — is directly from Archytas’ proof for doubling the cube, namely that if a cube has a side one then to double the volume to two the side must be cube root of two with the proportion 1:5/4::8/5:2. That’s the exact equation Simon Stevin used — only converted to logarithmics.

    As I discussed in my article above and in my previous blogbook chapter, the subject of several emails from math professor Joe Mazur, Archytas’ source for the cube root of two is from Babylon’s use of the equation, arithmetic mean x harmonic mean = geometric mean squared.

    So again Stevin ASSUMES the value of the cube root of two without discussing the ORIGINS of the square root of two and all this time no one has questioned that fact that the arithmetic mean x harmonic mean equation Archytas relied on to create geometric mean is based on the octave, not as a doubled value, but as a SQUARED valued.

    Again having arithmetic mean 3/2 x harmonic mean 4/3 = 2 with 2 as geometric mean squared so that 3/2 x 3/2 = 9/4, the major second above the octave or the 11th interval (with 4 as the octave “squared” not doubled) and then halved to the sixth root of two as 9/8, the major second, cubed as the tritone or the square root of two is the source for Simon Stevin’s equal-tempered tuning. The cube root is then just double the sixth root — or 10/8 as 5/4.

    So Simon Stevin used the geometric mean equation A/B = C/X with X=BC/A just as Archytas used the arithmetic mean equations with A = 1 and X = 2. Only with Archytas it’s not said what “X” is — so the Babylonian geometric mean is 6:8::9:12, ostensibly the same as Simon Stevin’s A:B::C:X but this time reduced to either 2:3::2:3 or 3:4::3:4. Archytas, in solving for the square root of two excludes the START of the octave — which again is now no longer doubled, as is the case in the harmonic series, but is squared. SIMON STEVIN CONVENIENTALLY IGNORES THIS HARMONIC SOURCE FOR THE SQUARE ROOT OF TWO. So Archytas converts this process that starts with 1:2 as 6:12 so that 6 would be A and X would be 12. Instead Archytas reduces this to a 3 term equation with no “X” so that A = 1 and C = 2 and then what had been 2:3 as 6:8 is now the RESULT of the octave as arithmetic mean (A + C divided by 2) equals 3/2. What had been 3:4 as 6:9 is now the RESULT of the octave as harmonic mean (2 x AC divided by A + C), thereby HIDING the complimentary opposite harmonics of 1:2:3:4 and replacing a doubling of the octaves with a squaring of the octaves (2 is now geometric mean squared with “half” of the octave now the square root of two as the tritone).

    • The positive (as well as the negative) square root of two is a solution of:

      x^2 – 2 = 0.

      Thus the square root of two is an algebraic number and not a transcendental number.

      Drew does not know what he’s talking about.

  19. Hi Bryan: My math research was called “very valuable” by math professor Joe Mazur this summer and he urged me to submit it to the most read math journal in the world. He reconfirmed this “very valuable” opinion this fall based on discussion about my math at

    Here’s the paper you’d need to read on the square root of two:

    Philosophy (1999), 74: 169-176 Cambridge University Press
    Copyright © The Royal Institute of Philosophy 1999
    Published online by Cambridge University Press 04Apr2001

    Copy and paste this link:

    Did the Greeks Discover the Irrationals?
    Philip Hugly and Charles Sayward

    A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.

    • Drew, I disputed your claim:
      It needs to be emphasized that the square root of two is actually a “transcendental” number [...]

      Now you seem to be arguing that the square root of two is irrational. Do you not know the difference between irrational and transcendental?

      Googling “transcendental number”, the top hit is Wikipedia. Check out the article’s example of a number that is irrational but not transcendental.

      Dropping a math professor’s name or quoting private e-mail does not help your case. Has that math journal notified you of whether they will publish your paper?

  20. I too find Drew’s writing hard to understand. In places where I know about the subject, I can tell that Drew does not.

    Drew Hemple says:
    It needs to be emphasized that the square root of two is actually a “transcendental” number although this is only acknowledged in number theory and not in standard math.

    Number theory is standard math. The reason standard math does not acknowledge that the square root of two is a transcendental number is that the square root of two is not a transcendental number.

    Drew goes on:
    There is no positive proof demonstrating that the hypotenuse of the 1-1 triangle equals the algebraic symbol the square root of two since there’s no algebric equation demonstrating the square root of two, only an infinite algebraic series that is not a closed set.

    Mathematical gibberish.

    The quote from is good. For more, look up “equal temperament”.

    • Hey, Brian.

      It’s been a long time since you commented. Have you visited the forums lately?

      • My visits are less and less frequent. My main interest was the testing, announced in:

        That seems to have concluded, and I think it is fair to say that the experiments found no evidence for psi.

        • I would even say that none of the three experiments were good enough to even be able to prove the existence of psi.

          These showed me how difficult it is to come with an experiment that could detect the possible existence of psi (especially an online experiment).

          I still would like to come with a good protocol for an experiment but I don’t currently have time to develop that site. When I opened it, it was during my unpaid vacation that I took for this reason. Took me a month to build the site and the first experiment and to promote it using sponsors, Press releases etc. Cost me a lot of money as such, so I’m now moving to other paths.

  21. This difference between squaring the octaves and doubling the octaves has to do with the symmetry of zero and one and the harmonic series. For example consider this excerpt from the above link on harmonic series. At first the harmonic is described as half the wavelength OF THE FUNDAMENTAL — notice zero is not used. So the graph shows – “fundamental” to one — and then half of that wavelength creating an octave….

    “The second harmonic always has exactly half the wavelength (and twice the frequency) of the fundamental; the third harmonic always has exactly a third of the wavelength (and so three times the frequency) of the fundamental, and so on.

    Figure 3: The second harmonic has half the wavelength and twice the frequency of the first. The third harmonic has a third the wavelength and three times the frequency of the first. The fourth harmonic has a quarter the wavelength and four times the frequency of the first, and so on. Notice that the fourth harmonic is also twice the frequency of the second harmonic, and the sixth harmonic is also twice the frequency of the third harmonic.”

    THEN the same page of harmonics converts this to “mathematics” by starting the octave instead of on the FUNDAMENTAL going to one – rather on ONE going to TWO:

    “A mathematical way to say this is ‘if two notes are an octave apart, the ratio of their frequencies is two to one (2:1)’. Although the notes themselves can be any frequency, the 2:1 ratio is the same for all octaves. And all the other intervals that musicians talk about can also be described as being particular ratios of frequencies.”

    This may seem like an inconsequential difference — until we apply equal-tempered tuning using the conversion of the “mathematical” octaves — A = 1 and C = 2 into geometric mean, from arithmetic mean (A + C divided by 2) and harmonic mean (2AC divided by A + C), equaling geometric mean squared.

    OK so in my article above I already applied Archytas’ use of this Babylonian equation to solve for the square root of two and now I’ve argued that when Archytas used the “double octave” — as 4 and not 2 — to create 9/4 (the source for this is in my “secrets of the Freemasonic Greek Miracle” article, or chapter 4 of he did not mean a “doubling” of two — but rather a SQUARING of two.

    So if A = 1 and C = 4 (the double octave) then B = 5/2 is the arithmetic mean while the harmonic mean is 8/5 and their product, the geometric mean is 4, which has the square root equaling two. This confirms that Archytas did not think of the octaves as a doubling but rather a SQUARING even though this clearly goes against the harmonic series.


    Archytas SQUARED the octave when he “doubled” it from two to four, or 1/2 converted to 3/2 as arithmetic mean so that 3/2 x 3/2 = 9/4. Half of the double octave (4) would now be the square root of two by converting 9/4 into 9/8 and then cubing it as the square root of two. Instead, the harmonic series states twice the string length, as two, being half of the octave or conversingly spliting the one string length in half gives the octave. Notice how there’s no start to this string length (0) and also this arithmetic series does not line up the fifths and octaves — so it’s not symmetric. Under the harmonic series 4 times the string length is 1/4th the frequency while under Archytas, and later Newton’s inverse square law of gravity, 4 times the string length is 1/2 the frequency, or one octave higher. This is, again, because since half an octave is the square root of two, an octave higher is two squared. Again the harmonic series doubles the octave as 2:4:8 while Archtyas’ system SQUARES the octave as square root of two:2:4.

  23. OK again here’s another MUSIC example — of the difference between arithmetic and geometric series.

    The issue again is that because the arithmetic series of octaves and fifths does not line up this does not justify the use of the square root of two as a geometric containment of time, but rather the difference between the fifth/fourth and octave reveals the inherent complimentary opposite harmonics of reality.

  24. Just because the octaves double so 1:2:4:8 does not mean the octaves expand as inverse squares — because after 4 would then come 16, not 8.

    The below link uses a “constant” to promote the “inverse square” expansion of octaves as the converging series, the square root of two.

  25. Here’s a good website describing the octaves AS NOT growing in a geometric series 2:4:8 which is how the octaves are described for equal-tempered tuning — a BIG difference.

    In contrast the Harmonic Series octaves are arithmetic — always 1:2 which again does not line up with 2:3 and 3:4, thereby causing an asymmetric or complimentary opposite dynamic creating transduction of sound into ultrasound which ionizes the body’s chemicals to create electromagnetic fields and finally light that bends spacetime to return to female formless awareness.

  26. OK I just found an excellent article on the “comma” called “Musimatics and the Nun’s Fiddle” by A. L. Leigh Silver (American Math Monthly, 1971).

    He notes that King-Fang in 40 B.C., China, calculated that the 53rd fifth exceeds the 31st octave by 3.6 cents (which is a logarithmic measurement).

    Leigh Silver then states:

    “We talk of two octaves as twice the size of one, three octaves as three times the size and so on. Yet the frequencies of these intervals are in the ratio 2:4:8….Base two logarithms are therefore naturally suited for musical purposes and were published in 1670, fifty-six years after Napiers tables.”

    Leigh Silver notes that in the 19th C. Bosanquet’s “Enharmonic Harmonium” used 53 notes.

    And of course this was Danielou’s final instrument, a 53 note per octave synthesizer.

    My point is that ALL OF THIS MISSES THE POINT.

    The crucial issue is defining frequency GEOMETRICALLY as a closed system — specifically the octaves as a geometric progression.

    The Law of Pythagoras states that string length is the inverse ratio of frequency, as per the DIVERGING harmonic series, not defining octaves as a geometric series.

    Descartes changed this harmonic series into momentum as DISTANCE using Archytas’ concept of geometric mean, thereby making time a CLOSED system.

    So Descartes states if you throw something up twice as fast then it takes twice as long to return with force = momentum x velocity. This is the equivalent of stating the string length is twice as long for half of the frequency or twice the time.

    So when velocity is measured as speed by Descartes then time is converted to distance through Archytas’ use of symmetric-based means from Babylon.

    So again if A is 1 and C is 2 as the string length or inverse ratio of the octave, then the arithmetic mean is 3/2 and the harmonic mean is 4/3 and the geometric mean squared is 2.

    Or mass equals the octave NOW as geometric mean with 4 times the weight creating an acceleration as velocity squared — time as distance. This is, again, where Newton got his inverse square law.

    4 times the weight stretches the string to twice the frequency – or an octave increase, thereby changing time to a measure of weight and distance (momentum) by utilizing Archytas geometric mean conversion of frequency as COMPLIMENTARY OPPOSITES.

    What had been frequency (A = 1 and C = 2) is now reduced to distance (A + C divided by 2) with the perfect fifth now converted to a logarithmic standard.

    This is often seen in the calculation of the Pythagorean comma. The 3/2 ratio is reduced to just 3 to a power and the “2″ is converted to the octave as 2 to a power.

    Now consider an essay called “The Nature of Time” — I’ll give the source details in a bit.

    “It will be noted that Hill and I spoke of there being ‘at least some aperiodic orbits which extend to infinity’ in the classical mechanics of open systems and that we were careful not to assert that every such allowed process extending to infinity is a de facto irreversible one. Instead, we affirmed the existence of a de facto irreversibility which is not predicated on Popper’s spontaneity proviso by saying: ‘there always exists a class of allowed elementary processes’ that are thus de factor irreversible. And for my part, I conceived of this claim as constituting an extension of Popper’s recognition of the essential role of coherence in de facto irreversibilty is not conditional on Popper’s finitist requirement of spontaneity, because there processes extend to ‘infinity’ in open systems and would hence have inverses in which matter or energy would have to come from ‘infinity’ coherently so as to converge upon a point.”

    from “Frontiers of Science and Philosophy” , the “Nature of Time” is by Adolf Grunbaum. U of Pittsburgh Press, 1962.

    OK so Grunbaum states that a “coherent” system can not need to rely on spontaneity but can LOGICALLY reverse time as an infinite open system.

    That’s EXACTLY what the complimentary opposites as resonance does — uses COHERENCE to resonate as an open system to time as logical inference of consciousness — a POSITIVE proof, not needing spontaneity.

    For Western science time has been converted to mass which enables greater force as defined by distance and weight and space.

    For nonwestern complimentary opposite harmonics time can be reversed with no need of force — just coherence.

  27. Hello Drew,
    While I don’t pretend to understand all of your copious display of concepts, a few do register. In case you missed it , you’ll enjoy this video, that will match much of what you are describing here.

    StumbleVideo: The Boy With The Incredible Brain

  28. OK now we can return to my previous comment as I stated I would do. I had typed: “For Descartes a body thrown up vertically with twice the velocity takes twice the time to descend. This is force(t) = momentum x velocity. To
    convert this back to Archtyas — mass = the square
    root of momentum x velocity just as string length or geometric mean equals the square root of arithmetic
    mean times harmonic mean.

    Leibniz used work as Fs = mv(squared) divided by two — which is equating arithmetic mean with harmonic mean as an average.”

    So, like the secret of bird song, this also is a
    “positive coupling” because normally the Law of Pythagoras
    states that twice the string length = 1/2 the
    frequency. So the longer the string the lower the
    frequency, as an inverse ratio.

    The secret of Archytas was that geometric mean changes
    the harmonic mean to arithmetic by using WEIGHT for
    geometric mean and in classical physics this means
    momentum and velocity HIDE the complimentary opposites
    because weight is the average of the extremes. So
    twice the velocity is Archytas’ “twice the string
    length” or 3/2 (Arithmetic Mean) x 4/3 (Harmonic Mean)
    = 2 which also means 1/2 the frequency.

    But since it’s also geometric mean squared the same
    new “positive coupling” creates an increase in
    string length as momentum or amplitude and an
    increase in frequency NOW AS TENSION from weight.
    This is where Newton got his “inverse square law”
    with 4 times the weight (or pressure) stretching
    the string length to twice the frequency, again
    in contrast to the law of Pythagoras stating that
    twice the string length would be half the frequency.

    What Einstein did was just consider that at the speed of light the change in velocity (c) now just converges
    at its own maximum (so it’s c squared) while the
    weight or momentum changes as mass.

    Again Einstein is relying on mass which was originally weight
    as frequency turned into amplitude as string length –
    the concepts combined as geometric mean to create mass
    while the concept of complimentary opposites was lost.

    But the “positive coupling” of natural resonance whereby
    subharmonics of frequency create a significant increase
    in amplitude, as biology professor Brian Goodwin
    details in his “Temporal Order of Cells” book, indicates
    not the use of weight as mass, with geometric mean,
    but instead the change of wave-form with nonlinear
    feedback that actually diverges to consciousness
    or what some biologists call “sound-pictures.” This
    positive coupling, instead of being based on weight,
    is simply using the complimentary opposite harmonics
    and is modeled in science as the Time-Frequency
    Uncertainty Principle.

  29. OK this computer is going to shut down and just locked up erasing my message.

    “How Birds Sing” by Crawford H. Greenewalt, Scientific American, Nov., 1969 has the secret of “positive coupling” so that an increase in membrane tension increases frequency but also retracts the membrane from the air tube thereby increasing amplitude.

    A normal resonant chamber has an exponential decrease in amplitude as harmonics increase exponentially with increase in tension or string length inverse to frequency.

    The secret again is waveform with it being a pyramid, tetrahedron, spirochete, vortex or full-lotus body position, the must be measured from logical inference complimentary opposites.

  30. Analogy is a visual concept — geometric mean is literally an analog or analogy. Alchemy works by what Buddhism teaches as the “inner ear method” just as science recognizes the inner ear can not be modeled with a deterministic equation due to the “time-frequency uncertainty principle.” While the technique of harmonics as mind-body transformation is rather primitive, the foundation of the process is beyond time, and therefore the “highest technology of all technologies,” as qigong master Yan Xin states.

    Qigong master Yan Xin describes the alchemical process as “synchronous resonance” but even quantum chaos relies on a deterministic equation in a closed trajectory.

    An excellent book on paranormal mind-body transformation in the context of modern society is “Qigong Fever: Body, Science and Utopia in China” by Dr. David A. Palmer (Columbia U Press, 2007).

    When we start relying on listening based on inferring the source of sound as consciousness then a physiological transformation takes place. This is facilitated by the discovery that the 12 notes of the music scale are mirrored on the outside of the body as alchemical pressure points. You can get a practice tape or c.d. from (level 1 sitting meditation, small universe cassette is only $3 at clearance while an hour long small universe c.d. is also available). Another excellent source on spring forest qigong is

    All of western science is based on “proof by contradiction” logic with a symmetric-based negative infinity (zero) using technology to replicate the results. Synchronous resonance creates blissful-heat and electromagnetic energy and light-information healing solely due to listening to the source of sound, consciousness, as modeled through complimentary opposite harmonics.

  31. Thanks for the reply. Often find it difficult to say in words, that which one is endeavering to convey. I need some kind of analogy to hang my cap on,in order to give me a comprehension starting point.
    Would a complex square wave, with multiple front and rear porches, be a crude simile to what you are expressing! How about the highly unlikely sine wave impressed with numerous complex square waves!

  32. I appreciate you mentioning heterodyne as a beat phenomenon though (although again it relies on symmetric-based resonance which is not alchemy). The below link to the 250 year old man in China gives further insights into this alchemy practice.

  33. Your close — but “heterodyne” is still a Western concept just as “frequency” is Western — measured by symmetric-based or one-to-one correspondence with algebra and geometry.

    This is more like wave-form — the sine-wave — creating resonance with the formless, as consciousness — beyond spacetime and most importantly the resonance is COMPLIMENTARY OPPOSITES or inherently gendered. The consciousness is FEMALE.

    So there’s no “objective” rational science — it’s pure music resonating throughout the whole energy spectrum to something that is beyond power, beyond matter yet can be logically inferred as the source of the I-thought.

    The practice is specifically called the “small universe” from a book called: TAOIST YOGA: ALCHEMY AND IMMORTALITY trans. by Charles Luk.

    The above article just focuses on how the complimentary opposites were covered up and therefore why Westerners are always using Western terms to approach this topic.

    Qigong master Chunyi Lin is an excellent teacher who transmits this free energy created from female formless awareness. You can listen to and watch his videos for free at — the “In the News” section. He works with the Mayo Clinic in treating people for all types of disease.

  34. Seems to me that your describing the heterodyne principle carried to the macro and micro extremes, that occur within the brain,perhaps using the earths frequency as the master oscillator, controlled by that frequency signal received from that unknown outer space source! Am I close or just beyond comprehending any of the write. Still deep.
    Merry Christmas to you.

  35. OK I just had a long conversation with my relative
    who is an engineer — about this research of mine.

    He described how in his work there are amazing
    examples of huge energy systems that have a “natural frequency” resonance which must be avoided since it
    will cause the system to self-destruct. He mentioned
    one engineer who specialized in harmonics and even
    used a tuning fork to fix a system that had always
    broken down every 3 years. Another system he
    described was designed to work by acoustics but
    never became commercially feasible.

    This type of analysis misses the point of what I’m
    getting at — the West defines frequency as a
    materialistic phenomenon, the vibration of an
    object, using symmetric-based measurements (divide
    and average, and doubling of harmonics for geometric
    mean). But if we are honest about frequency we need
    to recognize that the overtones are not based on
    symmetry but rather complimentary opposites. That
    in fact the West has a psychological imbalance —
    with a patriarchal use of technology, through
    left-brain and right hand dominance.

    It would appear that what I’m advocating is
    right-brain dominant — through nonwestern music, but in fact the logical inference of harmonics relies on left-brain analysis. Only the results of the logical inference
    are not repeatable because the results are beyond time itself!

    Pure consciousness is also not something “neutral”
    but in fact is female since it’s listened to as the complimnetary opposite source of the one or the

    So the West may seem practical but psychologically
    it’s cut off from this female formless awareness
    foundation and the trajectory of the West is
    increasing tension and materialistic destruction.
    This is not to say that technology is not part of
    Nature, only that there is also something beyond
    technology which is the foundation of Nature: pure consciousness.

    To resonate with consciousness is to go the
    opposite direction of Western technology and it’s an experiential practice of physiology and light synchronization and bending of spacetime.

    While this path is unrealistic in a modern environment since it involves a deeper transformation of psychology,
    the process of seeming results inherently relies on a foundation that is more solid than the structure of

  36. Thank you. Now I’ll get back to what I was working on before I posted here. Lets see one times three equals…… All kidding aside, Great read. But is still deep material.

  37. There’s a clarification that may help in understanding just how fundamental the above research is to classical science.

    Essentially Newton proved Kepler’s third law — that square of velocity = cube of mean distance so that weight is proportional to gravity. Kepler used 5/4 as the orbit ratios of Saturn and Jupiter which he then converted to the cube root of two — directly from Archytas’ proof for doubling the cube, as described above.

    The insight, again, is that the frequency arithmetic mean, as 3/2, is squared to 9/4 and then halved to fit back into the octave as 9/8 and then doubled to 10/8 or 5/4 as the cube root of two.

    The problem with this squaring and halving process is that it converts the complimentary opposite frequency 2/3 into 3/4 (which violates the commutative principle) into equating the arithmetic mean, 3/2 with the harmonic mean, 4/3, through the geometric mean. In my article above I described this equivalence error for gravity through the iteration of the square root of two series whereby 3/2 = square root of two.

    Another way to approach this is to analyze the difference between Descartes development of frequency as momentum and Leibniz development of work.

    For Descartes a body thrown up vertically with twice the velocity takes twice the time to descend. This is force(t) = momentum x velocity. To convert this back to Archtyas — mass = the square root of momentum x velocity just as string length or geometric mean equals the square root of arithmetic mean times harmonic mean.

    Leibniz used work as Fs = mv(squared) divided by two — which is equating arithmetic mean with harmonic mean as an average.

    Have to go to xmas dinner. So more later.

  38. Well presented information. Yes it is deep thought material and to many (I’m included)it will take a few reads to comprehend that depth.