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: http://hypertextbook.com/physics/waves/music/

“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.

We believe it

we could publish also a book

or to

add some emails here

Many Thanks for hosting my comments

Your Friend

TONI

I like also the WSEAS Conferences in the University of Harvard

The WSEAS Students opened a very nice Blog for Environment, but your blog is much better

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.

http://64.233.167.104/search?q=cache:jQEP3bAflG8J:www.midicode.com/tunings/Tuning10102004.pdf+9/8+cubed,+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 http://mothershiplanding.blogspot.com

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.

http://www.chrysalis-foundation.org/origins_of_length_ratios.htm

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

http://www.ex-tempore.org/means/means.htm

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”

http://books.google.com/books?id=2ietQ7tX8TEC&pg=PA174&lpg=PA174&dq=archytas+harmonic+mean&source=web&ots=0d2zwuHO_E&sig=dQZTj1jnN7BJGId1e0OiQJmMfAo#PPA159,M1

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.

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.

http://hanskayser.com/EZ/kayser2/kayser2/index.php

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.

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).

For the basic math I’d suggest more thorough and comprehensible sources. Once again, Wikipedia rocks.

Most people are familiar with “average”, which math-geeks call the “mean” or “arithmetic mean”. There are other ways to get a central value among several, such as the geometric mean and harmonic mean.

http://en.wikipedia.org/wiki/Arithmetic_mean

http://en.wikipedia.org/wiki/Geometric_mean

http://en.wikipedia.org/wiki/Harmonic_mean

Music is but one application of the mathematical results; the math would work the same were we all deaf. Contrary to Drew’s assertions, math is not based on music; quite the reverse. For the musical analysis, I suggest the articles:

http://en.wikipedia.org/wiki/Overtone

http://en.wikipedia.org/wiki/Equal_temperment

http://en.wikipedia.org/wiki/Piano_tuning

Drew Hemple wrote:

“So this is the equivalent of stating that 4/3 and 3/2 ARE the square root of two […]”

Since those are not the square roots of two, we can mark that stuff wrong and move on. Just to sure, let’s check. The square root of two squared is two, while:

(4/3)^2 = 16/9 = 1.77778

(3/2)^2 = 9/4 = 2.5

Hmmm… not close, but not all that far from 2 either. I cannot tell what Drew was on about.

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

theory).

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

COLLECTION OF

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.

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.

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?

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:

http://en.wikipedia.org/wiki/Algebraic_equation

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.

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:

http://en.wikipedia.org/wiki/Transcendental_number

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

http://en.wikipedia.org/wiki/Not_even_wrong