Continuum is external to the knowledge of reality: On the difference between the Pythagorean Comma and Philolaus irrational geometric magnitudes

So we have established that the "legend" of Pythagoras using anvil hammers was based on the same ratios that Philolaus used as irrational magnitude. Yet strangely some prominent academics have not discussed Philolaus using irrational magnitude harmonics. Boethius made a point to reject the irrational magnitude harmonics of Philolaus and Burkert did not understand them either. And it was Paul Tannery who realized this secret and then math professor Luigi Borzacchini picked up were Paul Tannery left off - realizing that indeed music theory had been the secret origin of irrational magnitude math as the Greek Miracle.

Borzacchini called this cover up "really astonishing" and "shocking" and the source of a "deep pre-established disharmony" that guides science - as a "negative judgement paradox."

And yet even Borzacchini had not zeroed in on the precise Boethius ratios discussed from Philolaus. Or maybe he had? I remember him now discussing whether the Philolaus text was genuine from Boethius and it is the text giving these semitone irrational magnitude ratios that was questioned as possible spurious. Borzacchini thought the text was genuine. That's how I remember it.

But before making that review - let's consider the fact that Plato's Timaeus which is supposedly Pythagorean is actually also based on irrational magnitude from Philolaus - and so is not based on the Pythagorean comma but actually the irrational magnitude difference derived from 9/8.

So Andrew Barker argues the fragment of Boethius is genuine and represents

"a new, though inevitably hypothetical perspective on Philolaus' bizarre approach to the quantification of intervals,....it represents an attempt to fuse two quite different ways of measuring intervals, both of which were current among theorists, but still in their infancy in Philolaus' time

The strange thing is Andrew Barker does not seem to realize the implications of what he has discovered. This is the origin of irrational magnitude from ratios of musical intervals as logistics!! It's not "bizarre" - its the Greek Miracle of the continuum as Professor Borzacchini has emphasized.

So then the pdf, "Approximating square roots in antiquity" notes that Andrew Barker then cites Theon explaining that Plato approximated the musical semitone, the square root of 9/8 as 17/16.

 And citing Boethius quoting Archytas, that the comma is NOT the Pythagorean Comma but rather the comma as 3 to the 12th/2 to the 19th which is exactly the same ratio that Philolaus derived.

I thought Carl Huffman's book was out before Andrew Barker's book on Philolaus. OOPS. So Barker's book was published the SAME year that Borzacchini's article was published. Amazing. Huffman's book was 1993 and Barker's book was 2007.

O.K. so no wonder Zhmud was not aware of the irrational magnitude use by Philolaus.

O.K. so it is Plato's book directly - Timaeus - that gives the WRONG ratio of the Pythagorean Comma! Too funny. So of course Plato got it from Philolaus and Archytas!! google

How could anyone dispute the authenticity of that Boethius fragment now? Crazy.

Here a 2014 MIT Press book by Frances Dyson promotes the fake Platonic Pythagorean Comma lie as the secret to the success of Western civilization!!

In other words the exact opposite of what Michael E. Hudson and I have been writing on. Hilarious. Leave it to MIT Press to crank out such propaganda.

The Tone of Our Times: Sound, Sense, Economy, and Ecology

By Frances Dyson

So I quoted before - how Carl Huffman was forced to react to Peter Kingsley completely debunking Huffman's Philolaus book

So Kingsley is corroborating the connection between Pythagoras and Zeus.

The Fifth Hammer: Pythagoras and the Disharmony of the World

https://books.google.com/books?isbn=193540816X

Daniel Heller-Roazen - 2011 - ‎No preview

From music to metaphysics, aesthetics to astronomy, and from Plato and Boethius to Kepler, Leibniz, and Kant, this book explores the ways in which the ordering of the sensible world has continued to suggest a reality that no notes or ...

So this dude makes the SAME error - he calls the comma of irrational magnitude from Philolaus, the "Pythagorean Comma"!!! When it is not!!!

Is he some kind of scholar?

Daniel Heller-Roazen | Comparative Literature

https://complit.princeton.edu/people/daniel-heller-roazen

Daniel Heller-Roazen. Arthur W. Marks '19 Professor of Comparative Literature and the Council of the Humanities. Professor of Comparative Literature. Phone: 609-258-2878. Email Address: dheller@princeton.edu. Office Location: 103 East Pyne. Office Hours Spring 2018: By Appointment Only (Spring Term Teaching ..

Oh he's just Comp-Lit at Princeton. No wonder - what a fake.

Dear Professor Heller:

  the Pythagorean comma  = 531,441/524,288 in your book, p. 36 The Fifth Hammer

Now compare his claim with a music professor!!

 results in the number 129.74632.
This ratio, 129.75632 : 128 is the Pythagorean comma. 12 perfect fifths do not equal up to 7 perfect octaves.

So Professor Heller - you are relying on Philolaus who used irrational magnitude which is not real Pythagorean harmonics. Busted!
https://archive.org/details/springer_10.1007-978-3-0348-0672-5

Sir James Jeans, Science and Music, archived.

"The value is 129.75 as 12 steps of the fifth while the frequency of the last C as the octave is "128 times that of our starting-point, so that our twelve steps slightly overshoot the mark."

SAME as the music professor but NOT the Philolaus irrational geometric magnitude.

Thanks for trying, hope Princeton is happy. haha.

Let me know any comments, criticisms or questions,

drew hempel

http://ecoechoinvasives.blogspot.com for my research  - ALL FREE!!

O.K. so now rereading Borzacchini - I see that it is his lack of knowing music theory that does not enable him to see the conundrum of Philolaus to produce the music intervals from 6:8:9:12.

The arithmetic mean in one octave 12:6 can be easily recognized in the fourth 12:9 or 8:6, in fact 12 - 9 = 9 - 6 (such as C-F). The harmonic mean in one octave 12:6 can be easily recognized in the fifth 12:8 or 9:6, in fact 12-8 : 12 = 8-6 : 6 (such as C-G). 

Notice how he just throws in the intervals with no recognition that they are noncommutative to each other!! In other words 0 to 9 is 3/4 and so can not be C to F since it would not allow the geometric magnitude to be created as the octave. He does state though the geometric mean is not rational since it would mean (6 x the square root of 2) on each side relative to the octave (meaning 12 notes in total). And he notes Aristotle rejected the geometric mean for this reason since it went against music harmonics.

Plutarchus (1976, de an. procr. in Tim., c17, 1020E) reminds us that the crucial problem was the division of the tone (9/8, i.e. the interval between the fourth and the fifth) in two 'equal', i.e. 'proportional', parts, and that the Pythagoreans discovered it to be impossible because 9/8 was superparticular.

Boethius’ text that I am going to analyze shows that the early musical theory of incommensurability was somehow known, even though it had been overcome by a sudden rupture.

It is easy to see that 17 is not a middle proportional between 16 and 18. In III,5 and III,8 Boethius describes Philolaus' attempt to cut the tone. Philolaus set out to solve the problem starting from purely 'numeric' considerations by displaying an idea of 'ratio' as a generic relation between two numbers and by analyzing the music intervals as ratios between two numbers or as differences between the same numbers or as single numbers. There we can find the statement of the problem in terms of dichotomy of the tone, of the comma and of the diesis (Diels and Kranz 1964, 44 A26, B6). 

Some authors (Burkert 1972, Huffman 1993) consider the naivety of this approach as a sign of Boethius' unreliability. Instead, I would like to point out how well it fits with Szabo's analysis of the ancient meaning and employment of words like di£sthma,  lÒgoj and ¢nalog…a (1978, 114-170), and how closely Philolaus' words agree with Plato's numerology in Timeus 35-36 (Plato 1964).

According to Burkert (1972, 399) “the material introduced by Boethius must have belonged to Pythagorean musicology before Archytas”. The fact that Boethius neither mentions incommensurability nor connects these results to arguments other than the defense of Pythagoreans’ music theory makes fake reconstructions or forgeries unlikely. The whole book is only about the Pythagorean theory, mostly against Aristoxenus, and is oriented toward the main results involving the 'cutting the superparticularis ratio'. I think it reports quite exactly the ancient Pythagorean mathematical theory of music since its beginning, and its conclusion is the negative face of the incommensurability, with a proof of superparticular ratios’ indivisibility whose original rigor, however, can not be ascertained.

The trouble with 17 [in Plato] could be connected not to some geometrical construction, but to the role this number played in the Pythagorean arithmetic, where it was called the 'obstacle', because it "broke the proportion of 9/8 in not equal intervals" (Plutarchus 1962, De Iside et Osiride, 367 f).

Why did Theaetetus mention 3-foot and 5-foot among the ‘powers not commensurable in lengths with the 1-foot’, but ignored 2-foot, the most important one? (Burnyeat 1976). If my hypoythesis is not wrong the answer is easy: 3 and 5 are the natural candidates as arithmetic means for the dichotomy of the octave (1:2) and of the fifth (2:3). In other words 3, 5 and 17 are the most natural values for cutting the most important musical ratios, whereas 2 plays no role in this musical problem.

An analogous 'perception' appears in Nichomachus’ (1866) Introductionis Arithmeticae I, VII, when he defines 'even' numbers by requiring that their splitting could be "greatest in size and smallest in quantity". From the context we realize that the 'linear number' was therefore seen as a sort of  'magnitude', whose division in parts produced a 'multitude' of '(integer) magnitudes', so that the even number could be divided in the smallest quantity ('two') of greatest (integer) equal magnitudes (the 'half').

These remarks raise the question of the difference between the ancient Pythagorean ‘musical’ perception as displayed in the Pythagorean idea of ‘linear number’ in Boethius or in Nicomachus, and the modern ‘geometrical’ perception of the linear numerical magnitudes. 

I have already pointed out that in modern perception the “number” is a “point” on the geometrical continuum. However, there is no trace of this perception among the Greek mathematicians. Euclid stated in both the Elements and the Sectio Canonis that a magnitude can be denoted by a single letter (if the magnitude is to be considered as a whole), or by two letters, its extremes (if the magnitude has been or is going to be divided), and thus the point can be only an extreme. Actually even in Archytas  (Diels and Kranz 1964, 47 A14) we find the same style, even though this could be a later restyling (by Eutocius?).

According to Barker (1989, II: 7-8), in all Pythagorean harmonics “notes are treated as entities one of whose attributes, that of pitch, varies quantitatively and can be expressed in numbers. Intervals between notes are to be expressed as ratios of numbers. Notes, then, are items possessing magnitudes of some sort. They are not points on a line”.

The discrete description of reality is grounded on such units, and science requires ‘minimums’ (like the letters of the alphabet and the quarter of tone in music), not connected to physical constraints of the organs of sense. Continuum is not only inexpressible, but also external to the knowledge of reality.

According to Aristoxenus, the infinite sounds are never ‘actual’, and the discrete perception of  sounds can be compared with the finite number of letters in the alphabet. Hence, it concerns immediately the basic structure of science.

Boethius Pythagorean Tetrad as the Tai Chi, (failed as geometric mean)