https://www.hrstud.unizg.hr/_download/repository/McKirahan,_Philosophy_Before_Socrates,_2nd_ed.pdf
So notice - no mention of the continuum as based on logarithms!!
So we know that the above visual definition does not apply when Philolaus developed the Subcontrary Mean that Archytas called the Harmonic Mean - because the 3/4x for F has a different value of "x" - as 0 to 1 (the root tonic wavelength or C). So it is the ratio of 0 to 8 so that 6/8 is 3/4x and the MAGNITUDE is not 4/3x of 6 to 12 (the 1/2x of C at 0 to 12 is NOT the 1/2 of 0 to 8).
The term subcontrary may refer to the fact that a tone based on this mean reverses the order of the two fundamental musical intervals in a scale. It is believed that Archytas or one of his contemporaries gave the name "harmonic"
to the subcontrary mean because....[source: Infinite Series cover up of the noncommutative "bait and switch"].
So if C is 1 and the octave is C as 2 what Plato and Archytas covered up is that 2/3 is C to F as the Perfect Fifth, at the same time that 3 is G as 3/2, Perfect Fifth.
So for Archytas the logarithmic equation is the arithmetic mean x harmonic mean = geometric mean squared, so that 2/3 could not be allowed since 3/2 x 4/3 = 2. But 4/3 as the Perfect Fourth is the "double" of 2/3 as C to F - yet 3 as the denominator is not part of the harmonic series with 1 as the root tonic.
“The Minor Sixth (8:5) in Early Greek Harmonic Science,” by Alan C. Bowen, The American Journal of Philology, 1978:
Any who doubt that the musical ratios are all of greater inequality, i.e., that the antecedent or first term in each is greater than the consequent or second term, should consult Archytas DK 47 B 2. This Fragment…requires that the ratios be of this form if the assertions about the three means [arithmetic, harmonic and geometric] are to be true. Accordingly, the ratios assigned to the octave, fifth, fourth and minor sixth, must be 2:1, 3:2, 4:3 and 8:5, and not 1:2, 2:3, 3:4 and 5:8, respectively, as Mosshammer and others would have them….Indeed, there is early proof deriving from the Pythagorean school that intervals, such as the fifths, which are represented by superparticular ratios cannot be partitioned into any number of equal subintervals because the terms of these ratios admit no number of geometric means….Consider now the question of the status of the ratio (8:5) in the Pythagorean harmonic science that dates from the late fifth century B.C. to the time of Apollodorus. One should not expect that this ratio was recognized as melodic by every school of Pythagorean musical theory. For example those who sought to derive all the musical ratios from the Tetrad of the decad by compounding and dividing the ratios of the primary and most familiar intervals, the concords of the octave, fifth and fourth, would find the minor sixth unascertainable….There is reason to believe that these were supplied by Archytas in the early fourth century B.C.
In music theory this is called the Phantom Tonic.
Nicolas Slonimsky once pointed out, in an effort to dissuade readers from the idea that Western tonality is the inevitable result of how we hear (as opposed to a largely artificial invention), that no matter how high one goes in the harmonic series, a fundamental pitch will not produce a perfect fourth above the fundamental.
Therefore 8/6 plus 12/8 = the octave as 12/6 as the first logarithm developed by changing the value of x to be an arbitrary value with the interval defined as a dynamic ratio between the arbitrary x as zero - negative infinity - and the actual geometric ratio.
To quote an expert:
“Orthodox Pythagorean theory recognizes five consonances: fourth, fifth, octave, twelfth, and double octave; and these are represented by the multiple and superparticular ratios [n + 1 : n] from the tetrad. The number 8 obviously does not belong to the tetrad.”
André Barbera, "The Consonant Eleventh and the Expansion of the Musical Tetractys: A Study of Ancient Pythagoreanism," Journal of Music Theory, 1984.
"Why these silences? And why this sudden and radical change?" (hiding the secret musical origins of western science!). "Why this sharp change? I think the first reason was that the musical proof was only negative, whereas the geometrical approach allowed the effective construction of incommensurable magnitudes."Math professor Luigi Borzacchini.
BEING AND SIGN I: Syntactic paradigm and negative judgement paradox. Luigi Borzacchini, Department of Mathematics, University of Bari, Italy. Knowledge representation and formal thinking have 'ancient roots', deeply embedded in our modern culture. This core can be revealed in the classic Greek philosophy and ...
I'll give you Borzacchini's research - it is free online - http://www.academia.edu/
So then the definition of infinity as formless Apeiron was changed into a materialistic ratio instead of an infinite non-material logical inference.
"Magnitudes have all the characteristics Plato attributes to Apeiron."
p. 394
Socratic, Platonic and Aristotelian Studies: Essays in Honor of Gerasimos Santas 2011
And so then you have Proclus directly stating if there were no Apeiron there would be no irrational magnitudes. From this googlebook review link
citing Vassliis Karasmanis, "Continunity and Irrationality in Ancient Greek Philosophy and Mathematics."
"...at the time of Archytas and Plato a sharp rupture occurred that fostered a shift from musical to geometrical incommensurability, so that we can find nothing about the geometric approach in the extant Pythagorean fragments as well as nothing about the musical approach in Plato or Aristotle."
Math professor Luigi Borzacchini
It was when the Attic alphabet arose that Number aligned with a Form as the phonic symbol enabled Number to no longer be inherently Cardinal but instead be just a "heap" of "substance" , as both integers and geometric units combined.
John J. Cleary, "Aristotle's Criticism of Plato's Theory of Form Numbers," in
Platon und Aristoteles, sub ratione veritatis: Festschrift für Wolfgang Wieland zum 70. Geburtstag
Vandenhoeck & Ruprecht, 2003
Colloquium 7: Philolaus on number
· January 2012 with 10 Reads
So we can prove this by reviewing McKirahan again.
https://sci-hub.tw/10.1163/22134417-90000137
and
So McKirahan argues that the Pythagoreans were wrong while Philolaus was correct to define music as based on the geometric ratio as "X" - an arbitrary number that has the same geometric magnitude. So instead of the root tonic being a certain pitch that is listened to - the home note - there no longer is a root tonic! There is simply this geometric ratio that does not refer to a physical size but instead an arbitrary number (in this case it is both 0 to 8 and 0 to 12 as the root tonic "x" so that 1/2x is both 4/8 and 6/12 - with neither number as physical size being consequential).
So Philolaus clearly, as McKirahan explains, is FLIPPING the Lyre around - in order to be a LIAR about the root tonic of the music instrument. As McKirahan stated:
So the fact that the pitch changes is then supposed to not matter because the magnitude ratio stays the same from arithmetic (even though the geometric distance as length changes).
So even though the pitches are different - the Ratio of the pitches then states the same:
So in other words C to F is a grasp and F to C' (octave) is a grasp therefore C' (octave) to G is a grasp and G to C is a grasp. Therefore C (octave) to G as a Perfect Fourth is the same as C to F as a Perfect Fourth.... (but the Pythagoreans would state that the relation of F or G to C as the root tonic are NOT the same, rather they are complementary opposites - this is also emphasized in Chinese and Indian tuning google book link as well).
The thesis can be expressed the following way: If two drones, either a fourth or fifth apart are sounded, one of these will 'naturally' sound like the primary drone. It is not always the lower of the two which will sound primary, but the one which initiates the overtone series to which the other note (or one of its octaves) belongs. for more details
So then
Correct - if we ignore what the root tonic value is then we can pretend that the different fraction values are the same (and therefore we can not use the 2/3 or 3/4 fraction values as the pitches - since they distinguish what the root tonic value is as absolute pitch).
So now the "bait and switch" is then to take the 0 to 8 as a new "root tonic" or absolute pitch of the home tone key.
We want to take 3/4 of that.
Yes we "want" to but it's a LIE - it's a different "that."
Essentially you confuse the ratio of the 3 to the 12th divided by 2 to the 19th with the Pythagorean Comma ratio between 7 octaves and 12 perfect fifths.
But then we get to Philolaus and Burkert gives his analysis from Boethius on Philolaus with the final ratio as 531441:524288. But as Andrew Barker points out - this is not a frequency/wavelength ratio but a geometric magnitude ratio (and Burkert did not realize this!).
The difference is 531441:
Wrong:
Thus, we must go up in perfect fifths 12 times. This would be 3/2 raised to the 12 power (1.5 to the 12th). This 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:
(3/2)^12 ≠ (2/1)^7 or you could say (3/2)^12 / (2/1)^7 ≠ 1.As Gareth Loy says in his great book Musicathics: "Contrary to the wishes of scale builders and musicians from antiquity to the present, the powers of the integer ratios 3/2 and 2/1 do not form a closed system."
The Science of Harmonics in Classical Greece
https://books.google.com/books?isbn=1139468626
Andrew Barker - 2007 - Philosophy
It is again obvious that Philolaus is not thinking in terms of ratios alone. The ratio of the komma can be computed;it is 531441:524288, but this–in Burkert's phrase– is 'pure frivolity'.21 Boethius has already told us, in fact (Inst. mus. 3.5), that Philolaus identified the komma [531441 : 524288 aka the Ditonic Comma] with the unit, 1, as being the difference between a diesis ...
No comments:
Post a Comment