Thank you very much
Professor Purves for considering this research. Below are some notes
that I just cut and paste from my blogposts. I hope you can piece
together the information. Please let me know if you have any comments or
questions, etc. take care, drew hempel
From your collaborator Daniel Bowling:
The most frequently used intervals—the octave, perfect 5th and 4th—correspond to those considered the most consonant by culturally diverse listeners (Bowling & Purves, 2015; Burns, 1999), an observation that is all the more impressive given that the frequency resolution of the human auditory system is sufficient to distinguish hundreds of unique intervals (Parncutt, 1989; Roederer, 2008).
The Nature and Nurture of Musical Consonance (PDF Download Available).
NOW: On the Perfect Fourth as the Phantom Tonic:
This is known by music theorists but NOT by scientists! Unless you study noncommutative phase logic.
Fields
Medal math professor Alain Connes on the noncommutative phase logic of
music theory (the Perfect Fifth is 2/3 as C to F subharmonic while 3/2
as C to G overtone harmonic so F=3=G at the same time! This was covered
up to create Western math as the Greek Miracle (using logarithms). The Harmonic Series already assumes the logarithmic math as ratios! That actual natural resonance of the modal scale is noncommutative.
Notes on the noncommutative phase origin that was covered up:
Sir James Jeans, the
quantum physicist, author of Science and Music he describes the Pythagorean Comma as the difference between 3 to the 12th and 2 to the 7th.
“To put the same thing in another way, we have just identified the frequency ratio 1.5 with the interval of a [perfect] fifth, although our table gave the value as 1.4983. The difference is only small – 1.13 parts in a thousand – but by the time we have taken the twelve steps needed to pass completely around the clock-face, it has been multiplied twelvefold into the difference of 13.6 parts in a thousand, which represents the aforesaid difference in pitch of almost a quarter of a semitone. When this is allowed for, the true clock-face is that shown in fig. 55; it extends to infinity in both directions and all simplicity has disappeared.” Sir James Jeans book Science and Music, (Dover Publications, 1968), p. 166- so that above definition of the Pythagorean Comma is not how it is defined based on the logarithmic math as per the "Pythagorean Comma" (as in the Wikipedia page).
1.05946309436
So that is the value of the 12th root of 2 for equal-tempered tuning.
So then you take that to the 7th as 7 half steps from the tonic and you get = 1.49830707688
And so that appears to be close enough to 1.5 for 3/2 as the Pythagorean harmonics.
But as Sir James Jeans points out - this "error" multiples.
So 2 to the 7th is 128 and 3 to the 12th is 531441
And
so the assumption is that the logarithms can be added so you get 2 to
the 19th=524288
And so the Pythagorean Comma is 531441/524288=1.01364326477 This is an
ERROR that covers up the noncommutative phase truth of the "Phantom
Tonic."
Whereas if you take 3/2 to the 12th instead of assuming that the octave can be divided out of the 3 though "halving" -
then you get 129.746337891/128= 1.013671875
So you have 8 to the 6th power and 9 to the 6th power (531441) based on the rule of doubling/halving for octaves but the ONLY way that the Perfect Fifth - the Perfect Fourth = 9/8 and the Perfect Fifth + the Perfect Fourth = the Octave is to define the Perfect Fifth and Perfect Fourth not as the fractions 3/2 or 2/3 but as the logarithms of 3/2 squared being 9/4 with the Octave 2 defined as the Geometric Mean Squared. This is the subtle difference between multiplying as doubling and squaring as geometric mean.
So - from Professor Richard McKirahan we learn how Philolaus had to flip his Lyre around so that 3/4 as the Perfect Fourth is the wavelength of 0 to 8 root tonic as 1 while 4/3 is the geometric magnitude of 8/6 as the root tonic of 12 to 0 in the opposite direction as 1. So the question is - is 1 a square inherently as geometry? or is 1 a number that can not be seen but rather listened to.
For the Harmonic Series to derive the standard Pythagorean Comma we:
(start counting at zero):Oh yeah? REALLY? Now the Harmonic Series in math starts at 1 and diverges but still defines frequency as square root of wavelength divided by PI (assuming commutative phase time at zero). This covers up noncommutative time-frequency energy.
So for Hertz the frequency is actually defined as the square root of the wavelength divided by PI - it is already assumed to have the symmetric logarithmic math.
Professor Richard McKirahan reveals the secret:Or as math Professor Luigi Borzacchini states:
The word translated epogdoic is not a musical term but a mathematical one. An epogdoic ratio is the ratio of 9 to 8. The occurrence of a mathematical term here [in Philolaus]is unexpected. It has been treated as an unimportant anomaly but in fact it is the key to the entire fragment....The word magnitude normally refers to physical size, but here it is given a new application, extending the notion of magnitude to include musical intervals.
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 [Philolaus] or in Nicomachus, and the modern ‘geometrical’ perception of the linear numerical magnitudes.
note that all is geometric magnitude so that 12:8 (3/2) plus 8:6 (4/3) = 2/1 as geometric magnitude from the double octave. This has to use 0 to 8 as one "root tonic" for 6/8 wavelength as 4/3 frequency and then 0 to 12 as the other root tonic for 12/8 frequency as 3/2 with 8/12 wavelength. This is called the "phantom tonic" in music theory since the Perfect Fourth can not be created from the harmonic series as 3 does not go into 2 from 1 as the root tonic denominator. In other words the root tonic changes due to complementary opposites.
And so the use of "zero" is to create the geometric mean as logarithm by covering up the noncommutative phase. So at "zero" energy there is STILL a noncommutative phase quantum energy. Professor Richard McKirahan:
So you have 8 to the 6th power and 9 to the 6th power (531441) so that the 9 is equated to 3/2 squared as 9/4 halved to 9/8 with the 9 being 3 squared and so 9 to the 6th = 3 to the 12th.
So instead of taking 12:9, which is 3/4 of 12, we take 8:6, which is 3/4 of 8. And so by adding the length 12 to 8 [as geometric magnitude not wavelength!!] with the length 8 to 6, [as geometric magnitude, not wavelength!!] we get the length 12 to 6, which corresponds to the ratio 2:1.
We are told that 6:8:9:12 is just multiplying (1:4/3:3/2:2) times 6 but in actually 9/8 is assuming that 3/2 is also the geometric mean squared.Yes Philolaus taught that a "double octave" is necessary to create music theory with 0 to 6 as first octave and 6 to 12 as second octave and then the wavelength is reversed. So 3/2 is the first harmonic as C to G overtone, Perfect Fifth but then 8/6 is the 2nd harmonic as 4/3 frequency from 6/8 or 3/4 wavelength that is really the 2nd octave (as 0 to 8 turned into geometric magnitude)! See the bait and switch! This then enabled Archytas to claim that Perfect Fifth plus Perfect Fourth = the Octave since the arithmetic mean x harmonic mean = geometric mean squared or 3/2 x 4/3 = 2. Notice that 2/3 is also the Perfect Fifth as C to F subharmonic - but that can no longer be used based on the lie of ARchytas. So everyone learns the wrong music theory based on symmetric math when the truth is noncommutative phase or G=3=F at the same time as complementary opposites of overtones and subharmonic undertones.
So Philolaus called 4/3 the "subcontrary mean" [the Phantom Tonic] because it was derived from the subharmonic of the Perfect Fifth as C to F - or 0 to 8 with 6/8 as the 3/4 of the root tonic of the Perfect Fifth! But Archytas changed the term to Harmonic Mean instead of subcontrary mean - thereby covering up the fact that Philolaus got it from the subharmonic as the Perfect Fifth, which was a different geometry as C to F, that was noncommutative phase with the fundamental pitch.262144 was then used by Aristotle (and Philolaus) as the starting root tonic value so that the octave was then 524288 as 8 to the 6th with the difference to the Perfect Fifth as 3 to the 12th being 531441 from 9 to the 6th. So that is the diatonic scale of 6 notes above the root tonic but with 12 notes - the error keeps multiplying for each note, as Sir James Jeans points out - so you have a quarter note error.So by defining the Tonic as a geometric mean "X" - the geometric magnitude is ALREADY ASSUMED as a commutative math value - from Philolaus covering up the CHANGE in the root tonic value by "flipping" the lyre around.
And so then when we turn to Chinese music tuning we get this strange "exception."
>With the exception of B#, or the ditonic comma,[10] ratio 531441/524288, Table 11.2 shows that the Chinese and Pythagorean progression of ratios are identical, which means that both sequences form an ascending spiral of “fifths.”
>In other words - as you claim - the Fifth and Octave both have a denominator two - and so can be "divided back" - if you use instead of 2 to the 7th power, rather 2 to the 19th power (by adding the logarithms of 12 plus 7 from the 3/2 and the 2/1).
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.
This musical property is the counterpart of the principle mathematical characteristic of the Pythagorean diatonic, very Pythagorean indeed, constituted by the fact that each interval of the scale is expressed by the ratios of type 2 to the m divided by 3 to the n OR 3 to the m divided by 2 to the n.Fabio Bellissma (acknowledging the noncommutative phase origin of music theory!!)The problem with just adding these logarithms is you have the SAME "bait and switch" that I am referring to - that is noncommutative. this is why the Chinese and Pythagorean tuning does not use the same definition of the Comma. Because 2/3 is ALSO the Perfect Fifth (as C to F subharmonic) - not just 3/2 as C to G overtone harmonic.
So this definition, as Alain Connes points out - as I quoted above - actually originates from the noncommutative quotient - the division is noncommutative since 2/3 AND 3/2 are both the Perfect Fifth.
And so rather it is just 3/2 to the 12th and 2/1 to the 7th that is the proper definition of the Pythagorean Comma since it does not assume you can just divide and "average" or then "halve" the fifth back into the octave.So we can see that already with Philolaus - the logarithms as geometric magnitude are assumed in the math.
Lore and Science in Ancient Pythagoreanism
Walter Burkert - 1972 - Literary Criticismto this, the apotome would be 2187: 2048, and the komma 531441 : 524288 — pure frivolity.41 Philolaus' treatment is different:42 He establishes as the basis of tone the number which first makes the cube of the first odd number and was highly honored among the Pythagoreans [i.e., 27] ... a number which is separated by a ...The Science of Harmonics in Classical Greece
Andrew Barker - 2007 - PhilosophyIt 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 ...From the Beginning to Plato - Page 269 - Google Books Result
Christopher Charles Whiston Taylor - 1997 - ReferenceThe 'diesis' should be 256:243 and the 'comma' 531441:524288. Neither of these intervals can be divided in half in the sense of the Sectio Canonis. Since Philolaus seems clearly to recognize that the tone cannot be divided in half, it is rather surprising that he apparently takes for granted—what is false in terms of the ..
So then it is assumed that the octave is already defined as the geometric mean squared - which is where Archytas got his equation (Arithmetic mean x harmonic mean = geometric mean squared). So that the 3/2 is actually the arithmetic mean and it can not be 2/3 even though both are the Perfect Fifth. And 4/3 is the Harmonic mean but it was DERIVED from 2/3 that was doubled - and so originally the Harmonic Mean was called the Subcontrary Mean by Philolaus.
So then you get 6 whole tones as 9/8 - being greater than the octave by the so-called Pythagorean Comma. But again this is not the original Pythagorean Comma! This already assumes the octave is a geometric mean squared definition - not the number 2 as a doubling.
So you have 8 to the 6th power and 9 to the 6th power (531441) as the secret of the Ditonic Comma as the fake Pythagorean Comma due to geometric mean. From Euclid and earlier from Philolaus and Archytas as 8 to the 6th compared to 9 to the 6th which assumes that the starting root tonic "frequency" as the 1 is 8 to the 6th which is already from assuming the Perfect Fifth is squared!
http://dspace.chitkara.edu.in/jspui/bitstream/1/546/1/32014_MJIS_Abdounur.pdf
On Mon, Mar 18, 2019 at 3:52 PM Dale Purves, M.D. <purves@neuro.duke.edu> wrote:
Thanks for your interesting note Drew. I did not know several of the things you mentioned and look forward to following up these issues.With best regards,Dale_____________Dale Purves M.D.Geller Professor of Neurobiology, EmeritusResearch ProfessorDuke Institute for Brain SciencesPO Box 90999Room B256, LSRC Bldg.450 Research DriveDuke UniversityDurham NC 27708
Office: 919 681 7414
Mobile: 919 260 4734
email:
purves@neuro.duke.edu
Website:
www.purveslab.netFrom: Voidisyinyang Voidisyinyang <voidisyinyang@gmail.com>
Date: Monday, March 18, 2019 at 2:28 AM
To: "purves@neuro.duke.edu" <purves@neuro.duke.edu>
Subject: Noncommutative phase and tonal modes as biological emotionDear Professor Dale Purves: Regarding your UCMerced lecture on youtube with the challenger re: the Harmonic Series, there is a secret to music theory that scientists often miss. So he was claiming the chromatic scale could derived from the Harmonic Series and you disagreed with this - correctly. The reason you are correct is that the Perfect Fourth is noncommutative phase to the Perfect Fifth and this basic fact was covered up to create the origin of the Greek Miracle as irrational magnitude symmetric commutative mathematical physics. So Alain Connes, the Fields Medal math professor, has a lecture on youtube on music theory and noncommutative geometry, confirming that this secret of music theory actually enables the "formal language" for a unified field science. So in nonwestern music - as you correctly point out - the sad/happy/scary modes are universal - as expressed in language also. There is a deeper meaning to this also as - the photon parameter that you mention is also based on this noncommutative time-frequency phase, as Louis de Broglie discovered. So I have been corresponding with Nobel physicist Brian Josephson about this and now he is focused on cymatics. But I agree with you that the emotional biological context is key - and as you know, the older language is, then the more musical it is. I have researched this topic for 30 years - and I did a master's degree studying this topic, before I had realized the noncommutative phase secret. So I'm just pointing out - the person who claimed there were thousands of studies about the harmonic series creating the chromatic music scale - yes that is only by assuming a Western symmetric math bias that covers up this deeper noncommutative phase truth of math. So until now Western math has relied on the symmetric logic - this is explained by Alain Connes. So in terms of biology - I did my master's thesis on music theory and "radical ecology" - in 2000 at University of Minnesota - the title is called "Epicenters of Justice: Music theory, radical ecology and sound-current nondualism." So again I had made errors since I did not understand this noncommutative phase secret of music theory - but I was close. So by scientists not understanding the noncommutative phase secret - that means it also changes how we define entropy in physics in relation to biology as well. I have details on my blog http://elixirfield.blogspot.com and http://ecoechoinvasives. blogspot.com So in music theory this is called the "Ghost Tonic" about the noncommutative phase of the Perfect Fourth - it was covered up by Philolaus when he literally "flipped" his lyre around to create a symmetric double octave - thereby converting the chromatic scale into a logarithmic symmetric equal-tempered scale. So then the Harmonic Series already assumes this logarithmic math! Math professor Luigi Borzacchini has a great article on this "cognitive bias" at the foundation of science - from music theory - on "the continuum and incommensurability." I've corresponded with him as well - starting in 2001. Take care and let me know if you want more info on this subject, comments or questions, etc.Thanks for your great research!drew hempel
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