Sunday, July 25, 2021

Grant Sanderson, Stanford trained youtube math teacher makes the SAME "Bait and Switch" Music error that covers up noncommutative phase!

 He plays 4:3 music Perfect Fourth as LOWER note in the scale

 

But he does NOT mention that this can ONLY be derived from noncommutative phase!! Oops.

 When you play the Perfect Fourth as 4:3 you are neglecting to mention that the Lower Frequency was derived from Noncommutative Phase of doubling the 2/3 as the Undertone of the 1:1 fundamental frequency ratio. See Professor Richard McKirahan's translation of Philolaus for details - the first Greek use of irrational magnitude math was from Philolaus. So you say the ratio is 4:3 but as an Overtone harmonic that is NOT the lower note in the scale. It is the 4:3 as G to C and so would have to be a perfect fifth to the octave higher. Sorry to expose your wrong music just starting out on the video. See Alain Connes, Fields Medal math professor talk on quantum music of the sphere as cited by Math PRofessor Micho Durdevich for details - as I quote Durdevich:  

"However, even in this case there is a highly non-commutative world of higher order collectivity algebras B(n). This can be used to capture the geometry of rotations, like those appearing in the classical Pythagorean octave versus perfect fifth considerations." 

Music of Quantum Circles October 2017 DOI:10.1007/978-3-319-47337-6_11 In book: The Musical-Mathematical Mind (pp.99-110) Professor Micho Đurđevich https://www.matem.unam.mx/~micho/inde... Institute of Mathematics, UNAM, Mexico citing 

Alain Connes:

 "a non-commutative algebra naturally engenders a one-parameter group of automorphisms that makes it evolve, that makes it rotate in such a way that the passage from xy to yx corresponds to what evolution yields for the purely imaginary value t = sqrt(-1) of the parameter t of the evolution group. We must ... introduce evolution only after choosing a state in the algebra, but the evolution obtained is modified only by inner automorphisms and these are in a certain sense invisible. Here, I believe, is the key link that Hamilton sought between time and algebra. ..." 

In the book "Triangle of Thoughts" (American Mathematical Society, 2001) Alain Connes, Andre Lichnerowicz, and Marcel Paul Schutzenberger discuss the interrelationships among Mathematics, Physics, and Philosophy.

 as "less" harmonious than 3/2 or 2/3 (C to the G above it), and I don't think that makes sense" 

Sorry but 2/3 is C to F below the "1". Please read Professor Richard McKirahan's translation of Philolaus. The 9/4 you refer to was derived by Philolaus flipping his lyre around so that 6:8 became the new 4:3 by changing the Root Tonic of the octave. So you are claiming 1:2 and 2:1 are the same but it's NOT the same if the octave as 6 is not used to create 4:3 as the new root tonic. Therefore the 9/8 can be derived as the first "irrational magnitude" ratio by considering the root tonic as just a geometric "x." And so the octave as 2 is actually the first "geometric mean" as a squaring process and not an algebraic doubling. This covered up the noncommutative phase of the Perfect Fifth to the root tonic. I have more details on my youtube channel, etc. It's quite hilarious that most mathematicians cover this up - ONLY Fields Medal math Professor Alain Connes reveals the truth.

 

Is there a way that this notion of rationalness can be further formalized? Like some sort of measure for which we can plug in 1/2 and it says it's very harmonious, whereas plugging in sqrt(2) will yield a number that we interpret as cacophonous?
yes he is saying that because square root of two converges much faster then it sounds more cacophonous as a probability. Whereas 8/5 is the Golden Ratio which is the slowest converging irrational number derived from 1 plus 1 divided by 1 as a continuous fraction. So 8/5 is in music theory the minor sixth as complementary opposite to the major third as 5/4 that approximated the Cube root of two. So that is how the Pythagorean Tetrad or tetractys could be justified as extended to 1:2:3:4:5 by Archytas (and his followers). Then Simon Stevin in the late 1500s took 5/4 and just converted it to cube root of two with actual logarithms. And so Simon Stevin dropped the rational fractions all together to justify the equal-tempered tuning. The reasoning being that it would take an infinite amount of time to achieve the Perfect Rational numbers as logarithmic repeating decimals of 1/3 as .33333 and therefore the rational numbers can not be the truth.
 
 As we get closer and closer to the square root of 2 over 2....the intervals placed on top of those rationals get really small really fast....for any sequence of rational numbers approaching... shrink faster than the sequence converges....

 

Topher The11th

14 minutes ago

 @Voidisyinyang Voidisyinyang  There is something seriously wrong with your math. If C is 2, the G above it is 3/2. That means that the C is 2/3 of the G. 2/3 can't be C to F below the "1", because if the C is higher than the F, the fraction must work out to being greater than 1. And it's wrong to say that 9/8 is irrational. That statement is simply false. And if two notes are 1 octave apart, one of them is DOUBLE the other one, not SQUARE, and the other note is HALF the one that is double of itself, not the SQUARE ROOT. The square root of 2 is the ration of F# to the C-natural below it. (I.e. the 12th root of 2, raised to the 6th power, is the square root of 2.) Your remark about 2:1 not being the same as 1:2 contains a lot of incoherent gibberish. You don't need a "6" anywhere. The only difference between 1:2 and 2:1 is that one of them is the idea of being one octave higher, and the other one an octave lower. Why bring in "6"? Your Comment is mostly a bunch of word salad that doesn't make sense.
1
 @Topher The11th  hey thanks for clarifying what I'm trying to communicate. So there is a difference between 2/3 of the 1 when the "one" is a sound that is listened to and not a physical vibration of a string. So this is why Western music tries to claim that undertones don't exist because you need a 3/2 wavelength in order to have a 2/3 frequency and the 3/2 wavelength is longer than a 1:1 fundamental frequency ratio. So you understand that 4:3 can NOT be an overtone because the denominator is not the same pitch of the root tonic as the one (or "C"). So that is the difference between the Harmonic Series (that assumes a closed geometric ratio) in contrast to the overtone series. For example in traditional Indian raga music if there are two notes played and it is a 4:3 as Perfect Fourth higher then it is well known that the Perfect Fourth is HEARD as the root tonic!! So this is also why in traditional Chinese music the octave is not used to try to "contain" the scale as a symmetric doubling - even though the octave was obviously known. Instead of constructing the scale as multiplying 3/2 to try to line up with the octave (as is done in the Pythagorean scale creating the Pythagorean Comma) the traditional Chinese scale takes 2/3 of the 1 and then 4/3 of the 2/3 for the Perfect Fourth.
Yes the square root of 2 was originally from music theory as 9/8 cubed (approximated). So this is from Archytas (not Orthodox Pythagorean music philosophy). 
 "the Pythagorean music school was rapidly able to calculate the tone fa-sol as the difference between the fifth do - sol and the fourth do - fa, and consequently as the ratio 3/2 : 4/3 = 9/8 [logarithm]....The Pythagorean tradition denied that it was possible to divide the tone into two equal parts (semitones [based on rational ratios])....Dividing the Pythagorean tone into two parts would mean admitting the existence of the proportional mean between 9 and 8, that is to say, 9 : [a] = [a] : 8, where 9:[a] and [a]:8 are the proportions of the required semitone....Clearly [a]= [square root of 9 x 8] and therefore [a] = (3 x 2) x [square root of 2]!"  
Math Professor Tito M. Tonietti, University of Pisa, Italy and 
 
"However, he [Archytas] noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5."  
A Truman State University review on Scriba, Christoph J. “Mathematics and music.” (Danish) and  
 
"The 'demusicalization' of the theory of proportions by Plato is shocking." (Borzacchini, p. 281 of his academic article on the topic, "Incommensurability, Music and Continuum: A Cognitive Approach").... "...this 'removal' seems really astonishing!"  
 
Then Math Professor Luigi Borzacchini pulls out his trump card: 
"However, I think I can prove that in the Platonic Academy there was a trace of this earlier approach, with a tight connection between music, numerical means and similarity, and without any reference to geometric figures, such as square or pentagon." Borzacchini, again, is revealing a cover-up: "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." 
 and  
“The sound of square roots Take two strings, one sounding an octave higher than the other, so that their lengths are in the ratio 2:1. Then find the geometric ratio (also called the mean proportional) between these strings, the length x at which 2:x is the same proportion as x:1. This means that 2:x = x:1; cross-multiplying this gives x squared =2. Thus, the “ratio” needed is √2:1 ≈ 1.414, in modern decimals. This is close to the dissonant interval called the tritone, which later was called the “devil in music,” namely the interval composed of three equal whole steps each of ratio 9:8. The tritone is thus 9:8 × 9:8 × 9:8 = 729:512 ≈ 1.424.” 
from “Scandal of the Irrational” M.I.T. Press. and 
 "... the tyrant in the Republic as the tyrant's suffering is exactly 729 times that of a philosopher, using the metaphor of ... The ratio 729/512, three whole tones (8:9) above 512....“Since 9 actually reduces to a wholetone of 9/8, its cube will reduce to (9/8)³ = 729/512, a [FAKE] Pythagorean approximation to the square root. of two, a problem which fascinated Socrates in the marriage allegory.”
 Ernest McClain, The Pythagorean Plato: Prelude to the Song Itself (Nicholas-Hays, 1978), p. 36
 
 " no matter how high one goes in the harmonic series, a fundamental pitch will not produce a perfect fourth above the fundamental. ..Just as Slonimsky opined, the perfect fourth above the tonic is nowhere to be found. ... Thus the perfect fourth above the tonic enters the scene, not as part of a stable major scale, but as a tempter, a seducer, a built-in modulation away from the true tonic. The perfect fourth, and not the tritone, is the true “devil in music.” It’s no “subdominant.” It’s the phantom tonic."
 
By Kenneth LaFave
on October 17, 2007

OK so yes that is 9/8 then derived by taking 4/3 of the 2/3 wavelength (3/2 frequency). But notice again that is NOT how the Western scale is derived but rather it is by multiplying as squaring so that you have 3 squared over 2 cubed. Now I'm glad you brought this up - as this refers to the noncommutative phase truth of the fractions. 

"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."

  Professor Fabio Bellissima,"Epimoric Ratios and Greek Musical Theory," in Language, Quantum, Music edited by Maria Luisa Dalla Chiara, Roberto Giuntini, Federico Laudisa, Springer Science & Business Media, Apr 17, 2013 

 So that also explains the noncommutative phase of music. So then as Fields Medal Math Professor Alain Connes explains: See Alain Connes lecture on Music of the Quantum Sphere for more details.

  "Nature is quantum. Nature is very quantum. ..Why should Nature require some noncommutativity for the algebra? This is very strange. For most people noncommutativity is a nuisance. You see because all of algebraic geometry is done with commutative variables.... And this answer is coming from quantum mechanics.....the way you can hear the music of shapes...which would be its scale in the musical sense; this shape will have a certain number of notes, these notes will be given by the frequency and form the basic scale, at which the geometric object is vibrating....]. And the only thing you need to know...is the relative position of the algebra...a "universal scaling system," manifests itself in acoustic systems....There is something even simpler which is what happens with a single string. ... They are isospectral [frequency with the same area], even though they are geometrically different [not isomorphic]....when you take the square root of these numbers, they are the same [frequency] spectrum but they don't have the same chords. There are three types of notes which are different....What do I mean by possible chords? ..Our notion of point will emerge, a correlation of different frequencies...The space will be given by the scale. The music of the space will be done by the various chords. It's not enough to give the scale. You also have to give which chords are possible....The only thing that matters when you have these sequences are the ratios, the ear is only sensitive to the ratio, not to the additivity...multiplication by 2 of the frequency and transposition, normally the simplest way is multiplication by 3...2 to the power of 19 [524288] is almost 3 to the power of 12 [531441]...... Musical shape has geometric dimension zero...There is a beautiful answer to that, which is the quantum sphere... .There is a quantum sphere with a geometric dimension of zero...I have made a keyboard [from the quantum sphere]....This would be a musical instrument that would never get out of tune....It's purely spectral....The spectrum of the Dirac Operator...space is not simply a manifold but multiplied by a noncommutative finite space......It is precisely the irrationality of log(3)/ log(2) which is responsible for the noncommutative [complementary opposites as yin/yang] nature of the quotient corresponding to the three places {2, 3,∞}. The formula is in sub-space..... the phase space of a microscopic system is actually a noncommutative space and that is what is behind the scenes all the time....A fascinating aspect of music...is that it allows one to develop further one's perception of the passing of time. ...time emerges from noncommutativity....What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers qn for the real number q=2 to the 12th∼3 to the 19th. [the 19th root of 3 = 1.05953 and 12th root of 2=1.05946] Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number....This means it is a zero dimensional object! But it has a positive volume!... .Algebra and Music...music is linked to time exactly as algebra is....So for me, there is an incredible collusion between music, perceived in this way, and algebra....I believe that this variability is more fundamental than the passing of time. And that it's behind the scene, meaning that the passing of time is a corollary of this..."

 Fields Medal math professor Alain Connes (compilation of quotes)

 "Ma [Perfect Fourth], although consonant to Sa (root tonic), is alien to the overtone series and is not evoked in the sound of Sa. On the other hand, Sa is evoked in the sound of Ma, since Sa is a fifth above Ma and is its second overtone. For this reason it can be argued that the tendency to view Ma [the Perfect Fourth] as the ground-note has a 'natural' basis. The same cannot be said for Pa as Sa is not part of its overtone series. The thesis can be expressed in 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. By amplifying a prominent overtone the secondary drone lends support to the primary and intensifies its 'primary' character."  

The Rāgs of North Indian Music: Their Structure and Evolution Front Cover Nazir Ali Jairazbhoy Popular Prakashan, 1995

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