Friday, February 18, 2022

My newly published article on viewzone! another look at the Pythagorean noncommutative phase music secret: Hugo Rieman Undertones

http://viewzone.com/tuning.html 

 the Pythagorean scale are 2 and 3, with their powers 2 2 = 4,
2 3 = 8, 3 2 = 9, etc. ; the numbers 5, 7, 11 do not occur at all.
But the frequency ratios of a note and its various harmonics
are represented by the complete sequence of numbers 2, 3,
4> 5? 6, 7, . . . , so that most of these harmonics are not
represented by notes on the Pythagorean scale at all. And
Helmholtz's theory of dissonance makes it clear that the
pleasurable consonances are the harmonics, and not the
notes of the Pythagorean scale.
To take the simplest instance, the fifth harmonic of G
has five times the frequency of the fundamental. But the
nearest note on the Pythagorean scale, e", has a frequency
f-| or 5-06 times that of the fundamental C, and so is about
a fifth of a semitone out of tune with the fifth harmonic of
C. If we sound C, this latter insists on sounding anyhow,
for we have seen (p. 83) that the natural harmonics alone
are forced by resonance, and makes considerable discord
with the Pythagorean e". Thus the note that our ear wants
to hear sounding with C is not the Pythagorean e but the
harmonic e.
It will be clear from what has already been said that
these complications are absolutely fundamental ; they arise
out of the laws of arithmetic, which the musician is
completely powerless to alter. If we visited another planet,
we should find the same laws there as on earth. Here, 3 to the 12th
is very nearly equal to 2 to the 19 th, which means that 12 fifths are
very nearly equal to 7 octaves, and the same would be
true there, so that if the inhabitants were at about the
same musical level as we are, we might expect to find
them employing the same diatonic scale as ourselves.

 

 

 https://ia800504.us.archive.org/27/items/sciencemusic00jean/sciencemusic00jean.pdf

So this later commentary by Sir James Jeans is a bit different than his first presentation a couple pages back.

 

 Infinity is Infinity - as simple as that!! All simplicity has disappeared.

So how does this connect to the noncommutative Pythagorean exponentiation?


So he compares the expansion or spiral of fifths to the doubling of the octave as also an exponentiation or squaring, cubing, etc.

He is stating that we want to hear the harmonic series of integers. So that we prefer the 5/4 as the major third E over the Perfect Fifth major third C to E as 81/64 aka 3/2 squared to 9/4 HALVED to 9/8 and then squared to 81/64.

So what gets left out of this discussion is that the 9/8 actually already assumes the Perfect Fourth of 8/6 from the root tonic of 8 and not the 12 and therefore reveals the noncommutative phase of 2/3 and 3/2 as part of the double octave (NOT the octave as the geometric mean squared)....

So therefore the major third must be related to the undertone... And this was the claim of Rieman I think.

So essentially the Perfect Fifth as the overtone is inverted in direction while the root tonic increases in step to create the Perfect Fourth and so at the minor fifth chord there is a major Fourth chord above.

This is similar to what Erno Lendvai claimed about Bartok.

https://core.ac.uk/download/pdf/210597688.pdf

Yeah this thesis is WAY better than the Wiki page on Hugo Rieman...


So the 3rd is G as overtone and F as undertone. The 4th overtone/undertone as the double octave is the SAME note. Why? Because he starts with a four octave difference...of highest and lowest as the "harmonic mean" so to speak...  The fifth overtone/undertone is this the Major third as 5/4 and minor third... and then the sixth overtone/undertone is a G and F or 9/4 difference between them.... hmm.

Right so once again the Perfect Fourth is never an overtone but if it is a drone to the root tonic then it is a Perfect Fifth as an octave below - as the noncommutative phase undertone creating a NEW root tonic!

So that corroborates Riemann's claim.


Right so he applied this concept to the chords....based on frequency and wavelength!

It's not arbitrary at all if you know the secret of the Double Octave covering up noncommutative phase!

Wow - so the "primitive time" that Connes refers to directly goes back to the Riemannian musical isomorphism!

So he says BECAUSE there is a "minus" sign in front of the one for the absorption spectra this is due to the noncommutative negative frequency of the imaginary primitive time negative frequency axis.

So then the square root of minus one is noncommutative as the cross product to the square root of the wavelength based on the noncommutative inverse or negative frequency as the wavelength. So for music then it is 2 to the (1/12) or 3 to the (1/19) as the noncommutative negative frequency or inverse wavelength square root rational approximation to the Perfect Fifth and Octave as the Spectrum Scale that is commutative and isomorphic.


fascinating! Now it's starting to make better sense!

So there's a complex number plus the inverse frequency....

Since the inverse frequency is noncommutative then it converges to 1.

So it's the square root of negative one that is then inverted as a noncommutative exponent... so that the inverse frequency has the exponent as the wavelength inherent to it.


Wow I didn't know he had a formal proof for this....and it's based on primes!!

So that is 3 to the (1/19) x 2 to the (1/12) = the noncommutative space... 

So the Quotient Space is (3/2) to the (1/7) or the Perfect Fifth as the convergence of 19-12 or the 7th note of the scale... = 1.0596

Fascinating! So since it's greater than 1 then the sum to the negative 1 = always less than any positive number.

 1.0595 is 3 to the (1/19)

 1.0594 is 2 to the (1/12)....

3 to the 12th/2 to the 19th = 1.0136

hmmm

So the noncommutative "double quotients" as inverse frequency then ... due to the imaginary "primitive time" phase is always LESS than any positive number because it's an eternally converging noncommutative phase  (hence a positive number) to the negative one power!

citing p. 385

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 is almost 3 to the power of 12...time [spacetime] 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 1/12th~3 to the 1/19th. 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. (Connes, 2012)

Here there's the Infinite Spiral of Fifths again! 

Didn't notice it at first.














 

 

 

 

 

 

 

 

 

 

 

 

 

 

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