the Pythagorean Komma, 82944,

neutrino oscillation & Real trinary logic

On September 28, 2011 Mr. Tom Mellett communicated the following [text edited]. 

“The Pythagorean komma (http://en.wikipedia.org/wiki/Pythagorean_comma) arises as the discrepancy between the ‘circle of 5ths’ and the run of octaves in the Western major scale. Since the 5th is a 3/2 harmonic ratio, then when 12 5ths are played up the scale, the resulting frequency goes up by a factor of (3/2)¹² = 129.7463379….  When 7 octaves are played the fundamental frequency goes up by a factor of 2⁷ = 128.  The ratio of 129.7463379… to 128 yields the Pythagorean komma of 1.0136432647705078125….  It is instructive to transform 129.7463379… to an integer ratio: (3/2)¹² = 3¹²/2¹².  When 3¹²/2¹² (= 129.7463379…) is divided by 2⁷ (= 128) we get the Pythagorean komma expressed in the form of its prime factors:

 (3¹²) / (2¹⁹) = 1.0136432647705078125….

“What’s remarkable is that the prime factors of 82944 are 3⁴ and 2¹⁰.  I’m not sure there’s anything more to this than a coincidence of prime factors being restricted to 2 and 3 for both this komma and 82944, but who knows?  Maybe there is a deeper significance.”

The writer thanks Mr. Mellett for his communication which has prompted the following observations.

“Deeper Significance”

The proof for the uniqueness of the absolute dead center cube (where [x/6]⁴ = 82944) is that among all cubes it is the only one where [(x/6)⁴/x³]² = (8x³/x⁴)² = 162⁻¹ (see, on this web, Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube). 

Multiplying inversely the two sides of the equation whose equality is unique to the absolute dead center cube produces a result shared with all cubes without exception:

 [([x/6]⁴/x³)² × (8x³/x⁴)²]⁻¹ = 162². 

Integrating this result with the prime factors of the Pythagorean komma and with those of 82944 and the 7 octaves prime frequency factor gives—  

3¹²/3⁴ × 2¹⁹/2¹⁰ = 2⁷ × [([x/6]⁴/x³)² × (8x³/x⁴)²]⁻¹.

Moreover, where P is the perimeter of the absolute dead center cube the unique measure of all other cubes (ibid.) and 3/2 is the harmonic fifth “used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to a given initial note, the frequency ratio of any other note” (http://en.wikipedia.org/wiki/Pythagorean_comma), then, exactly,

(3¹² × 2¹⁹) / (3⁴ × 2¹⁰) = 1832.8207768355312…² = (3/2 × P)².¹

Neutrino Oscillation & (3¹² × 2¹⁹) / (3⁴ × 2¹⁰)  

Nature News Mar 8, 2012

Neutrino oscillations measured with record precision 

“An elusive parameter quantifying the rate of oscillation of ghostly subatomic neutrinos from one type to another has been measured with precision for the first time.

“In a paper released online on 8 March, the Daya Bay Reactor Neutrino Experiment in southern China reports a measurement of the disappearance of antineutrinos produced in the world’s fifth-largest nuclear power plant as they travelled about one kilometre between two sets of three 20-tonne, water-filled detectors. It finds that a parameter known as sin² 2θ₁₃ is 0.092. Physicists had speculated that the quantity, the last of three 'mixing angles' that quantify rates of neutrino oscillation to be measured precisely, might be as low as zero. That would have made several future neutrino experiments that plan to compare the oscillation rates of neutrinos to those of antineutrinos virtually impossible to carry out; the positive result suggests that those are on firm territory to proceed.”

F. P. An, et al., “Observation of electron-antineutrino disappearance at Daya Bay” March 8, 2012

(Online. Available at: http://dayabay.ihep.ac.cn/docs/DYB_rate_prl_APS.pdf. March 2012)

Abstract: “The Daya Bay Reactor Neutrino Experiment has measured a non-zero value for the neutrino mixing angle θ₁₃ with a significance of 5.2 standard deviations. Antineutrinos from six 2.9 GW_th reactors were detected in six antineutrino detectors deployed in two near (flux-weighted baseline 470 m and 576 m) and one far (1648 m) underground experimental halls. With 55 days of data, 10416 (80376) electron antineutrino candidates were detected at the far hall (near halls). The ratio of the observed to expected number of antineutrinos at the far hall is R = 0.940 ± 0.011(stat) ± 0.004(syst). A rate-only analysis finds sin² 2θ₁₃ = 0.092 ± 0.016(stat) ± 0.005(syst) in a three-neutrino framework.”

sin² 2θ₁₃ = 0.092 & Real Trinary Logic
 

Where sin² 2θ₁₃ = 0.092, θ₁₃ ≈ (.86̅6̅6/.57̅6̅9̅2̅3̅0         )⁴/.57̅6̅9̅2̅3̅0          ≈ 8.8 degrees,      where .86̅6̅6 is the ratio at infinity of the total arithmetic value to the total number of logical digits of the whole infinitely expanding triangle of digits produced by the iterative sequential summing of the real trinary logic geometric series (see, on this web, Real Trinary Logic Geometric Series Matrix of the Numeric Geometric Series & the Series of Perfect Numbers) and .57̅6̅9̅2̅3̅0          is the ratio at infinity of the total arithmetic value of the edge-sharing triangles to the total arithmetic value of the whole infinitely expanding triangle.  

The latter ratio reappears when, following the Beginning/Power Body sequence, 1, 1, 1, 1, (1), 1, (1), (1), 1, the cornerstone is constructed in the order here illustrated,²

where the sum of the corners of each of the alternate diagonals = 10, the sum of each of the center vertical and horizontal poles = 11, and the square root of the ratio of the latter to the former, 1.1^1/2, is the body- face-diagonal ratio of the absolute dead center triple-cube (see, on this web, Report on a Dream & Related Matters & the God Particle, Sorted Diagonals, and Maxwell's Constant & the Trinary Logic Triple-cube).  The ratio of the rational product of the sums of the squares on the successive diagonals beginning with 1 and ending with 9 to that of the sums of the squares on the successive diagonals beginning with 8 and ending with 2 in the Beginning/Power Body ordered cornerstone,

[1/(4+5) × (8+3+2)/(6+7) × 9]/[8/(4+6) × (1+3+9)/(5+7) × 2] = .57̅6̅9̅2̅3̅0          

(1/9 × 13/13 × 9)/(8/10 × 13/12 × 2) = .57̅6̅9̅2̅3̅0         .

Note that in the construction order of the real trinary logic cornerstone

the product of the numbers in the two 3-valued diagonals (1 × 3 × 9) (2 × 3 × 8) divided by the product of the numbers in the two 2-valued diagonals (5 × 7) (4 × 6) — the number of numbers in the division in the ratio of the harmonic 5th 3/2 — when raised to the inverse 2⁵ power

[(1 × 3 × 9 × 2 × 3 × 8) / (5 × 7 × 4 × 6)](25)−1 = 1.013643357169400995…

the Pythagorean komma 1.0136432647705078125… with a difference of 0.0911 ppm (~R_∞⁻¹).

With reference to the exponent 2⁵, the number of edges of a voluminous 4D hypercube surface, note that

(3/2 × P)² × 2⁵ = 107495424

the absolute dead center hypercube volume 107495424 among all natural numbers the number uniquely in the ratio 82944/784 to its integral product 1016064 (see, on this web, Measure Beyond Beyond Reach). 

Symmetry Breaking & Real Trinary Logic

That θ₁₃ ≠ zero is required to explain the breaking of the matter-antimatter symmetry in the universe (see above).  In real trinary logic 10 = , that is, unlike Boolean and Peircean 0, in this logic 0 ≠ zero.³  The following substantive analogy relates the 3 elements of real trinary logic to the 3 neutrino mixing angles:

0 (≠ zero) : θ₁₃ (≠ zero) ::  : θ₁₂ (~ 33.4°) :: 1 : θ₂₃  (~ 45°).⁴

 

The fact that in real trinary logic 0 ≠ zero is the reciprocal of the fact that in this logic not only does 0 = 1 (as is the case in binary based ternary logics) but so also 00 = 1 just as  = 1.  Nor is this fact unrelated to the marvelous correspondence between the unique mathematics of the absolute dead center cube and these operations unique to real trinary logic as succinctly demonstrated on this web at Epitome or Food for Thought.

Neutrino (ν_τ ν_μ ν_ε) Oscillation

From Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Neutrino_oscillation):  “Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvo whereby a neutrino created with a specific lepton flavor (electron, muon or tau) can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates. Neutrino oscillation is of theoretical and experimental interest since observation of the phenomenon implies that the neutrino has a non-zero mass, which is not part of the original Standard Model of particle physics.”

Real Trinary Logic (01) Operations Analogue

Where (as above) 10 =  → 0 ≠ zero.
 

When a real trinary logic element is identified with either of the two different elements that are the constituents of its basic identity as one of three at once different and identical the other of the two different constituent elements in the mode of its triplet identity with the first constituent ‘ceases’ in accord with the real trinary logic cancellation rule.⁵   When this occurs the identity of a real trinary logic element with one or the other of its constituents is manifest.

Notes

¹  Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), pp. 434ff., for 1832.8207768355312... as “transcendental constant.”

²  For the ‘cornerstone’, cf. Leahy, Foundation, Section III.1.  For the construction here illustrated, cf. D.G. Leahy, Beyond Sovereignty: A New Ethics and Morality (Aurora, 2010), Backnote 3, passim, and, in particular, pp. 307f., n. 5.

³  Cf. Leahy, Foundation, Section III.1.

⁴  Cf., also, L. Everett, “Neutrinos from Particle Physics, Theory Overview,” PATRAS 2010: The 6th Patras Workshop on Axions, WIMPs, and WISPS, Zurich, Switzerland, July 2010. Online. Available at: http://axion-wimp2010.desy.de/e80839/e80847/e93683/100708_ everett.pdf.  October 2011.

⁵  Cf. Leahy, Foundation, III.1; see  also, on this web, Ontic Ontological Logic of Human Sexuality & Word's Body The Medium The Message & The Messenger, for the application of this rule in a very different context but one in which the equivalence of ‘ceasing’ ‘cancelling’ and ‘creating’ is amply illustrated.