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 2^{7} = 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 }=
3^{12}/2^{12}. When 3^{12}/2^{12 }(=
129.7463379…)^{ }is divided by 2^{7} (= 128) we get the
Pythagorean komma expressed in the form of its prime factors:

(3^{12})
/ (2^{19}) = 1.0136432647705078125….

“What’s
remarkable is that the prime factors of 82944 are 3^{4} and 2^{10}.
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}/*x*^{3}]^{2}
= (8*x*^{3}/*x*^{4})^{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}/*x*^{3})^{2}
× (8*x*^{3}/*x*^{4})^{2}]^{−1}
= 162^{2}.

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

3^{12}/3^{4}
× 2^{19}/2^{10} = 2^{7}
×
[([*x*/6]^{4}/*x*^{3})^{2} × (8*x*^{3}/*x*^{4})^{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,

(3^{12}
× 2^{19})
/ (3^{4} ×
2^{10})
= 1832.8207768355312…^{2} = (3/2 × *P*)^{2}.^{1}

**Neutrino Oscillation **&
**(3 ^{12}
× 2^{19}) / (3^{4} × 2^{10})**

**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 sin^{2 }2*θ*_{13} 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 *
θ*_{13 }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 sin^{2} 2*θ*_{13} = 0.092 ±
0.016(stat) ± 0.005(syst) in a three-neutrino framework.”

**New results from Daya Bay neutrino experiment announced**

“Daya
Bay's first results were announced in March 2012 and established the
unexpectedly large value of the mixing angle theta one-three, the last of
three long-sought neutrino mixing angles. The new results from Daya Bay put
the precise number for that mixing angle at sin^{2} 2*θ*_{13}
= 0.090 plus or minus
0.009. The improvement in precision is a result of having more data to
analyze and having the additional measurements of how the oscillation
process varies with neutrino energy.”

** **

**
sin ^{2} 2θ_{13} =
0.092883782 & Real Trinary Logic**

Where sin^{2}
2*θ*_{13} = 0.092883782, *θ*_{13 }=
(.86̅6̅6/.57̅6̅9̅2̅3̅0 ** **)^{4}/.57̅6̅9̅2̅3̅0 ** **
=
8.82711567 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 2^{5}
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 2^{5},
the number of edges of a voluminous 4D hypercube surface, note that

(3/2 × *P*)^{2} × 2^{5}
= 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 *θ*_{13} ≠ 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) : *θ*_{13} (≠
zero) :: :
*θ*_{12} (~ 33.4°) :: 1 : *θ*_{23} (~ 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}**)

** **

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.

1 = 0 → 1 |

1 = 0 → 0|000 → 0 |

1 = 0 → | → |

0 = 1 → 000|1 → 1 |

0 = 1 → 0 |

0 = 1 → |111 → |

= 01 → |1 → 1 |

= 01 → 0|111 → 0 |

= 01 → |

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}**

^{
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.