Orbital resonance and the celestial origins of Earth’s climatic changes – Why Phi?

Posted: October 30, 2021 by tallbloke in Analysis, Astrophysics, Celestial Mechanics, climate, COP26, Cycles, Ice ages, modelling, moon, Natural Variation, Phi, research, Solar physics, solar system dynamics
Tags: ,

A year after I wrote the original ‘Why Phi’ post explaining my discovery of the Fibonacci sequence links between solar system orbits and planetary synodic periods here at the Talkshop in 2013, my time and effort got diverted into politics. The majority of ongoing research into this important topic has been furthered by my co-blogger Stuart ‘Oldbrew’ Graham. Over the last eight years he has published many articles here using the ‘Why Phi’ tag looking at various subsystems of planetary and solar interaction periodicities, resonances, and their relationships with well known climatic periodicities such as the De Vries, Hallstatt, Hale and Jose cycles, as well as exoplanetary systems exhibiting the same Fibonacci-resonant arrangements.

Recently, Stuart contacted me with news of a major breakthrough in his investigations. In the space of a few hours spent making his calculator hot, major pieces of the giant jigsaw had all come together and brought ‘the big picture’ into focus. In fact, so much progress has been made that we’re not going to try to put it all into a single post. Instead, we’ll provide an overview here, and follow it up with further articles getting into greater detail.

One of the longest known climatic periods is the ~413,000 year cycle in the eccentricity of Earth’s orbit. This period has been found in various types of core sample data and discussed in many paleoclimatic science papers, along with cyclicities around 95, 112 and 124kyr, and shorter periods such as Earth’s obliquity variation, ~41Kyr and Earth’s equinoctial-precession periods of ~19 and ~23kyr. Stuart has discovered how all of these periods are related to each other and to the planetary orbits and their synodic conjunctions.

We’ve also been able to link these Earth Orientation Parameters and climatic periodicities to the planetary orbital and synodic conjunction periods which we believe are key to modulating solar activity. The basis for these were laid out in my 2011 post on Jupiter and Saturn’s motion and further developed with the valuable input of many Talkshop contributors, culminating in the solar variation models published by Rick Salvador and Ian Wilson in the 2013 special issue of Pattern Recognition in Physics.

Solar Total Solar Irradiance (TSI) prediction model hindcast created by Rick Salvador using planetary periods discussed at the Talkshop in 2013

Figure 1 below scratches the surface of what we have discovered. These relationships are all precise whole number ratios, not approximations. The red ‘Graham Cycle’ is a novel addition to previously known cyclic periods which connects the three areas of the figure; Solar-Planetary at the top, climatic periods bottom left, and Earth Orientation Parameters bottom right. Of note, are the ratios between the 60kyr Graham Cycle period and the periods in the three groups. They are mostly ratios of Fibonacci numbers or combinations of them. We know from a previous investigation that Fibonacci and phi (Golden Section) related periodicities tend to be stable and minimally resonant. It could be that the reason the 60kyr period hasn’t been found previously is due to it not showing up strongly in periodograms and other spectral analyses. Nonetheless, it’s an important period for our ‘Why Phi’ investigation and has a lot more connections than we wanted to clutter up Figure 1 with, as it already looks pretty busy!

Figure 1. Spatio-temporal diagram showing solar system dynamical arrangement with particular reference to solar modulation, climatic periods and Earth Orientation Parameters affecting Milankovitch cycles.

Solar cycles

Starting with the upper ‘Solar planetary’ section of figure 1, Ian Wilson’s 2013 PRP paper noted that the Hale cycle and Jupiter-Saturn synodic (J-S) have a 193 year beat period, which is evident in Oxygen18 isotope data as well as Group Sunspot Numbers and 10Be ice core data. This was picked up by the Helmholtz Institute research lab and covered in our earlier post on the Solar Magnetic cycle. What they didn’t pick up on is the fact that the same 193year beat period can also be derived from the 178.8yr Jose cycle and the 2403yr Solar Inertial Motion (SIM) period.

This second route to the 193 year solar magnetic cycle is a novel result revealed in this post. Using the beat period formula of (A*B)/(A-B) = period, the solar inertial motion cycle (A) proposed by Charvatova of ~2403 tropical years and the Jose cycle (B) produces the same 193 year result. It was then possible to tie all this together in the 60 kyr cycle shown in the diagram.

There are 336 Jose and 25 SIM in 60 kyr which means the beat period produces 336-25 = 311 solar magnetic cycles of 193 years each. The number of Hale cycles in 60 kyr is given by the number of J-S minus the number of solar magnetic cycles. i.e. 3024-311 = 2713. It’s notable that 311 and 2713 are both prime numbers. Coupled with the fact that the number of J-S in 60Kyr is the Fibonacci multiple 144×21, we think this is a strong indicator that both 193yr and 60kyr periods are significant solar-planetary cyclic periods.

Support for the 60kyr period comes from Russia, where in 2017 A. S. Perminov and E. D. Kuznetsov produced a paper at at Ural Federal University, Yekaterinburg, entitled ‘Orbital Evolution of the Sun–Jupiter–Saturn–Uranus–Neptune Four-Planet System on Long-Time Scales’. This paper shows inter-related variations in the orbital parameters of the gas giants including antiphase changes in the eccentricities and orbital inclinations of Jupiter and Saturn at ~60kyr and in-phase changes in those parameters at ~400kyr, antiphase to Uranus. These ~400kyr variations are likely to be drivers of Earth’s 413kyr eccentricity cycle.

ISSN 0038-0946, Solar System Research, 2018, Vol. 52, No. 3, pp. 241–259. © Pleiades Publishing, Inc., 2018.
Original Russian Text © A.S. Perminov, E.D. Kuznetsov, 2018, published in Astronomicheskii Vestnik, 2018, Vol. 52, No. 3, pp. 239–259
.

Planetary-climatic cycles

Moving on to the lower left ‘climatic and planetary cycles’ section of Figure 1,

The de Vries cycle is half of 21 J-S and is a prominent climatic cycle. It also links to other cycles through resonant harmonics: Hallstatt = 11 de Vries, J-S synodic precession cycle = 12 de Vries. 6 de Vries is 7 Jose cycles. 33 de Vries is 7 Eddy cycles. See also Why Phi? – Jupiter, Saturn and the de Vries cycle.

The lunar-terrestrial year (L-T) is 13 lunar months. Earth’s tropical year is used throughout this post. Whole numbers of both occur at 353 tropical years and 363 lunar years, forming 10 beats (363-353) of 35.3 years. An important period is 13 L-T, which is 2 Hallstatts and 11 de Vries cycle pairs (22 de Vries). This is 1/9th of the obliquity cycle. It is also 3x7x11 J-S. It follows that the 41kyr obliquity cycle is 3x7x11 Jose cycles, because the Jose cycle is 9 J-S. 3,7 and 11 are all Lucas numbers. We will post a separate article on the inter-relation of the Fibonacci and Lucas series, as they relate to orbital resonance. See also Sidorenkov and the lunar or tidal year (2016)

An explanation for the effect of the motion of the gas giants on these and other climatic periods is found in Nicola Scafetta’s 2020 paper ‘Solar Oscillations and the Orbital Invariant Inequalities of the Solar System’ discussed here at the talkshop.

EOP

At the lower right of Figure 1 we find Earth orientation parameters and associated cycles. To understand how these link to planetary periods we need to look at the motions of Jupiter and Saturn in particular. Kepler gives us this useful graphic in his book De Stella Nova (1606).

Kepler’s trigon showing the ~60 year cycle in the longitude of the Jupiter-Saturn synodic conjunctions. This may be linked to the ~60yr cycle of Earth’s major oceans, giving rise to the 30 year global cooling scare 1960-90 and the subsequent global warming scare 1990-2020.

From an earlier post: ‘As successive great conjunctions occur nearly 120° apart, their appearances form a triangular pattern. In a series every fourth conjunction returns after some 59.8 years to the vicinity of the first. These returns are observed to be shifted by some 7–8°’. Wikipedia. [2019 version]. After 3 J-S the conjunctions have nearly described an exact triangle, but the start position has moved (precessed) slightly, by 60/7 degrees of precession of the J-S conjunction axis. It takes 42 of those (42*3 J-S) to complete the precession cycle in 2503 years. (41×61.051 y = 41×360 degrees movement of the axis).

The 413kyr eccentricity cycle is equivalent to 55*3 of these J-S synodic precession periods, and 6765 or 55×123 (Fibonacci and Lucas numbers) of the 61.051 360 degree periods. Additionally 413 kyr = 10 obliquity periods.

In the brown triangle: the 19 kyr and 23 kyr periods have a beat period of the 112kyr perihelion precession.
23 kyr is 10 Hallstatt cycles.

In the blue triangle: the 95 kyr (5×19 kyr) and 124 kyr (3 obliquities) have a beat period of 413 kyr i.e. Earth’s eccentricity cycle (mentioned in various research papers). Since our 95 kyr = 353×270 and our 124 kyr = 353×351, we find: (351×270) / (351-270) = 1170, and 1170*353 = 413010 years (the obliquity period).

Discussion

The 95 and 124kyr eccentricity cycles are linked with glacial periods. From Park and Maarsch (1993) paper ‘Plio—Pleistocene time evolution of the 100-kyr cycle in marine paleoclimate records’: “The DSDP 607 time scale is more favorable to an abrupt jump in amplitude for the 95-kyr δ18O envelope, but not in the 124-kyr envelope. Rather, long-period δ18O fluctuations appear phase-locked with the 124-kyr eccentricity cycle some 300-400 kyr prior to its growth in amplitude and phase-lock with the 95-kyr eccentricity cycle in the late Pleistocene.” Because the 124kyr period is 3x41kyr (obliquity period), this may help explain the change from glacial periods around 41kyr to around 100kyr.

The bi-modality of glacial cycles and the 95 and 124kyr cycles is one of the modes of variation mirrored between celestial cyclic motion and Earth climatic events. There are also many periods which are ‘quasi-cyclic’ and vary in length within bounds whose attractor nodes fit our phi-Fibonacci scheme. We are not claiming to have elucidated a deterministic and predictable system with our precise whole-number orbitally resonant ratios. We are offering this scheme as a potentially useful roadmap for further investigations into the intriguing numerical links between planetary orbits, synodic timings, planar inclinations, eccentricities, energy transfers and other celestial mechanical and orientation data.

As an example of how our scheme links shorter to longer term cycles, there are exactly 9 Jupiter Saturn conjunctions in the period of the Jose cycle of 178.8 years. There are 55x21x2 Jose cycles in the 413kyr eccentricity period. Experienced researchers like Paul Vaughan will immediately see that this product of multiple Fibonacci numbers resolves to the product of the first 6 prime numbers 1,2,3,5,7,11.

The solar system is organised by the forces of gravity and electro-magnetism into a log-normal distribution of which the Fibonacci series and Lucas series are examples which maintain the stability of the system. Resonance is minimised, but also utilised to transfer energy between orbits in order to resolve inequalities through resonance-forced changes to the eccentricity and inclination of orbits. These changes give rise to the cyclic changes in climatic factors on Earth observed at all timescales from the ~22yr Hale and ~60yr J-S trigon to the ~100kyr and 413kyr glaciation in core sample data and other indices.

Data sources and acknowledgements

Planetary data used is from NASA JPL which gives the Seidelmann values for orbital periods. Our thanks to Paul Vaughan for insisting on their use.

The periods we have calculated can all be reproduced using the ratios we have provided on Figure 1 and the NASA JPL values for the Jupiter, Saturn and Uranus orbital periods.

Comments
  1. Paul Vaughan says:

    2317.99883398204 = 80*(√28+√163+√43+√19) ~= 2318
    4635.99766796409 = 160*(√28+√163+√43+√19) ~= 2318 * 2
    2384035.65182465 = beat(4635.99766796409,4627)
    2384110.34604552 = beat(74626.0277273697,72361.0252351259) = 360*60*60 / 0.543599
    La2011 Table 5

  2. Paul Vaughan says:

    moderators: calculations caught in filter

    _
    easy hindsight:
    https://tallbloke.wordpress.com/2013/01/09/tim-cullen-solar-system-holocene-lawler-events/

  3. Paul Vaughan says:

    algebraic review
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    49962.9015304794 = 1/(2/11.8627021700857-2/29.4701958106261-2/11.8619993833167+2/29.4571726091513)
    4669.65169718707 = 2/(-3/11.8627021700857+1/29.4701958106261+1/84.0331316671926+1/164.793624044745+4/11.8619993833167-4/29.4571726091513)

    simple alternate perspective
    anomalistic
    19.8549641949401 = 1/(1/11.8627021700857-1/29.4701958106261)
    13.8125825263028 = 1/(1/11.8627021700857-1/84.0331316671926)
    12.7828803855253 = 1/(1/11.8627021700857-1/164.793624044745)
    9.95061383963391 = 2/(3/11.8627021700857-1/29.4701958106261-1/84.0331316671926-1/164.793624044745)
    synodic
    9.92945505108639 = 2/(+4/11.8619993833167-4/29.4571726091513)
    4669.65169718707 = 2/(-3/11.8627021700857+1/29.4701958106261+1/84.0331316671926+1/164.793624044745+4/11.8619993833167-4/29.4571726091513)

    link to net search result for perihelion 4670

    _
    moderators: other calculations caught in filter

  4. Paul Vaughan says:

    4670 years

    This is a test comment.
    3 comments have vanished. Cause unknown

    headin’ up the mountain now.

    [mod] nothing in the WP spam filter now

  5. Paul Vaughan says:

    Mayan 36750

    15009.1608487337 = 5482096 / 365.25
    36135.2404360745 = beat(25672.5169367299,15009.1608487337)

    anomalistic
    171.471519050756 = beat(164.793624044745,84.0331316671926) — Standish

    tropical
    171.444289533663 = beat(163.7232045,83.74740682) — Seidelmann
    171.444286952825 = beat(163.723203285421,83.7474058863792) — ‘factsheet’

    1079630.33859387 = beat(171.471519050756,171.444289533663) ; * 2 = 2159260.67718775
    1079528.00372651 = beat(171.471519050756,171.444286952825) ; * 2 = 2159056.00745301

    36750.2561948093 = beat(2159260.67718775,36135.2404360745)
    36750.3154881725 = beat(2159056.00745301,36135.2404360745)

    compare
    Berger 1988 Table 4 (based on Berger 1978)
    2166101.14285714 = beat(75259,72732)
    36748.2810485504 = beat(2166101.14285714,36135.2404360745)

  6. Paul Vaughan says:

    supplementary
    lunisolar with general precession
    13374613.0030966 = beat(25771.4533429313,25721.8900031954)
    25672.5169367299 = axial(13374613.0030966,25721.8900031954)

  7. Paul Vaughan says:

    no mystery left here
    36135.2438440821 = axial(2159056.00745389,36750.3190131859)
    15009.1614366987 = axial(36135.2438440821,25672.5169367299) = 5482096.21475421 / 365.25
    bias hindsight “120k orbital solutions” tunes with round-off a typo f(UN)code’n’PRrhymesnot:
    30031.0042303539 = beat(36750.3190131859,16526.3120307908)
    15009.1624987455 = axial(36135.25,25672.5169367299) = 5482096.6026668 / 365.25
    weather influence campaign or entertainment: unknown

  8. Paul Vaughan says:

    Seidelmann short-model sidereal UJS bias clarification
    30031 = 59*509 (lowest primorial+1 that’s not prime)
    33052.4924754047 = harmean(36750,30031) ~= 33052.5
    36750.0186045651 = beat(30031,16526.25)
    36750.0196670027 — La(2004a,2010a)average

  9. Paul Vaughan says:

    systematic bias review
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)
    21.1746788367349 = beat(84.016845922161,16.9122914926352)
    321.183589283115 = slip(21.1746788367349,19.8650360864628)
    1908.55545325512 = slip(321.183589283115,19.8650360864628)
    33052.6240615815 = slip(1908.55545325512,321.183589283115)

    compare
    15009.1608487337 = 5482096 / 365.25
    36135.2404360745 = beat(25672.5169367299,15009.1608487337)
    36750.3154881723 = beat(2159056.00745389,36135.2404360745)
    33052.6200736723 = harmean(36750.3154881723,30031)

    again: the bias in the short-duration models is systematic
    it could easily be corrected by experts before publication
    it isn’t

    why?
    unknown

  10. Paul Vaughan says:

    supplementary
    30031 = 13*11*7*5*3*2 + 1 = 13# + 1 = 59 * 509
    36750.3253483715 = beat( 30031 , 33052.6240615815 / 2)
    36135.2499689834 = axial(2159056.00745389,36750.3253483715) ~= 36135.25
    15009.1624933944 = axial(36135.2499689834,25672.5169367299) = 5482096.6007123 / 365.25
    recall:
    33053√Φ/8 ~= 5256; 5256√Φ/8 ~= 836

  11. Paul Vaughan says:

    typo: “/√Φ” not “√Φ”
    serpent no. anomalistic UN guidance

  12. Paul Vaughan says:

    Lunisolar Bias

    general & lunisolar precession
    25746.6478202264 = harmean(25771.4533429313,25721.8900031954)

    NASA ‘factsheet’ tropical
    11.8619854620833 = beat(25746.6478202264,11.8565229295003)
    29.4571820908507 = beat(25746.6478202264,29.4235181382615)

    19.858866774147 = beat(29.4571820908507,11.8619854620833)
    60.9467636123559 = slip(29.4571820908507,11.8619854620833)
    883.349939238609 = slip(60.9467636123559,19.858866774147) ; / 2 = 441.674969619304

    600.349139225674 = harmean(936.955612197409,441.674969619304) ; * 4 = 2401.3965569027
    835.54616509501 = beat(936.955612197409,441.674969619304)

    biased (short-duration) Seidelmann sidereal model:
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    61.0464822565173 = slip(29.4474984673838,11.8626151546089)
    835.546575435631 = slip(61.0464822565173,19.8650360864628)

    2401.00140862743 = harmean(2401.3965569027,2400.60639037357)
    2401.00140862743 = 7.00000102669615 ^ 4

    supplementary anomalistic (Standish 1992 Table 2a) review
    936.955612197409 = 1/(-2/11.8627021700857+5/29.4701958106261)
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    1536.74746987137 = harmean(4270.51884168654,936.955612197409)
    2400.60639037357 = beat(1536.74746987137,936.955612197409)

  13. Paul Vaughan says:

    Why aren’t the biased models unbiased before publication?
    Unknown.

    25770.0359146014 = 360*60*60/50.290966 — widely cited general precession rate
    25722.1631216381 = 360*60*60/50.38456501 — W94 lunisolar precession

    25746.0772642216 = harmean(25770.0359146014,25722.1631216381)

    13846300.2974074 = beat(25770.0359146014,25722.1631216381)
    25674.4678646892 = axial(13846300.2974074,25722.1631216381)

    36131.3759941719 = beat(25674.4678646892,15009.1608487337) — spicy whether
    15009.1608487337 = 5482096 / 365.25

    11.8619855831937 = beat(25746.0772642216,11.8565229295003)
    29.4571828377274 = beat(25746.0772642216,29.4235181382615)

    19.858866774147 = beat(29.4571828377274,11.8619855831937)
    60.9467604151707 = slip(29.4571828377274,11.8619855831937)
    883.351954142505 = slip(60.9467604151707,19.858866774147)
    835.54977055391 = beat(936.955612197409,441.675977071252)

    36130.8944646852 = beat(29.447498973306,29.4235181382615) — Sidorenkov
    36131.2752839187 = harmean(36131.65611118,36130.8944646852) —- or
    36131.65611118 = beat(29.4474984673838,29.4235181382615) — Seidelmann

    29.447499867163 = beat(36131.3759941719,29.42351935)
    11.862615400484 = 2 * beat( 835.54977055391 , 29.447499867163 / 5 )

    blur 64k mixmmod11sample
    23093.6961437637 = beat(11.862615400484,11.85652502)
    63999.6554631887 = beat(36131.3759941719,23093.6961437637) ~= 64k
    un weather seidelenkov or sidormann

    11.8626149080812 = 2 * beat( 835.54977055391 , 29.4474986534485 / 5 )
    29.4474986534485 = beat(36131.3759941719,29.4235181382615)

    Bound 2401 = 7^4 with surely enough error = 0.000014667% J-aims.

  14. Paul Vaughan says:

    2362 notes

    comparing systematically biased models
    (which can be systematically unbiased)

    Seidelmann (1992) & Sidorenkov (2017) share superficial similarity
    however, lots of detail differences are noteworthy
    here’s one to start with

    Seidelmann (1992) sidereal review
    =
    Jovian V-E 5256 Ratio Summary
    J:U:S:N = 1:2:16:31

    J: 1 = ⌊0.999878363433384⌉ = ⌊5256 / 5256.6393995685⌉
    S: 16 = ⌊16.0035845857963⌉ = ⌊5256 / 328.426420457382⌉
    U: 2 = ⌊1.99911615294986⌉ = ⌊5256 / 2629.16188848974⌉
    N: 31 = ⌊31.0045597971227⌉ = ⌊5256 / 169.52345185329⌉

    J: 5256.6393995685 = 1 * 5256.6393995685
    S: 5254.8227273181 = 16 * 328.426420457382
    U: 5258.32377697949 = 2 * 2629.16188848974
    N: 5255.227007452 = 31 * 169.52345185329
    =

    Sidorenkov analogs:
    5256 / x =
    1.11245625125401
    16.039288833368
    1.98119882107214
    31.0141327829003

    4724.68017872629 = 1 * 4724.68017872629 — carefully take note of something about this one
    5243.12523289982 = 16 * 327.695327056239
    5305.87838443763 = 2 * 2652.93919221881
    5253.60490137049 = 31 * 169.471125850661

  15. Paul Vaughan says:

    even more perfect jupiter-saturn framing — part i

    19.8650352019356 = beat(29.447498973306,11.8626149212868)
    9.9325176009678 = 19.8650352019356 / 2
    4.9662588004839 = 19.8650352019356 / 4

    16.9122913389518 = harmean(29.447498973306,11.8626149212868)
    8.4561456694759 = axial(29.447498973306,11.8626149212868)
    4.22807283473795 = 8.4561456694759 / 2
    2.11403641736897 = 8.4561456694759 / 4

    6.56961469713012 = axial(14.723749486653,11.8626149212868)

    10.7425999661684 = beat(16.9122913389518,6.56961469713012)
    5.3712999830842 = 10.7425999661684 / 2
    2.6856499915421 = 10.7425999661684 / 4

    4.73161069513687 = axial(16.9122913389518,6.56961469713012)
    2.36580534756843 = 4.73161069513687 / 2
    1.18290267378422 = 4.73161069513687 / 4

    131.716314078385 = slip(19.8650352019356,10.7425999661684)
    65.8581570391927 = slip(19.8650352019356,5.3712999830842)
    50.0715810605726 = slip(19.8650352019356,2.6856499915421)

    100.143162121145 = slip(19.8650352019356,4.73161069513687)
    50.0715810605727 = slip(19.8650352019356,2.36580534756843)
    96.1826326350372 = slip(19.8650352019356,1.18290267378422)

    supplementary
    61.0464717290582 = slip(29.447498973306,11.8626149212868)
    835.550927105133 = slip(61.0464717290582,19.8650352019356)

  16. Paul Vaughan says:

    even more perfect jupiter-saturn framing — part ii

    basic
    2432.00637869116 = slip(100.143162121145,19.8650352019356)
    1216.00318934558 = slip(100.143162121145,9.9325176009678)
    1216.00318934592 = slip(50.0715810605726,9.9325176009678)
    608.00159467279 = slip(100.143162121145,4.9662588004839)
    608.00159467296 = slip(50.0715810605726,4.9662588004839)
    608.001594672738 = slip(96.1826326350372,19.8650352019356)
    304.000797336369 = slip(96.1826326350372,9.9325176009678)
    compound
    1216.00318934576 = slip(356.529955086161,131.716314078385)
    1216.00318934575 = slip(208.887731776281,65.8581570391927)
    1216.00318934571 = slip(208.887731776281,50.0715810605726)
    1216.00318934591 = slip(104.44386588814,50.0715810605726)
    608.00159467288 = slip(178.26497754308,65.8581570391927)
    1216.00318934554 = slip(104.443865888141,50.0715810605726)

    a little more wholesome than “perfect jupiter-saturn framing”
    2432.00637869017 = 2/(27/11.8626149212868-67/29.447498973306)
    608.001594672542 = 1/(54/11.8626149212868-134/29.447498973306)
    19.0000498335169 = 1/(1728/11.8626149212868-4288/29.447498973306)

    19 ~= 1/(1728/j-4288/s)
    431 ~= 1/(-89/j+221/s)

    431.004429758615 = 1/(-89/11.8626149212868+221/29.447498973306)
    862.008859517229 = 2/(-89/11.8626149212868+221/29.447498973306)

    basic
    862.008859517104 = slip(131.716314078385,4.22807283473795)
    431.004429758552 = slip(131.716314078385,2.11403641736897)
    431.004429758552 = slip(65.8581570391927,2.11403641736897)
    compound
    862.008859517073 = slip(356.529955086161,65.8581570391927)
    862.008859517088 = slip(310.94560772299,131.716314078385)
    862.008859517088 = slip(155.472803861495,131.716314078385)
    431.004429758544 = slip(155.472803861495,65.8581570391927)

  17. Paul Vaughan says:

    Theorrery Skeptic

    Doubtful weather there’s a trustworthy political party (or orrery).

    JEV ingredients (Bollinger 1952 method)
    1.59868960469858 = beat(1.00001741273101,0.615197262149213)
    0.799344802349289 = 1.59868960469858 / 2
    0.399672401174645 = 1.59868960469858 / 4

    0.761766202327597 = harmean(1.00001741273101,0.615197262149213)
    0.380883101163799 = axial(1.00001741273101,0.615197262149213)
    0.190441550581899 = 0.380883101163799 / 2
    0.0952207752909496 = 0.380883101163799 / 4

    0.814040380789063 = beat(11.8626149212868,0.761766202327597)
    0.407020190394532 = 0.814040380789063 / 2
    0.203510095197266 = 0.814040380789063 / 4

    0.715800563194352 = axial(11.8626149212868,0.761766202327597)
    0.357900281597176 = 0.715800563194352 / 2
    0.178950140798588 = 0.715800563194352 / 4

    44.2785528962259 = slip(1.59868960469858,0.814040380789063)
    22.1392764481129 = slip(1.59868960469858,0.407020190394532)
    11.0696382240565 = slip(1.59868960469858,0.203510095197266)

    6.84872428662588 = slip(1.59868960469858,0.715800563194352)
    3.42436214331294 = slip(1.59868960469858,0.357900281597176)
    24.1185188101801 = slip(1.59868960469858,0.178950140798588)

    thus:

    350.939503542819 = slip(22.1392764481129,0.761766202327597)
    207.994354394449 = slip(44.2785528962259,0.399672401174645)
    207.994354394443 = slip(73.0136987128728,44.2785528962259)
    146.027397425746 = slip(44.2785528962259,1.59868960469858)
    73.0136987128728 = slip(22.1392764481129,0.799344802349289)

    89.3948929753992 = slip(44.2785528962259,0.190441550581899)
    88.5571057924501 = harmean(89.3948929753992,87.7348758857048) ; 44.2785528962251 = axial
    87.7348758857048 = slip(22.1392764481129,0.190441550581899)

    4724.68017871817 = beat(89.3948929753992,87.7348758857048)
    4724.68017872629 = slip(44.2785528962259,0.0952207752909496)
    4724.68017870897 = slip(350.939503542819,44.2785528962259)
    4724.68017872737 = slip(89.3948929753992,44.2785528962259)
    4724.68017870897 = slip(87.7348758857048,44.2785528962259)
    2362.34008935448 = slip(350.939503542819,22.1392764481129) ——— 700,44
    2362.34008936368 = slip(89.3948929753992,22.1392764481129) ——— notice
    2362.34008935448 = slip(87.7348758857048,22.1392764481129) ——— anything
    1181.17004467724 = slip(350.939503542819,11.0696382240565)
    1181.17004468184 = slip(89.3948929753992,11.0696382240565)
    1181.17004467724 = slip(87.7348758857048,11.0696382240565)

    980.292586153988 = slip(292.054794851491,88.5571057924518) ; 292 = 163+67+43+19
    490.146293076994 = slip(146.027397425746,44.2785528962259)
    245.073146538497 = slip(73.0136987128728,22.1392764481129)

  18. Paul Vaughan says:

    with NASA Horizons 1929.72222222222 sidereal
    JSUN orbital invariant (long-run wide-guassian central limit)
    2320.40158057186 = axial(130705.116382548,2362.3400894)
    compare (Horizons 1929.72222222222 JSUN orbital invariants only — without
    Sidorenkov JEV)
    2320.22208286955 = axial(130705.116382548,2362.15404490606)

    notin’ a few interresstin’ properties not shared by Seidelmann & Sidorenkov
    ˚Knot f(eel)in’PRsure’22DC(44)ide weather appearance orrery’a11 IT

  19. Paul Vaughan says:

    /a typo h/tM11
    130704.452624679 = beat(173901.37537739,74619.9907876555)
    73: lowest prime congruent to 1 mod 24
    73 = average(19,43,67,163) ; 19 = x mod 24 for x = 19, 43, 67, 163
    5256 = 7920 – 2400 – 240 – 24

  20. Paul Vaughan says:

    Perfect Contrusst?

  21. Paul Vaughan says:

    Simple Reason: Center Unknown

    Here’s the backstory on how the lunisolar bias stood out distinctly.
    One – and only one – of the many parameter lists fits the criterion.

    “[…] the largest known number not of the form a^2+s with s a semiprime”

    883.339228237648 = harmean( 936.716909730743 , 835.716909730743 )
    883.339228237648 = harmean ( 101 + C√φ , C√φ ) = 2 / ( 1/(101+C√φ) + 1/(C√φ) )

    101: lowest odd prime Mertens zero-crossing

    _____________
    supplementary

    1/(
    5/beat( harmean(general,lunisolar precession) , Saturn tropical) –
    2/beat( harmean(general,lunisolar precession) , Jupiter tropical)
    )

    low
    Seidelmann (1992) tropical
    883.332192747065 — LLR
    883.334207569974

    ~Center:
    883.342073334256 = grand harmonic mean

    high
    NASA ‘factsheet’ tropical
    883.349939238609 — LLR
    883.351954142505

  22. Paul Vaughan says:

    Diversifying Bidecadal Chandler Resonance Diagnostics

    Seidelmann (1992) short-duration (biased) sidereal
    6.46811773461345 = beat(1.18290267954072,1.00001743371442)

    Standish (1992) nodal
    19.8630730877524 = beat(29.4511026866654,11.862499899747)

    bias diagnostics
    280.076577904465 = slip(19.8630730877524,6.46811773461345)
    2790.5838772567 = slip(280.076577904465,19.8630730877524)

    Standish (1992) anomalistic orbital invariant (0=-2+5+3-6)
    2790.83682251396 = 1/(-2/11.8627021700857+5/29.4701958106261+3/84.0331316671926-6/164.793624044745)

    reverse-engineering sidereal earth estimate from jovian anomalistic:
    1.00001743139444 = 14/(2/11.8627021700857-5/29.4701958106261-3/84.0331316671926+6/164.793624044745+84/11.8626151546089+140/29.4474984673838+43/11.862499899747-43/29.4511026866654)

    Experts failed us severely more than a decade ago.
    They harassed us viciously and relentlessly without ever volunteering the most important information.
    Why? Unknown.

    compare – using Standish (1992) short-duration (biased) sidereal earth
    6.46811767349981 = beat(1.18290267954072,1.0000174322536)
    280.076463317364 = slip(19.8630730877524,6.46811767349981)
    2790.74314407649 = slip(280.076463317364,19.8630730877524)

  23. Paul Vaughan says:

    Anomalistic QBO Halstatt

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) —– Standish (1992) anomalistic

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    8.45806059760692 = axial(29.4701958106261,11.8627021700857)

    6.57189983390616 = axial(14.735097905313,11.8627021700857)

    9.46619163320184 = harmean(16.9161211952138,6.57189983390616)
    4.73309581660092 = axial(16.9161211952138,6.57189983390616)
    2.36654790830046 = 4.73309581660092 / 2
    1.18327395415023 = 4.73309581660092 / 4

    203.72293652095 = slip(19.8549641949401,9.46619163320184)
    50.9307341302375 = slip(19.8549641949401,2.36654790830046) ——————–

    4724.17556802657 = slip(203.72293652095,16.9161211952138)
    2362.08778401328 = slip(203.72293652095,8.45806059760692)

    4724.17556802739 = slip(50.9307341302375,16.9161211952138)
    2362.0877840137 = slip(50.9307341302375,8.45806059760692) ——————– 2362

    _
    2320.15794438421 = axial(130704.452624679,2362.0877840137)
    2320.15815353613 = axial(130705.116382548,2362.0877840137)

  24. Paul Vaughan says:

    anomalistic JEV intro

    0.761769224080824 = harmean(1.0000262476142,0.615197860179071)
    0.715803548953639 = axial(11.8627021700857,0.761769224080824)
    6.84967828238651 = slip(1.59867106414771,0.715803548953639)
    835.563824740778 = slip(6.84967828238651,0.761769224080824)

    1.59867106414771 = beat(1.0000262476142,0.615197860179071)
    24.0670906604158 = slip(6.84967828238651,1.59867106414771)
    24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12

  25. Paul Vaughan says:

    Anomalistic “4670 years”

    19.8549641949401 = beat(29.4701958106261,11.8627021700857)

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)
    19.9477072615632 = beat(111.307357343015,16.9161211952138) ; / 2 = 9.97385363078158
    4270.51884168622 = slip(19.9477072615632,19.8549641949401)

    8.45806059760692 = axial(29.4701958106261,11.8627021700857)
    55.6536786715076 = axial(164.793624044745,84.0331316671926)
    9.97385363078158 = beat(55.6536786715076,8.45806059760692)
    2135.25942084311 = slip(19.8549641949401,9.97385363078158)

    4670.79911370059 = slip(2135.25942084311,19.8549641949401)

    invariant 0=1-3+1+1
    2135.25942084327 = 1/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)

  26. Paul Vaughan says:

    What’s in anomalistic “mode 11”? by US Diss Cover:

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857) ; / 2 = 9.92748209747005
    60.8544553085225 = slip(29.4701958106261,11.8627021700857)
    thus base what follows with Standish (1992) anomalistic

    _

    compare & contrast

    —-

    1.

    Seidelmann (1992) short-duration sidereal
    11.8626151546089
    29.4474984673838
    84.016845922161
    164.791315640078
    29.3625733662893 = harmonic mean

    19.8060547427555 = axial(60.8544553085225,29.3625733662892)
    66.1523612443866 = slip(28.3699051701599,19.8549641949401)
    8040.33760745923 = slip(19.8549641949401,19.8060547427555)

    172978.748617708 = slip(8040.33760745923,19.8549641949401) —– further notes be low
    86489.3743088542 = slip(8040.33760745923,9.92748209747005)

    56.7398103403199 = beat(60.8544553085225,29.3625733662892) ; / 2 = 28.3699051701599
    1182607.44767058 = slip(172978.748617708,66.1523612443866) ; * 2 = 2365214.89534115

    202615.037187354 = slip(86489.3743088542,66.1523612443866) ; * 2 = 405230.074374708

    —-

    2.

    Sidorenkov (2017)
    11.8626149212868
    29.447498973306
    84.0168377823409
    164.791321013005
    29.3625729287642 = harmonic mean

    analogUS (to 1. above) calculations rightly left as an exercise in undersstandin’ 4 those curryUS’n’willin’ (who’ll thus ‘no. who’ tune$ what – CO[$] ITsnot what y/n0boughtWHOyen owe…)

    —-

    supplementary

    2364963.50364963 = beat(74619.9907876555,72337.575351641)
    1182481.75182481 = beat(73001.7461837436,68756.9632341238)

    405629.613215262 = beat(304406.35241565,173901.37537739)

    172826.54615749 = beat(68756.9632341238,49188.0779029847)
    173901.37537739 = 1 / g_2

    in agree mint with us?
    Weather 5256 & 4724 are (no. doubt?) in agree mean T.

  27. Paul Vaughan says:

    JPLat0˚Know? Anomalistic weather variant. Mnemonic in-put taxicab:

    Seidelmann (1992) tropical
    4.72860952101702 = 2/(3/11.85652502+5/29.42351935) = 1727.12462755147 d ~= 1727
    2.36430476050851 = 1/(3/11.85652502+5/29.42351935) = 863.562313775734 d
    1.18215238025426 = 1/(6/11.85652502+10/29.42351935) = 431.781156887867 d
    0.591076190127128 = 1/(12/11.85652502+20/29.42351935) = 215.890578443933 d

    “Euler started to use the letter e for the constant in 1727 or 1728 […]”

    Sidorenkov (2017)
    4.73161069513687 = 2/(3/11.8626149212868+5/29.447498973306) = 1728.22080639874 d
    2.36580534756843 = 1/(3/11.8626149212868+5/29.447498973306) = 864.110403199371 d
    1.18290267378422 = 1/(6/11.8626149212868+10/29.447498973306) = 432.055201599685 d
    0.591451336892109 = 1/(12/11.8626149212868+20/29.447498973306) = 216.027600799843 d

    Seidelmann (1992) sidereal short-duration model
    4.73161071816289 = 2/(3/11.8626151546089+5/29.4474984673838) = 1728.220814809 d
    2.36580535908144 = 1/(3/11.8626151546089+5/29.4474984673838) = 864.110407404498 d
    1.18290267954072 = 1/(6/11.8626151546089+10/29.4474984673838) = 432.055203702249 d
    0.591451339770361 = 1/(12/11.8626151546089+20/29.4474984673838) = 216.027601851124 d

    Seidelmann (1992) synodic
    4.73208801967701 = 2/(3/11.8619993833167+5/29.4571726091513) = 1728.39514918703 d
    2.36604400983851 = 1/(3/11.8619993833167+5/29.4571726091513) = 864.197574593514 d
    1.18302200491925 = 1/(6/11.8619993833167+10/29.4571726091513) = 432.098787296757 d
    0.591511002459627 = 1/(12/11.8619993833167+20/29.4571726091513) = 216.049393648379 d

    Horizons 1929.72222222222 sidereal (wide-Guassian sample-center)
    4.73208337117502 = 2/(3/11.8619848807702+5/29.4571542179636) = 1728.39345132168 d
    2.36604168558751 = 1/(3/11.8619848807702+5/29.4571542179636) = 864.196725660838 d
    1.18302084279375 = 1/(6/11.8619848807702+10/29.4571542179636) = 432.098362830419 d
    0.591510421396877 = 1/(12/11.8619848807702+20/29.4571542179636) = 216.049181415209 d

    Standish (1992) nodal
    4.73181582715489 = 2/(3/11.862499899747+5/29.4511026866654) = 1728.29573086832 d
    2.36590791357745 = 1/(3/11.862499899747+5/29.4511026866654) = 864.147865434162 d
    1.18295395678872 = 1/(6/11.862499899747+10/29.4511026866654) = 432.073932717081 d
    0.591476978394361 = 1/(12/11.862499899747+20/29.4511026866654) = 216.03696635854 d

    Standish (1992) anomalistic
    4.73309581660092 = 2/(3/11.8627021700857+5/29.4701958106261) = 1728.76324701349 d ~= 1729
    2.36654790830046 = 1/(3/11.8627021700857+5/29.4701958106261) = 864.381623506743 d
    1.18327395415023 = 1/(6/11.8627021700857+10/29.4701958106261) = 432.190811753372 d
    0.591636977075115 = 1/(12/11.8627021700857+20/29.4701958106261) = 216.095405876686 d

  28. Paul Vaughan says:

    Solves Longstanding Curiosity

    This is highly technical but precise.

    26256.8112288057 = slip(8040.33760745923,16.9161211952138)
    306006.586378586 = slip(26256.8112288057,66.1523612443866)

    The original curiosity? Systematically be low.

    397906.462823915 = beat(306006.586378586,172978.748617708) ; / 4 =
    99476.6157059787
    99476.8155050052 = beat(16.9161211952138,16.9132450828034)
    16.9132450828034 = harmean(29.4571309198874,11.861990807677) —- Standish (1992) sidereal

    (further) clarifies lunisolar bias; well-paid technicians “No!” managers can unbias (hierarchically nested) models for clean, simple public presentation
    _
    supplementary
    398.773134809729 = slip(56.7398103403199,19.8549641949401)
    935.113131399583 = slip(398.773134809729,16.9161211952138)
    936.033465118244 = harmean(936.955612197393,935.113131399583) ~= 936.0
    for future reference
    2365.09305036237 = slip(398.773134809729,9.92748209747005)
    4730.18610072473 = slip(398.773134809729,19.8549641949401)

  29. Paul Vaughan says:

    Adjust Sidorenkov (2017) earth sidereal year estimate (which looks like a long-duration estimate) to 1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6) to match SUNEV 5256 (but not JEV 5256) more closely than Seidelmann (1992).

    At this point a very large volume of calculations needs to be presented to crystallize the lunisolar bias in detail. Presently this isn’t feasible.

  30. Paul Vaughan says:

    To complete the transformation:

    11.862615918328 = 2/(1/((φ^22+1/11)^(e/11+1/22))+5/((1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6))-3/0.615197262149213)

    0.761766208470514 = harmean(1.00001743390371,0.615197262149213)
    0.814040383108898 = beat(11.862615918328,0.761766208470514)
    1.59868955058708 = beat(1.00001743390371,0.615197262149213)

    44.2784630136991 = slip(1.59868955058708,0.814040383108898)
    351.268593378986 = slip(44.2784630136991,0.761766208470514)
    5256.18499467858 = slip(351.268593378986,44.2784630136991)

    alternately:
    0.380883104235257 = axial(1.00001743390371,0.615197262149213) ; / 4 = 0.0952207760588143
    5256.18499465625 = slip(44.2784630136991,0.0952207760588143)

  31. Paul Vaughan says:

    100ka note

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) —- Standish (1992) anomalistic
    19.8589101021728 = beat(29.4571726091513,11.8619993833167) —– Seidelmann (1992) synodic
    99925.8030607636 = beat(19.8589101021728,19.8549641949401)

    99972.391587704 = harmean(100019.023576957,99925.8030607636) ———— v ————–

    100019.023576957 = beat(16.9161211952138,16.9132606717144)
    16.9132606717144 = harmean(29.4571726091513,11.8619993833167) —- Seidelmann (1992) synodic
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857) — Standish (1992) anomalistic
    8.45806059760692 = axial(29.4701958106261,11.8627021700857) —– Standish (1992) anomalistic
    _

    55.6536786715076 = axial(164.793624044745,84.0331316671926) —– Standish (1992) anomalistic
    111.307357343015 = harmean(164.793624044745,84.0331316671926) — Standish (1992) anomalistic
    171.471519050756 = beat(164.793624044745,84.0331316671926)

    19.9477072615632 = beat(111.307357343015,16.9161211952138) —– Bollinger (1952) method
    9.97385363078158 = beat(55.6536786715076,8.45806059760692)

    4270.51884168622 = slip(19.9477072615632,19.8549641949401) —- orbital invariant
    2135.25942084311 = slip(19.8549641949401,9.97385363078158) —- orbital invariant

    49831.9228185121 = slip(4270.51884168622,19.8549641949401)
    4670.79911370059 = slip(2135.25942084311,19.8549641949401) —- “4670 years” (review)

    45051.2652209889 = slip(4270.51884168622,8.45806059760692)
    99973.0172242633 = slip(45051.2652209889,4270.51884168622) —————- ^ ————-

  32. Paul Vaughan says:

    Selected Algebra

    Seidelmann (1992) synodic:
    11.8619993833167, 29.4571726091513
    Standish (1992) anomalistic:
    11.8627021700857, 29.4701958106261, 84.0331316671926, 164.793624044745
    _

    “4670 years”

    4669.65169718707 = 2/(-3/11.8627021700857+1/29.4701958106261+1/84.0331316671926+1/164.793624044745+4/11.8619993833167-4/29.4571726091513)

    4670.79911381622 = 1/(107/11.8627021700857-323/29.4701958106261+108/84.0331316671926+108/164.793624044745)
    _

    “100 kiloyears”

    99972.3915878471 = 4/(1/11.8627021700857-3/29.4701958106261-1/11.8619993833167+3/29.4571726091513)

    99973.0183503139 = 1/(2766/11.8627021700857-8342/29.4701958106261+2777/84.0331316671926+2777/164.793624044745)
    _

    “41 kiloyears”

    derivation
    8.45806059760692 = axial(29.4701958106261,11.8627021700857) ; / 4 =
    2.11451514940173
    45051.2652209889 = slip(4270.51884168622,8.45806059760692) ; / 4 =
    11262.8163052472 = slip(4270.51884168622,2.11451514940173)
    41002.979235297 = slip(11262.8163052472,2135.25942084311)

    summary:
    41002.9789528914 = 1/(-5029/11.8627021700857+15167/29.4701958106261-5049/84.0331316671926-5049/164.793624044745)
    _

    4724 years &
    2362 years

    derivations

    14.6844304038441 = axial(111.307357343015,16.9161211952138) ; / 4 =
    3.67110760096102
    48.6117359109827 = slip(19.8549641949401,3.67110760096102)
    384.86912152873 = slip(48.6117359109827,16.9161211952138)
    4724.17556801612 = slip(2135.25942084311,384.86912152873)

    19.9477072615632 = beat(111.307357343015,16.9161211952138) ; / 4 =
    4.98692681539079
    1067.62971042156 = slip(19.8549641949401,4.98692681539079)
    4724.17556801612 = slip(1067.62971042156,384.86912152873)

    192.434560764365 = slip(48.6117359109827,8.45806059760692)
    2362.08778400806 = slip(1067.62971042156,192.434560764365)

    algebra:
    2362.08778401334 = 1/(31/11.8627021700857-77/29.4701958106261)
    4724.17556802668 = 2/(31/11.8627021700857-77/29.4701958106261)

  33. Paul Vaughan says:

    tropical
    4724 years &
    2362 years (orbital invariant: 0=-2+5+3-6)

    Seidelmann (1992) tropical
    2361.92512664087 = 1/(-2/11.85652502+5/29.42351935+3/83.74740682-6/163.7232045)
    4723.85025328173 = 2/(-2/11.85652502+5/29.42351935+3/83.74740682-6/163.7232045)

    NASA ‘factsheet’ tropical
    2362.05130068208 = 1/(-2/11.8565229295003+5/29.4235181382615+3/83.7474058863792-6/163.723203285421)
    4724.10260136417 = 2/(-2/11.8565229295003+5/29.4235181382615+3/83.7474058863792-6/163.723203285421)

  34. Paul Vaughan says:

    adjusting Seidelmann (1992) short-duration sidereal earth to a long-duration 4724 year tuning:
    4723.99999667307 = 1/(-693.5/0.615197263396975+1166.5/1.00001741532595-465/11.8626151546089)
    1.00001741532595 = 365.256360947803 days ; compare:
    1.0000174152119 = 365.256360906146 days —- Standish (1992) sidereal

  35. Paul Vaughan says:

    easy hindsight

    supplementary notes on geophysical 64 year structure previously illustrated & explored
    Standish nodal with day
    3.93839482956483 = slip(1.00001071395229,0.0027378507871321)
    63.8858575898369 = slip(3.93839482956483,1.00001071395229) ~= 64 year
    Meeus & Savoie tropical with day
    4.12891838459878 = slip(0.999978614647502,0.0027378507871321)
    32.0054607170862 = slip(4.12891838459878,0.999978614647502) ~= 32 year

    supplementary note on 26 (& 52) year geophysical structure(s) previously noted
    Standish anomalistic with day
    3.85237502099482 = slip(1.0000262476142,0.0027378507871321)
    26.077823955957 = slip(3.85237502099482,1.0000262476142) ~= 26 year

  36. Paul Vaughan says:

    JSEV tropical (Seidelmann 1992)

    1.59868953279706 = beat(0.99997862,0.61518257)
    0.799344766398529 = 1.59868953279706 / 2
    0.399672383199264 = 1.59868953279706 / 4

    0.761743683794994 = harmean(0.99997862,0.61518257)
    0.380871841897497 = axial(0.99997862,0.61518257)
    0.190435920948749 = 0.380871841897497 / 2
    0.0952179604743743 = 0.380871841897497 / 4

    61.0914225103732 = slip(29.42351935,11.85652502)

    0.771361726706669 = beat(61.0914225103732,0.761743683794994)
    0.385680863353334 = 0.771361726706669 / 2
    0.192840431676667 = 0.771361726706669 / 4

    1.50472508020829 = harmean(61.0914225103732,0.761743683794994)
    0.752362540104143 = axial(61.0914225103732,0.761743683794994)
    0.376181270052072 = 0.752362540104143 / 2
    0.188090635026036 = 0.752362540104143 / 4

    12.800522799798 = slip(1.59868953279706,0.752362540104143)
    6.40026139989901 = slip(1.59868953279706,0.376181270052072)
    3.2001306999495 = slip(1.59868953279706,0.188090635026036)

    1859.26427454788 = slip(12.800522799798,1.59868953279706)
    1859.26427454788 = slip(6.40026139989901,1.59868953279706)
    1859.26427454788 = slip(3.2001306999495,1.59868953279706)

    929.632137273939 = slip(12.800522799798,0.799344766398529)
    929.632137273939 = slip(6.40026139989901,0.799344766398529)
    929.632137273939 = slip(3.2001306999495,0.799344766398529)

    59.0007181299284 = slip(12.800522799798,0.190435920948749)
    29.5003590649642 = slip(12.800522799798,0.0952179604743743)
    29.5003590649642 = slip(6.40026139989901,0.0952179604743743)

  37. Paul Vaughan says:

    JSUN tropical (Seidelmann 1992) 1470 year & 2402 year

    8.45107360405992 = axial(29.42351935,11.85652502)
    16.9021472081198 = harmean(29.42351935,11.85652502)
    19.8588720868409 = beat(29.42351935,11.85652502)

    3635.42278750964 = slip(163.7232045,83.74740682)
    17760.7407596846 = slip(3635.42278750964,171.444289533663)

    16.9182475901445 = beat(17760.7407596846,16.9021472081198)
    8.45912379507225 = 16.9182475901445 / 2
    4.22956189753613 = 16.9182475901445 / 4

    33.7721548821359 = harmean(17760.7407596846,16.9021472081198)
    16.886077441068 = axial(17760.7407596846,16.9021472081198)
    8.44303872053399 = 16.886077441068 / 2
    4.22151936026699 = 16.886077441068 / 4

    114.253729166588 = slip(19.8588720868409,16.9182475901445)
    57.1268645832942 = slip(19.8588720868409,8.45912379507225)
    67.1361846326796 = slip(19.8588720868409,4.22151936026699)

    475.496204649037 = slip(114.253729166588,16.9021472081198)
    237.748102324519 = slip(114.253729166588,8.45107360405992)
    118.874051162259 = slip(57.1268645832942,4.22553680202996)

    2939.57946202776 = slip(114.253729166588,4.22553680202996) ~= 2940
    2939.57946202744 = slip(475.496204649037,114.253729166588)
    2939.57946202744 = slip(237.748102324519,114.253729166588)
    2939.57946202744 = slip(118.874051162259,114.253729166588)

    1469.78973101388 = slip(114.253729166588,2.11276840101498) ~= 1470
    1469.78973101372 = slip(475.496204649037,57.1268645832942)
    1469.78973101372 = slip(237.748102324519,57.1268645832942)
    1469.78973101372 = slip(118.874051162259,57.1268645832942)

    2402.06517243955 = slip(67.1361846326796,16.9021472081198) ~= 2402

  38. Paul Vaughan says:

    Amicable Bonds (tropical version)

    “consensus” of Seidelmann (1992) & NASA ‘factsheet’
    29.4235187441307 = harmean(29.42351935,29.4235181382615)
    11.8565239747501 = harmean(11.85652502,11.8565229295003)
    0.999978358596783 = harmean(0.99997862,0.999978097193703)
    0.615181976306751 = harmean(0.61518257,0.615181382614647)

    explore what others might have in mind:
    0.615181976285804 — tuned to “4670 years” (remember Bond pub. with 1800 & 4670 graph)
    0.999978614647502 — Meeus & Savoie (1992) rounded-off value used by so many

    1791.85469669027 = 1/(-8/11.8565239747501+16/29.4235187441307+13/0.615181976285804-21/0.999978614647502)

    1799.99949076613 = 221/220/(-8/11.8565239747501+16/29.4235187441307+13/0.615181976285804-21/0.999978614647502)

    1791.85469669027 = axial(395999.88796855,1799.99949076613)
    1791.85469669027 = axial( 220 * 1799.99949076613 , 1799.99949076613 )

    gives:

    1.59868553691508 = beat(0.999978614647502,0.615181976285804)
    0.799342768457541 = 1.59868553691508 / 2
    0.399671384228771 = 1.59868553691508 / 4

    0.761743227089166 = harmean(0.999978614647502,0.615181976285804)
    0.380871613544583 = axial(0.999978614647502,0.615181976285804)
    0.190435806772291 = 0.380871613544583 / 2
    0.0952179033861457 = 0.380871613544583 / 4

    0.771361261986248 = beat(61.0913999839243,0.761743227089166)
    0.385680630993124 = 0.771361261986248 / 2
    0.192840315496562 = 0.771361261986248 / 4

    1.50472418232299 = harmean(61.0913999839243,0.761743227089166)
    0.752362091161495 = axial(61.0913999839243,0.761743227089166)
    0.376181045580747 = 0.752362091161495 / 2
    0.188090522790374 = 0.752362091161495 / 4

    22.0353413519587 = slip(1.59868553691508,0.771361261986248)
    11.0176706759793 = slip(1.59868553691508,0.385680630993124)

    12.8009052113462 = slip(1.59868553691508,0.752362091161495)

    304.013638441169 = slip(22.0353413519587,0.761743227089166)
    152.006819220584 = slip(22.0353413519587,0.380871613544583)
    76.0034096102922 = slip(22.0353413519587,0.190435806772291)
    76 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24 = 19 + 19 + 19 + 19

    1791.85469667879 = slip(12.8009052113462,1.59868553691508)
    1799.99949075461 = 221 / 220 * 1791.85469667879

    164.876143595902 = slip(22.0353413519587,0.399671384228771) ———– z
    4670.79912511218 = slip(164.876143595902,11.0176706759793)

    perfect: s(496) = 496 = s(652) = s(s(608))

    608.027276881365 = 2/(-29/11.8565239747501+58/29.4235187441307-44/0.615181976285804+72/0.999978614647502)

    1216.05455376273 = 4/(-29/11.8565239747501+58/29.4235187441307-44/0.615181976285804+72/0.999978614647502)

    2432.10910752546 = 8/(-29/11.8565239747501+58/29.4235187441307-44/0.615181976285804+72/0.999978614647502)

  39. Paul Vaughan says:

    1800 & 4670 years:
    weather myth or math?

    lunisolar precession with tropical “Consenzus”
    11.8619917685154 = beat(25721.8900031954,11.8565239747501) — LLR
    11.8619917394734 = beat(25722.0265616918,11.8565239747501) — harmean(LLR,W94)
    11.8619917104315 = beat(25722.1631216381,11.8565239747501) — W94

    with anomalistic heavyweight Jupiter
    198078.497253751 = beat(11.8627021700857,11.8619917685154) ; * 2 = 396156.994507502
    198070.399443988 = beat(11.8627021700857,11.8619917394734) ; * 2 = 396140.798887976
    198062.302295769 = beat(11.8627021700857,11.8619917104315) ; * 2 = 396124.604591539

    1799.99857738802 = beat(396156.994507502,1791.85700697991)
    1799.9989117563 = beat(396140.798887976,1791.85700697991)
    1799.99924612472 = beat(396124.604591539,1791.85700697991)
    gives 4672.37005613987 (using 0.615181976306751 = harmean(0.61518257,0.615181382614647))

    1799.9962460564 = beat(396156.994507502,1791.85469669027)
    1799.99658042381 = beat(396140.798887976,1791.85469669027)
    1799.99691479137 = beat(396124.604591539,1791.85469669027)
    gives 4670.7991 (using 0.615181976285804)

    a reverse view bringz estimate ll’un ‘uz ole lore PR’ cz!sun from tropical!Jupiter “consensus”

    with 0.615181976306751 = harmean(0.61518257,0.615181382614647) & 4672.37005613987 :
    396088.097411778 = beat(1800,1791.85700697991) ; / 2 = 198044.048705889
    11.8619916449528 = axial(198044.048705889,11.8627021700857)
    25722.4710182008 = beat(11.8619916449528,11.8565239747501)

    with 0.615181976285804 & 4670.7991 :
    395975.242590358 = beat(1800,1791.85469669027) ; / 2 = 197987.621295179
    11.8619914424619 = axial(197987.621295179,11.8627021700857)
    25723.4232260154 = beat(11.8619914424619,11.8565239747501)

    also note
    Standish sidereal with Standish anomalistic
    197810.936263778 = beat(11.8627021700857,11.861990807677) ; / 2 = 395621.872527555
    Standish sidereal with NASA ‘factsheet’ tropical
    25721.4885660692 = beat(11.861990807677,11.8565229295003)
    25721.8900031954 = LLR lunisolar precession
    11.8565229295003 = NASA ‘factsheet’ tropical Jupiter
    11.8565230147972 = axial(25721.8900031954,11.861990807677)

    4 those who (anomalistically) are averse (“Can˚T get from the Cab to the C[I]RB” — The ‘PR 10’ drs)
    2 amicable bonds

  40. Paul Vaughan says:

    Enough $aid Sir Pentagon

    terre$ketchof whether
    loony so lure

    buy USamicably
    COMBINE Sidorenkov (2017) short-duration sidereal
    WITH Standish long-duration anomalistic
    883.352600477237 = harmean(936.955612197409,835.550927105136)
    984.598022541318 = beat( 25771.4533429313 / 3 , 883.352600477237 )
    ˚T?hird harmonic of General precession arises from opposing trigon COefficients (5 – 2 = 3) in sidereal-tropical conversion (use to detect bias inside pub.lists)
    979.992217251443 = beat(984.598022541318,491.144860474028) ; * 75 / 2 =
    36749.7081469291 ; compare:
    36749.7014379182 — La2011 Table 6 La2010a

    lunisolar bias estimate using Seidelmann (1992) short-duration sidereal JSU
    491.144860474028 = 1/(-3187.5/11.8626151546089+10462.5/29.4474984673838-7275/84.016845922161-37.5/30031-3/25771.4533429313+1/883.339228237648)

    noteworthy: 50482100 (rounded-off version of 50482096)
    systematically links NA!SA ‘factsheet’ & Seidelmann (1992) tropical (no. mystery in such hindsite)

    0.999978614647502 widely-cited rounded-off Meeus & Savoie (1992) tropical
    0.99997861640616 = 31/(-3187.5/11.8626151546089+10462.5/29.4474984673838-7275/84.016845922161-37.5/30031+21/0.0745030006844627+10/0.0754402464065708-31/0.0808503463381246-3/25770.7446092762+1/883.339228237648)
    off by 1 minute (time no. T˚angle) per century
    0.999978616353183 unrounded Meeus & Savoie (1992)

    supplementary

    0 = -42.5 + 139.5 – 97
    33052.6240611784 = 1/(-42.5/11.8626151546089+139.5/29.4474984673838-97/84.016845922161)
    36750.3253473747 = 1/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031)
    36135.2499680199 = 1/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031+0.5/83.7474058863792-0.5/163.723203285421-0.5/84.0331316671926+0.5/164.793624044745)
    15009.1624932282 = 1/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031+0.5/83.7474058863792-0.5/163.723203285421-0.5/84.0331316671926+0.5/164.793624044745+2/25721.8900031954-1/25771.4533429313)
    5482096.60065159 = 365.25/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031+0.5/83.7474058863792-0.5/163.723203285421-0.5/84.0331316671926+0.5/164.793624044745+2/25721.8900031954-1/25771.4533429313)

    0 = 31 – 21 – 10
    491.132481368366 = 1/(31*(1/0.999978614647502+1/0.0808503463381246)-21/0.0745030006844627-10/0.0754402464065708)
    491.145236548486 = 1/(31*(1/0.999978616353183+1/0.0808503463381246)-21/0.0745030006844627-10/0.0754402464065708)

    derive
    0.0748024157879311 = axial(0.999978616353183,0.0808503463381246)
    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)

    18.6129703214384 = beat(0.0748024157879311,0.0745030006844627)
    8.84735306511776 = beat(0.0754402464065708,0.0748024157879311)

    16.8627866218841 = beat(18.6129703214384,8.84735306511776)
    179.333487684639 = slip(18.6129703214384,8.84735306511776)
    ace shown
    491.145236537217 = slip(179.333487684639,16.8627866218841)

    984.581409347557 = beat( 25771.4533429313 / 3 , 883.339228237648 )
    984.584512785494 = beat( 25770.7446092762 / 3 , 883.339228237648 )
    984.587616242996 = beat( 25770.0359146014 / 3 , 883.339228237648 )

    980.01017321043 = beat(984.581409347557,491.145236537217)
    980.007098542353 = beat(984.584512785494,491.145236537217)
    980.004023893569 = beat(984.587616242996,491.145236537217)

    36750.3814953911 = 75 / 2 * 980.01017321043
    36750.2661953382 = 75 / 2 * 980.007098542353
    36750.1508960089 = 75 / 2 * 980.004023893569

    36135.3042523091 = axial(2159056.00745389,36750.3814953911)
    15009.1718585901 = axial(36135.3042523091,25672.5169367299)
    5482100.02135004

    review
    Berger 1988 Table 4 (based on Berger 1978)
    2166101.14285714 = beat(75259,72732)
    36748.2810485504 = beat(2166101.14285714,36135.2404360745)
    note$ trees in llUNe^don foresst matrix calllculaceyen
    2159056.00745301 = 2 * beat(171.471519050756,171.444286952825)
    SSTand!sh anomalistic with ‘factsheet’ tropical
    PC in
    fact ch.UK IT
    luke dawn sank$yuan tyrant$
    R out of ˚T ou ch! within UKquality

    36750.3190131859 = 1/(g_3+g_4) — La2021 Table 2 with explore a tory add “just” (weather mint or) mean˚T
    36135.2438440821 = axial(2159056.00745389,36750.3190131859)
    15009.1614366987 = axial(36135.2438440821,25672.5169367299)
    5482096.21475421 —- pub.list.cz!is note$baseofllreflect$yen=peace$ together quickly

  41. Paul Vaughan says:

    120,000 “noteworthy” Typo Solutions

    won over UN √8
    1800 ~= 1 / √8 * 5090
    4670 = s(4370) + (378-178)*2

    owe bowl’n’Valley
    323 = 196883-196560

    “bye don!” Jail˚Talk
    25746 = Σs(5090) – ΣΦ(323)
    25746 = Σs(5090) – ΣΦ(936)

    0 LA˚Table
    25722 = Σs(5090) – (378-178)*2 – 71
    25746 = Σs(5090) – (378-178)*2 – 47
    25770 = Σs(5090) – (378-178)/2 – 196883 + 196560

    too Sell a con
    25808 = Σs(5090) – 11#/3#
    25808 = Σs(5090) – 7*(28^2-27^2)
    25808 = Σs(5090) – 7*(1^2+2^2+3^2+4^2+5^2)

    ——————–

    s(5090) = 4090
    s(4090) = 3290
    s(3290) = 3622
    s(3622) = 1814
    s(1814) = 910
    s(910) = 1106
    s(1106) = 814
    s(814) = 554
    s(554) = 280
    s(280) = 440
    s(440) = 640
    s(640) = 890
    s(890) = 730
    s(730) = 602
    s(602) = 454
    s(454) = 230
    s(230) = 202
    s(202) = 104
    s(104) = 106
    s(106) = 56
    s(56) = 64
    s(64) = 63
    s(63) = 41
    s(41) = 1
    s(1) = 0

    th borg herd sum??un
    26193 = Σs(5090)
    ABout what nos.˚TemperUN

  42. Paul Vaughan says:

    317 years
    anomalistic JSU
    the very top-level cycle is a familiar one
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    21.1796477480038 = beat(84.0331316671926,16.9161211952138)
    317.450267066043 = slip(21.1796477480038,19.8549641949401)

  43. Paul Vaughan says:

    ok

    2545 = ΣΣδ(220)
    5090 = ΣΣδ(220) * 2

    unI˚Que.T˚A CO$metallica
    400 = s(2401) = 2*(378-178) = s(836-42) = s(496+28) = 744-104-240
    400 = ΣΦ(323) – Σφ(323) ; 447 = ΣΦ(323) ; 47 = Σφ(323)
    400 = Σs(242) ; 242 = ΣΣΔ(178)-ΣΣφ(178) = average(ΣΔ(220),Σδ(220)) = ΣΦ(163)-Σφ(163)
    993 = (σ^2)(400) ; Σδ(Σδ(993)) = Σδ(Σδ(Σδ(378))) = 902 = Σδ(894) ; Σδ(902) = 894
    σ(894) = σ(1691) = 1800 ; ΣΦ(1800) = 735 ; 600 = σ(216)
    216 = 378 – (163+67+43+19+28) + (10+13+18+22+37+58)

    J[amai$SUNami]11buy
    100 = average(-ΣΦ(220),Σφ(220)) = average(-178,378) = 71-Σδ(42)
    100 = average(-Σφ(323),ΣΦ(323))/2 = 4370-s(4370) = s(194)
    100 = 2+3+5+7+11+13+17+19+23 = Σ(primes up to 23)
    100 = 2+3+5+7+11+13+17+19+23 = 41+59 = 29+71 = 31+47+(378 mod 178)
    178 = 2+3+5+7+11+13+17+19+23 + 31+47
    200 = 2+3+5+7+11+13+17+19+23 + 41+59
    300 = 2+3+5+7+11+13+17+19+23 + 41+59 + 29+71
    378 = 2+3+5+7+11+13+17+19+23 + 41+59 + 29+71 + 31+47

    done “ch.op.eur.bri˚k
    25722 = Σs(5090) – 400 – 71
    25746 = Σs(5090) – 400 – 47
    25770 = Σs(5090) – 100 – 196883 + 196560
    f(sst) 0˚NA(!w)ru|ur”

    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5

  44. Paul Vaughan says:

    25722 = Σs(5090) – s(2401) – 71
    25746 = Σs(5090) – ΣΦ(196883-196560) + Σφ(196883-196560)) – 47
    25770 = Σs(5090) – 4370 + s(4370) – 323 ; 4370 – s(4370) = s(194)

    Standish (1992) anomalitic:
    317.450267066044 = 1/(1/2/11.8627021700857-3/2/29.4701958106261+1/84.0331316671926)

  45. Paul Vaughan says:

    ˚Too *$ the conCOIIUSh’n’writeSAMicAB11y
    25674 = Σs(5090)-Σs(242)-47-average(59*59,-47*71)
    242 = 71 + 171
    714463914.954807 = beat(29.42351935,29.4235181382615) —– (Seidelmann1992,NASA’factsheet’)
    1714009140.222 = beat(29.447498973306,29.4474984673838) — (Seidelmann1992,Sidorenkov2017)
    IT$UNknown ‘weather friendly’ giants are in2cn moonSST!R UShine
    1225154850 = beat(1714009140.222,714463914.954807)
    2450309700, 4900619400, 9801238800, …
    3675464550, 7350929100, 14701858201, …
    (m)essturn wise sh!owe D-IT B(!w)(un)k(umou)
    242 = ΣΣΔ(178)-ΣΣφ(178) = average(ΣΔ(220),Σδ(220)) = ΣΦ(163)-Σφ(163)
    29.4474987203449 = harmean(29.447498973306,29.4474984673838) — (Seidelmann1992,Sidorenkov2017)
    29.4235187441307 = harmean(29.42351935,29.4235181382615) —– (Seidelmann1992,NASA’factsheet’)
    36132.1889074394 = beat(29.4474987203449,29.4235187441307)
    25674 = beat(36132.1889074394,15009.1608487337) ; 15009.1608487337 = 5482096/365.25
    142 = ΣΔ(220) = 2*71
    242 = average(142,342) = 2*11^2 = 71+171
    342 = Σδ(220) = 2*171

  46. Paul Vaughan says:

    4724 to 5256

    25761.5669315114 = beat(1.00001743371442,0.999978616353183) —- Seidelmann1992
    25768.5314808954 = harmean(25775.4997969807,25761.5669315114) —————————-
    25775.4997969807 = beat(1.00001741273101,0.999978616353183) —- Sidorenkov2017

    25773.8517155112 = beat(1.0000174152119,0.999978616353183) — Standish1992 (mediates)

    5256.6393995685 J
    5254.8227273181 S
    5258.32377697949 U
    5255.227007452 N
    5256.25286945517 = JSUN harmean
    5256.24218758401
    = beat( beat( 25808.2447032344, 25773.8517155112 / 2 ) / 6 , 2362.08778401782 )

  47. Paul Vaughan says:

    anomalistic-nodal 2320 years

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857) — anomalistic
    61.0124738503575 = slip(29.4511026866654,11.862499899747) —– nodal
    851.495746676794 = slip(61.0124738503575,19.8630730877524) — nodal
    derivation (via generalized Bollinger method)
    16.5866057119762 = axial(851.49574667679,16.9161211952138)
    100.762038265834 = slip(19.8549641949401,16.5866057119762)
    2320.03347068461 = slip(100.762038265834,16.9161211952138)

    2320.03347068424 = 1/(-7/2/11.8627021700857+17/2/29.4701958106261-12/11.862499899747+30/29.4511026866654)

    purely anomalistic 2362 years derived above
    2362.08778401334 = 1/(31/11.8627021700857-77/29.4701958106261)

    combine
    130310.598028785 = beat(2362.08778401334,2320.03347068424)
    130310.598021243 = 1/(-69/2/11.8627021700857+171/2/29.4701958106261-12/11.862499899747+30/29.4511026866654)

    compare
    130192.356944535 = beat(2362.05130068208,2319.96076275948) — NASA ‘factsheet’ tropical
    orbital invariant with harmonic means of Seidelmann & NASA ‘factsheet’ tropical:
    130476.603432851 = 1/(3/11.8565239747501-8/29.4235187441307-2/83.7474063531896+7/163.72320389271)
    130762.093817962 = beat(2361.92512664087,2320.01916295313) — Seidelmann tropical

    130704.452624679 = beat(173901.37537739,74619.9907876555) — La(2004a,2010a)average

  48. Paul Vaughan says:

    nodal-anomalistic 836 years

    9.93153654387618 = 19.8630730877524 / 2
    19.8630730877524 = beat(29.4511026866654,11.862499899747) —————– nodal
    16.912768715208 = harmean(29.4511026866654,11.862499899747) ————- nodal

    6.57189983390616 = axial( 29.4701958106261 / 2 , 11.8627021700857 ) —- anomalistic

    10.7485186386365 = beat(16.912768715208,6.57189983390616)
    130.662968425853 = slip(19.8630730877524,10.7485186386365)
    835.601801700049 = slip(130.662968425853,9.93153654387618)

    836 = 11 * ( mod(163,24) + mod(67,24) + mod(43,24) + mod(19,24) )
    836 is the smallest weird number that is also an untouchable number”

  49. Paul Vaughan says:

    JS Heart

    4.96576827193809 = 19.8630730877524 / 4
    9.93153654387618 = 19.8630730877524 / 2
    19.8630730877524 = beat(29.4511026866654,11.862499899747) —————– nodal

    2.114096089401 = 8.456384357604 / 4
    4.228192178802 = 8.456384357604 / 2
    8.456384357604 = axial(29.4511026866654,11.862499899747)
    16.912768715208 = harmean(29.4511026866654,11.862499899747) ————- nodal

    6.57189983390616 = axial( 29.4701958106261 / 2 , 11.8627021700857 ) —- anomalistic

    set up generalized Bollinger method

    10.7485186386365 = beat(16.912768715208,6.57189983390616)
    5.37425931931824 = 10.7485186386365 / 2
    2.68712965965912 = 10.7485186386365 / 4

    4.73283332391597 = axial(16.912768715208,6.57189983390616)
    2.36641666195799 = 4.73283332391597 / 2
    1.18320833097899 = 4.73283332391597 / 4

    130.662968425853 = slip(19.8630730877524,10.7485186386365)
    65.3314842129265 = slip(19.8630730877524,5.37425931931824)
    50.6802153502585 = slip(19.8630730877524,2.68712965965912)

    100.895781229192 = slip(19.8630730877524,4.73283332391597)
    50.4478906145961 = slip(19.8630730877524,2.36641666195799)
    93.4596178908348 = slip(19.8630730877524,1.18320833097899)

    derive 317, 836, 1470, 1800, 4670, 100ka, & more (some omitted for now)

    2937.90591403587 = slip(100.895781229192,16.912768715208)
    1468.95295701793 = slip(100.895781229192,8.456384357604)
    734.476478508967 = slip(100.895781229192,4.228192178802)
    734.476478508967 = slip(50.4478906145961,4.228192178802)

    1798.69939643745 = slip(93.4596178908348,8.456384357604)
    899.349698218727 = slip(93.4596178908348,4.228192178802)

    317.021047394046 = slip(100.895781229192,4.96576827193809)
    317.021047394066 = slip(93.4596178908348,19.8630730877524)

    835.601801700049 = slip(130.662968425853,9.93153654387618)
    835.601801700049 = slip(309.763646655713,130.662968425853)
    835.601801700042 = slip(225.988187105058,130.662968425853)
    835.60180170002 = slip(130.662968425853,112.994093552529)
    where
    309.763646655713 = slip(130.662968425853,19.8630730877524)
    225.988187105058 = slip(65.3314842129265,19.8630730877524)
    112.994093552529 = slip(50.6802153502585,19.8630730877524)

    4670.02054486764 = slip(1344.24031754827,130.662968425853)
    where
    1344.24031754827 = slip(130.662968425853,4.228192178802)
    1344.24031754825 = slip(144.731137488337,130.662968425853)
    144.731137488337 = slip(65.3314842129265,4.228192178802)

    198956.792259811 = slip(14755.2250793229,130.662968425853)
    99478.3961299054 = slip(14755.2250793229,65.3314842129265)
    where
    14755.2250793229 = slip(50.6802153502585,16.912768715208)

    compare sidereal vs. anomalistic
    16.9132450828034 = harmean(29.4571309198874,11.861990807677)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    99476.8155050052 = beat(16.9161211952138,16.9132450828034)
    99477 review

  50. Paul Vaughan says:

    Seidelmann (1992) sidereal earth

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)

    1.00001743390371 = (1-(1/σ(σ(σ(73))))^1)^(0/1)/(1-(1/σ(σ(σ(73))))^2)^(2/2)/(1-(1/σ(σ(σ(73))))^3)^(3/3)/(1-(1/σ(σ(σ(73))))^4)^(2/4)/(1-(1/σ(σ(σ(73))))^5)^(5/5)/(1-(1/σ(σ(σ(73))))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(209))^1)^(0/1)/(1-(1/σ(209))^2)^(2/2)/(1-(1/σ(209))^3)^(3/3)/(1-(1/σ(209))^4)^(2/4)/(1-(1/σ(209))^5)^(5/5)/(1-(1/σ(209))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(47+59+71))^1)^(0/1)/(1-(1/σ(47+59+71))^2)^(2/2)/(1-(1/σ(47+59+71))^3)^(3/3)/(1-(1/σ(47+59+71))^4)^(2/4)/(1-(1/σ(47+59+71))^5)^(5/5)/(1-(1/σ(47+59+71))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(average(320,158)))^1)^(0/1)/(1-(1/σ(average(320,158)))^2)^(2/2)/(1-(1/σ(average(320,158)))^3)^(3/3)/(1-(1/σ(average(320,158)))^4)^(2/4)/(1-(1/σ(average(320,158)))^5)^(5/5)/(1-(1/σ(average(320,158)))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^1)^(0/1)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^2)^(2/2)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^3)^(3/3)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^4)^(2/4)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^5)^(5/5)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^6)^(1/6)

    keywords: Schneider, E8, monster, Ramanujan
    73: lowest prime congruent to 1 mod 24

  51. Paul Vaughan says:

    another 317 year note

    744 = σ(743)
    743.744122286576 = slip(317.021047394066,130.662968425853)
    744 = σ(240)
    240 = σ(σ(σ(73))) = σ(average(320,158)) = σ(average(28+163+67+43+19,10+13+18+22+37+58))

    examples:

    104 = d(2,1/2,58) = R(2,1/2,58) – R(1,1/2,58)
    104 = d(4,1/2,58) = R(4,1/2,58) – R(1,1/2,58)
    R(4,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π
    R(2,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π
    R(1,1/2,58) = 0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π

    744 = d(3,1/2,28) = R(3,1/2,28) – R(1,1/2,28)
    R(3,1/2,28) = 744.01187441498 = ⌊(e^√28π)^(1/3)⌉^3 – e^√28π
    R(1,1/2,28) = 0.0118744149804115 = ⌊(e^√28π)^(1/1)⌉^1 – e^√28π

    The middle argument “1/2” corresponds to “√” — i.e. raise to power 1/2; e.g. 28^(1/2) = √28

  52. Paul Vaughan says:

    744 = σ(240)
    240 =σ(209) = σ(47+59+71) = σ(average(ΣΦ(216),Σφ(216)))
    216 = 378-ΣΔ(378) = Φ(ΣΦ(323)+Σφ(323))

    1.00001743390371 = (1-(1/σ(average(ΣΦ(216),Σφ(216))))^1)^(0/1)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^2)^(2/2)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^3)^(3/3)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^4)^(2/4)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^5)^(5/5)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^6)^(1/6)

    Seidelmann (consciously? or not?) simply substituted 240 for Schneider‘s phi.

  53. Paul Vaughan says:

    Seidelmann (1992) synodic helps clarify bias structure.

    11.8619992845449 = 1/(-1/1.0920848733744+1/1.00001743301243) — note sidereal earth period
    29.4571720000365 = 1/(-1/1.03515920602327+1/1.00001743301243)

    936.955612647599 = 1/(178/11.8619992845449-442/29.4571720000365)
    1800.77215359294 = 1/(29/11.8619992845449-72/29.4571720000365)

    11.8619992845449 = beat(1.0920848733744,1.00001743301243)
    29.4571720000365 = beat(1.03515920602327,1.00001743301243)

    set up generalized Bollinger method

    19.8589101021728 = beat(29.4571720000365,11.8619992845449)
    9.92945505108639 = 19.8589101021728 / 2
    4.9647275255432 = 19.8589101021728 / 4

    16.9132604709107 = harmean(29.4571720000365,11.8619992845449)
    8.45663023545537 = axial(29.4571720000365,11.8619992845449)
    4.22831511772768 = 8.45663023545537 / 2
    2.11415755886384 = 8.45663023545537 / 4

    10.7442781148351 = beat(16.9132604709107,6.57038853131229)
    5.37213905741754 = 10.7442781148351 / 2
    2.68606952870877 = 10.7442781148351 / 4

    9.46417591360268 = harmean(16.9132604709107,6.57038853131229)
    4.73208795680134 = axial(16.9132604709107,6.57038853131229)
    2.36604397840067 = 4.73208795680134 / 2
    1.18302198920033 = 4.73208795680134 / 4

    derive 936 & 1800

    130.930052601617 = slip(19.8589101021728,10.7442781148351)
    65.4650263008087 = slip(19.8589101021728,5.37213905741754)

    506.04166179034 = slip(130.930052601617,16.9132604709107)
    703.826579261491 = slip(130.930052601617,9.92945505108639)

    1800.77215358773 = beat(703.826579261491,506.04166179034)

    351.913289630745 = slip(130.930052601617,4.9647275255432)
    936.955612639497 = slip(351.913289630745,65.4650263008087)

    3747.82245055826 = slip(130.930052601617,4.22831511772768) ; / 4 = 936.955612639565

  54. Paul Vaughan says:

    Introducing Moonshine Bias

    1.0000262476142 —- Standish (1992) anomalistic earth period
    1.00002624761586 = 30/(15/0.0745030006844627+16/0.0754402464065708-31/0.0808503463381246-1/2364963.50364963+8/7201)
    2364963.50364963 = beat(74619.9907876555,72337.575351641) — La(2004a,2010a)average

    7201: 1, 19, 379, 7201; harmean = 3.79
    7201 = 19 + ( 19 * 378 )

    7200.99967683779 = 8/(30/1.0000262476142-15/0.0745030006844627-16/0.0754402464065708+31/0.0808503463381246+1/2364963.50364963)

    7201.000000 = 8/(30/1.00002624761586-15/0.0745030006844627-16/0.0754402464065708+31/0.0808503463381246+1/2364963.50364963)
    _____________________________________________________________________________________________
    monstrous error:
    -0.000023823837 seconds ( temporal (not angular) ) per century

  55. Paul Vaughan says:

    supplementary art dove wiser
    118239.762151809 = beat(1814.75583949423,1787.32381267774)

    1800.249870433 = harmean(2364795.24303618,900.467685151888)
    1800.2498895435 = harmean(2364861.19623035,900.467685151888)
    1800.24991918585 = harmean(2364963.50364963,900.467685151888)

    scale(un)80/4
    84.021214079097 = beat(1.0120629705681,1.00001743390371)
    164.770564556546 = beat(1.00612375085558,1.00001743390371)

    111.291642790288 = harmean(164.770564556546,84.021214079097)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)

    788287.065410296 = beat(111.307357343015,111.291642790288)
    2364861.19623089 = 3 * 788287.065410296

  56. Paul Vaughan says:

    Swiss bunkers watch M11

    0 = ⌊(70^2-55)/2^x-2^(2^2^2-x)*27/365.25⌉, x=0,1,2,…

    317.44 = s(s(608))*(2/(Φ+φ))^4
    317.44 = 496*(4/5)^2 = 496*16/25

    What Fairbridge said was:
    ‘[…] 317.749 years […]. A storminess record in geomorphic (that is, physical) form is preserved in a “staircase” of 184 isostatically uplifted beach lines on Hudson Bay (Fairbridge and Hillaire-Marcel” 1977, Nature. Vol. 268), which date back to more than 8,000 years. Their extraordinary regularity is duplicated in other parts of the Arctic, which denies any theory of randomness in storminess cycles. […]’

    25684 = Σs(5090) – 509 ; 509 * 59 = 30031; 30031 lowest primorial+1 not prime
    25808 = beat(beat(25746,25684),25746) = 1/(2/25746-1/25684)

  57. Paul Vaughan says:

    Phi(un)ally Perfect
    sidereal JS estimates

    x = average(Φ,φ)√√(y/496); search “why?” = 317.45026706604

    106975.540995836 = beat(1.00001743390371,1.00000808573393)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857) — anomalistic
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)

    with sense(un)a dove mercury:
    231842.576028623 = 1 / g_1
    σ(25808) = 50034; * 2 = 100068
    198928.033687742 = beat(231428.571428571,106975.540995836)
    16.9132447127788 = axial(99464.016843871,16.9161211952138)
    19.8589044939136 = beat(100068,19.8549641949401)
    14.7285651846795 = beat(19.8589044939136,8.45662235638942)
    5.93099526646578 = axial(19.8589044939136,8.45662235638942)

    19.8589044939136 = beat(29.4571303693591,11.8619905329316) — compare Standish sidereal

    4.73208318183218 = axial(16.9132447127788,6.57038170393041)
    2.36604159091609 = 4.73208318183218 / 2
    1.18302079545805 = 4.73208318183218 / 4

    100.985212897808 = slip(19.8589044939136,4.73208318183218)
    50.4926064489041 = slip(19.8589044939136,2.36604159091609)
    93.0623385488049 = slip(19.8589044939136,1.18302079545805) ~= 744/8
    ⌊744.498708390439⌉ = 744

    3455.67836213557 = slip(50.492606448904,16.9132447127788)
    enlightenin’12sense: aro(un)d‘ve 3456 sing un
    1727.83918106778 = slip(50.492606448904,8.45662235638942)

    1186.17557935921 = slip(100.985212897808,19.8589044939136)

    3455.67836213692 = slip(100.985212897808,16.9132447127788)
    3455.67836213692 = slip(50.4926064489041,16.9132447127788)
    1727.83918106846 = slip(100.985212897808,8.45662235638942)
    1727.83918106846 = slip(50.4926064489041,8.45662235638942)

    ⌊1727.83918106846⌉ = 1728
    no. won nos. weather myth or math

  58. Paul Vaughan says:

    at the Heart of Fleur Dehli nos.

    317.021047394046 = “why?”
    2876.43697242977 = beat(1.00001743390371,0.999669890283597)
    2876.43340223995 = (5+Φ)*2^9

  59. Paul Vaughan says:

    100, 496, 836, 73500

    19.8589044939136 = beat(28.8814946499969,11.7675450749242)
    9.92945224695679 = 19.8589044939136 / 2
    4.9647261234784 = 19.8589044939136 / 4

    16.7218853102037 = harmean(28.8814946499969,11.7675450749242)
    8.36094265510182 = axial(28.8814946499969,11.7675450749242)
    4.18047132755091 = 8.36094265510182 / 2
    2.09023566377546 = 8.36094265510182 / 4

    10.5902904904416 = beat(16.7218853102037,6.48390755374098)
    5.29514524522082 = 10.5902904904416 / 2
    2.64757262261041 = 10.5902904904416 / 4

    159.124845970951 = slip(19.8589044939136,10.5902904904416)

    1246.02534363448 = slip(159.124845970951,2.09023566377546); coefficient 76
    7350.70156103038 = slip(1246.02534363448,159.124845970951)

    2492.05068726896 = slip(159.124845970951,4.18047132755091)
    7350.70156103038 = slip(2492.05068726896,159.124845970951)

    79.5624229854757 = slip(19.8589044939136,5.29514524522082)
    3675.35078051519 = slip(1246.02534363448,79.5624229854757)

    3115.06335908839 = slip(79.3092165212675,4.9647261234784)
    7350.70156127398 = slip(3115.06335908839,159.124845970951)

    12460.2534363458 = slip(159.124845970951,19.8589044939136)
    30082.0055397318 = slip(12460.2534363571,4.18047132755091)
    323010.783812806 = slip(30082.0055397318,79.5624229854757); coefficient 378

    simple enough

  60. Paul Vaughan says:

    plate 000˚k (Alive in 1728 over herd “he’s just taco˚k shun”)

    quote:
    12 = 71-59 = 59-47
    3^3 = 27
    4^3 = 64
    5^3 = 125; 125+64+27 = 216; 216000 = 125*64*27
    6^3 = 216

    supplement:

    Φ(25771) = 25770
    σ(σ(25770)) = 216216

    σ(σ(25770)) = (3^3*4^3*5^3)+(6^3)
    σ(σ(25770)) = (3^3*4^3*5^3)+(3^3+4^3+5^3)

    1 / g_3 =
    74619.9907876555 — La2011 Table 6 La2004a
    74619.9907876555 — La2011 Table 6 La2010a
    74621: σ(σ(74621)) = σ(σ(Φ(25771))) = σ(σ(25770)) = 216216
    74626.0277273697 — La2011 Table 5

  61. Paul Vaughan says:

    with NASA ‘factsheet’ synodic
    1.00001743356471 = 1/(1/100/(φφ/(1/11.8629550321199+1/29.4600280504908)+1/(1/11.8629550321199-1/29.4600280504908))+1/√5/√√(1/(1/11.8629550321199-1/29.4600280504908)/496))

  62. Paul Vaughan says:

    blaring Soundgarden’s “black hole sun”
    speedboat cruised at sunset
    past an island up the inlet between the mountains
    starboard featured nice blue pentagon design

    understand how whoever came with that for the ‘factsheet’?

    above used right side of illustration
    x = average(Φ,φ)√√(y/496)

    left side:
    z = 2*average(Φ,φ)√√(u/496) = (√5)*(√√(u/496))
    ‘factsheet’ synodic: substitute u =
    19.8602908360448 = beat(29.4600280504908,11.8629550321199)

    4200.19506151223 = beat(1.00025558289712,1.00001743371442)
    underscore this: try same calculation on EVERY other parameter list ever featured at the talkshop — you won’t find ANYTHING even remotely close — TUNING is the obvious interpretation

    compare:
    8.45735138020641 = axial(29.4600280504908,11.8629550321199)
    22.1416333681812 = φφ * 8.45735138020641
    42.001924204226 = 22.1416333681812 + 19.8602908360448 —- review
    4200.1924204226 = 100 * 42.001924204226

    a little algebra, arrive (it isn’t really curious) at a very precise estimate of their (lunisolar-biased) sidereal earth year-length

    recommendation: stop the sanctions now (PLEASE: don’t delay)

  63. Paul Vaughan says:

    Mayan Sun

    review — notes shared a few years ago now connected with k & s_3
    25684.9315068493 = 360*60*60/50.4576
    68756.6342763388 = beat(41001.6165713381,25684.9315068493)
    68756.6342763387 = 1/(1/360/60/60*50.4576-1/2^9/5^3/13/(5256.63940169013)*(44.2784629967671)*(73.0002008969005)*11*3)
    41001.6165713381 = 2^9*5^3*13*(5256.63940169013)/( 44.2784629967671)/(73.0002008969005)/11/3
    5125.20207141727 = 41001.6165713381 / 8
    394.246313185944 = 5125.20207141727 / 13
    143998.465891166 = 394.246313185944 * 365.25

    NASA ‘factsheet’ synodic
    carefully scrutinize 365.256 days in concert with scaling featured in last few comments
    11.8631499061245 = beat(1.09207392197125,1.00001642710472)
    29.4605119934568 = beat(1.03515400410678,1.00001642710472)
    83.9387085475763 = beat(1.01207392197125,1.00001642710472)
    164.501359353944 = beat(1.00613278576318,1.00001642710472)
    68753.7838334262 = 1/(-2067/11.8631499061245+6233/29.4605119934568-2075/83.9387085475763-2075/164.501359353944)
    41002.6302294426 = beat(68753.78398,25684.9315068493)
    5125.32877868033 = 41002.6302294426 / 8
    394.256059898487 = 5125.32877868033 / 13
    144002.025877922 = 394.256059898487 * 365.25

    baktun balance with Standish (1992) long-duration sidreal earth year-length
    11.8632889636452 = beat(1.09207392197125,1.0000174152119)
    29.4613695891215 = beat(1.03515400410678,1.0000174152119)
    83.9456708028269 = beat(1.01207392197125,1.0000174152119)
    164.528101664775 = beat(1.00613278576318,1.0000174152119)
    5125.26068671143 = 1/(32.5/11.8632889636452-81.5/29.4613695891215+1.5/83.9456708028269+1.5/164.528101664775)
    5125.26068671143 = 2/(65/11.8632889636452-163/29.4613695891215+3/83.9456708028269+3/164.528101664775)
    68755.3156707909 = beat(41002.0854936914,25684.9315068493)
    41002.0854936914 = 5125.26068671143 * 8
    5125.26068671143 = 41002.0854936914 / 8
    394.250822054725 = 5125.26068671143 / 13
    144000.112755488 = 394.250822054725 * 365.25

    generalized Bollinger (1952) method used to derive all of the preceding

  64. Paul Vaughan says:

    typo near end of lunisolar 1800 comment

    supplementary
    68952.9911154182 = slip(1800.93537030388,0.999978616353183) — M&S unrounded
    68961.1021023274 = slip(1800.93537030388,0.999978614647502) —— M&S rounded

  65. Paul Vaughan says:

    The Real Thing

    generalized Bollinger setup

    NASA ‘factsheet’ “Perihelion (10^6 km)”
    18.5132744565067 = beat(27.1859355652811,11.0133345026533)
    9.25663722825335 = 18.5132744565067 / 2
    4.62831861412668 = 18.5132744565067 / 4
    15.6761007011677 = harmean(27.1859355652811,11.0133345026533)
    7.83805035058385 = axial(27.1859355652811,11.0133345026533)
    3.91902517529192 = 7.83805035058385 / 2
    1.95951258764596 = 7.83805035058385 / 4

    crossed with Standish (1992) anomalistic
    11.3158491444058 = beat(15.6761007011677,6.57189983390616)
    5.65792457220292 = 11.3158491444058 / 2
    2.82896228610146 = 11.3158491444058 / 4
    9.26121549052347 = harmean(15.6761007011677,6.57189983390616)
    4.63060774526174 = axial(15.6761007011677,6.57189983390616)
    2.31530387263087 = 4.63060774526174 / 2
    1.15765193631543 = 4.63060774526174 / 4

    hierarchy top levels
    50.8673777758235 = slip(18.5132744565067,11.3158491444058)
    68.039402479501 = slip(18.5132744565067,5.65792457220292)
    40.616453731329 = slip(18.5132744565067,2.82896228610146)

    9362.47281505814 = slip(18.5132744565067,4.63060774526174)
    4681.23640752907 = slip(18.5132744565067,2.31530387263087)
    2340.61820376454 = slip(18.5132744565067,1.15765193631543)

    33052.6407001566 = slip(9362.47281505814,18.5132744565067)

  66. Paul Vaughan says:

    Standish anomalistic JS
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    crossed with nasa ‘factsheet’ synodic (using Standish long-duration earth sidereal)
    10.7452986346853 = beat(16.9161211952138,6.57120187976141)
    130.437274099443 = slip(19.8549641949401,10.7452986346853)
    451.069113778482 = slip(130.437274099443,16.9161211952138)
    984.586552944021 = slip(451.069113778482,130.437274099443)
    980.005077317815 = beat(984.586552944021,491.145236537217)
    73500.3807988361 = 980.005077317815 * 75
    36750.1903994181 = 73500.3807988361 / 2

  67. Paul Vaughan says:

    33052.6407001566 ; / 8 / √φ = 5255.45105486596 ; / 8 / √φ = 835.629626106117
    36133.4834429326 = beat(29.4474984673838,29.42351935)
    36133.4519209772 = axial( 500 * 835.629626106117 , Φ * 64000 )
    36133.2446397442 = axial( 500 * 835.546575435627 , Φ * 64000.2003306117 )

    11.8619906635942 = axial( 250 * 835.629626106117 , 11.8627021700857 )
    11.861990807677 — compare Standish sidereal

  68. Paul Vaughan says:

    JS Seidelmann short-duration
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    9.93251804323141 = 19.8650360864628 / 2
    4.9662590216157 = 19.8650360864628 / 4
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)
    8.4561457463176 = axial(29.4474984673838,11.8626151546089)
    4.2280728731588 = 8.4561457463176 / 2
    2.1140364365794 = 8.4561457463176 / 4

    crossed with Standish anomalistic
    10.7487113950462 = beat(16.9122914926352,6.57189983390616)
    5.37435569752312 = 10.7487113950462 / 2
    2.68717784876156 = 10.7487113950462 / 4
    9.46559190444059 = harmean(16.9122914926352,6.57189983390616)
    4.73279595222029 = axial(16.9122914926352,6.57189983390616)
    2.36639797611015 = 4.73279595222029 / 2
    1.18319898805507 = 4.73279595222029 / 4

    generalized Bollinger top-level
    130.804508068333 = slip(19.8650360864628,10.7487113950462)
    65.4022540341667 = slip(19.8650360864628,5.37435569752312)
    50.6080139018586 = slip(19.8650360864628,2.68717784876156)

    100.676696558682 = slip(19.8650360864628,4.73279595222029)
    50.3383482793411 = slip(19.8650360864628,2.36639797611015)
    94.2634877779342 = slip(19.8650360864628,1.18319898805507)

    ~ 2 * 1470 :
    2940.22366441706 = slip(639.79104597475,65.4022540341667)
    639.79104597475 = slip(94.2634877779342,8.4561457463176)

    astronomical delight
    23098.5346098618 = slip(2081.08715717028,130.804508068333) ~= 23.1 ka
    2081.08715717028 = slip(130.804508068333,4.2280728731588)

    concise lunisolar review
    29976.553895592 = beat(179.333323110834,178.266850068779) — M&S rounded tropical
    29971.9562539596 = beat(179.333487684639,178.266850068779) — M&S unrounded tropical
    contains
    29972.4308734593 = slip(6642.042967895,130.804508068333)
    6642.042967895 = slip(50.6080139018586,16.9122914926352)
    50.6080139018586 = slip(19.8650360864628,2.68717784876156)

  69. David A says:

    Your foray into orbital resonance is fascinating. To even understand orbital resonance I found this video https://youtu.be/Qyn64b4LNJ0 very explanatory to me. You all may enjoy it, not for what you already know, but for the last third or so, where the video gets into converting orbital resonance into sound. Very interesting and cool.

  70. oldbrew says:

    David A – in reality there are no *exact* orbital resonances that we know of. The true resonances are synodic, i.e. based on periods when two bodies are in line with the body they’re orbiting.

    However the Galilean moons of Jupiter are very close to an exact orbital resonance, but see here:

    Why Phi? – the resonance of Jupiter’s Galilean moons

  71. Paul Vaughan says:

    David A: reference framing & sampling/aggregation biases (not resonance) have been the more memorable exploration spices. (Look for “anomalistic periods” on the net, find little, wonder why.)

  72. Paul Vaughan says:

    no. fancy policy

    132942.038841156 = slip(2493.45557752384,50.8673777758235)
    2493.45557752384 = slip(50.8673777758235,3.91902517529192)

    anomalistic purpose?

    132942.038841155 = beat(2361.4060221632,2320.1931882465)
    2340.61820376454 = harmean(2361.4060221632,2320.1931882465)

    Dow

    non-Mayan


    pleas

  73. Paul Vaughan says:

    28.8 & 29.8 ka note

    background

    28861.8479491577 = slip(521.471474393677,65.3314842129265)
    521.471474393677 = slip(93.4596178908348,4.96576827193809)

    1268.08418957619 = slip(100.895781229192,19.8630730877524)
    59572.1478647727 = slip(1268.08418957619,50.6802153502585) ; / 2 =
    29786.0739323864

    image disappeared from here

  74. Paul Vaughan says:

    Vert

    anomalistic
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    9.92748209747005 = 19.8549641949401 / 2
    4.96374104873503 = 19.8549641949401 / 4
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    8.45806059760692 = axial(29.4701958106261,11.8627021700857)
    4.22903029880346 = 8.45806059760692 / 2
    2.11451514940173 = 8.45806059760692 / 4

    cross with nodal
    10.7419198566295 = beat(16.9161211952138,6.56993811757712)
    5.37095992831474 = 10.7419198566295 / 2
    2.68547996415737 = 10.7419198566295 / 4
    9.46415641396783 = harmean(16.9161211952138,6.56993811757712)
    4.73207820698391 = axial(16.9161211952138,6.56993811757712)
    2.36603910349196 = 4.73207820698391 / 2
    1.18301955174598 = 4.73207820698391 / 4

    top level
    130.937221254586 = slip(19.8549641949401,10.7419198566295)
    65.4686106272931 = slip(19.8549641949401,5.37095992831474)
    50.4636420418041 = slip(19.8549641949401,2.68547996415737)

    101.392224425295 = slip(19.8549641949401,4.73207820698391)
    50.6961122126475 = slip(19.8549641949401,2.36603910349196)
    91.6213954766362 = slip(19.8549641949401,1.18301955174598)

    so what?
    323.049985568367 = slip(130.937221254586,19.8549641949401)
    220.179370109776 = slip(65.4686106272931,19.8549641949401)
    110.089685054889 = slip(50.4636420418041,19.8549641949401)
    504.343543575558 = slip(130.937221254586,16.9161211952138)

    no. doubt:
    2998.18929479554 = slip(50.4636420418041,16.9161211952138)
    1499.09464739777 = slip(50.4636420418041,8.45806059760692)

    86900.0492950508 = slip(504.343543575558,50.4636420418041)
    86900 = 11 * 25 * 316

  75. Paul Vaughan says:

    con$hiver˚T review:

  76. Paul Vaughan says:

    JEV nodal-anomalistic (casual intro)

    nodal

    320.347986973937 = slip(44.2875117414503,0.761762061330659)
    320.347986973937 = slip(22.1437558707251,0.761762061330659)
    23.7880850683634 = slip(11.0718779353626,0.761762061330659) ———-

    160.173993486969 = slip(44.2875117414503,0.38088103066533)
    160.173993486969 = slip(22.1437558707251,0.38088103066533)
    160.173993486969 = slip(11.0718779353626,0.38088103066533)

    99.0755539755134 = slip(44.2875117414503,0.190440515332665) ———-
    80.0869967434843 = slip(22.1437558707251,0.190440515332665)
    80.0869967434843 = slip(11.0718779353626,0.190440515332665)

    anomalistic

    575.799247511725 = slip(44.2411450188424,0.761769224080824)
    575.799247511725 = slip(22.1205725094212,0.761769224080824)
    23.0043337056467 = slip(11.0602862547106,0.761769224080824) ———–

    287.899623755862 = slip(44.2411450188424,0.380884612040412)
    287.899623755862 = slip(22.1205725094212,0.380884612040412)
    287.899623755862 = slip(11.0602862547106,0.380884612040412)

    143.949811877931 = slip(44.2411450188424,0.190442306020206)
    143.949811877931 = slip(22.1205725094212,0.190442306020206)
    143.949811877931 = slip(11.0602862547106,0.190442306020206)

    reorganizing

    320 = 28+163+67+43+19 (the 744 levels)
    320.347986973937 = slip(44.2875117414503,0.761762061330659) — n
    575.799247511725 = slip(44.2411450188424,0.761769224080824) — a
    576 = 320+256 = 28+163+67+43+19 + 2^8 = 4 * 12^2

    320.347986973937 = slip(22.1437558707251,0.761762061330659) — n
    575.799247511725 = slip(22.1205725094212,0.761769224080824) — a

    23.7880850683634 = slip(11.0718779353626,0.761762061330659) — n
    23.0043337056467 = slip(11.0602862547106,0.761769224080824) — a

    160.173993486969 = slip(44.2875117414503,0.38088103066533) — n
    287.899623755862 = slip(44.2411450188424,0.380884612040412) — a
    288 = 160+128 = 2 * 12^2

    160.173993486969 = slip(22.1437558707251,0.38088103066533) — n
    287.899623755862 = slip(22.1205725094212,0.380884612040412) — a

    160.173993486969 = slip(11.0718779353626,0.38088103066533) — n
    287.899623755862 = slip(11.0602862547106,0.380884612040412) — a

    99 = 163-64 —————————————————————————————— note well
    99.0755539755134 = slip(44.2875117414503,0.190440515332665) — n
    143.949811877931 = slip(44.2411450188424,0.190442306020206) — a

    80 = 144-64
    80.0869967434843 = slip(22.1437558707251,0.190440515332665) — n
    143.949811877931 = slip(22.1205725094212,0.190442306020206) — a
    144 = 80+64

    80.0869967434843 = slip(11.0718779353626,0.190440515332665) — n
    143.949811877931 = slip(11.0602862547106,0.190442306020206) — a
    144 = 59*59 – 47*71 = 12^2

  77. Paul Vaughan says:

    Tortoise

    576 = 24^2
    23 = 99 – 76
    “Experts” misled us severely last decade.
    76 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24 = 19 + 19 + 19 + 19

    JEV nodal

    417.865531943236 = slip(160.173993486969,44.2875117414503)
    417.865531943179 = slip(99.0755539755134,44.2875117414503) ————
    417.865531943236 = slip(80.0869967434843,44.2875117414503)

    208.93276597159 = slip(99.0755539755134,22.1437558707251) ————-
    208.932765971618 = slip(80.0869967434843,22.1437558707251)

    compare with

    JEV anomalistic

    417.781912370365 = slip(49.480945018333,44.2411450188424)
    208.890956185182 = slip(49.480945018333,22.1205725094212)
    104.445478092591 = slip(49.480945018333,11.0602862547106)

    46 = 209 – 163 = 2 * 23
    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5

    835.563824740778 = slip(6.84967828238651,0.761769224080824)
    417.781912370389 = slip(6.84967828238651,0.380884612040412)
    208.890956185194 = slip(6.84967828238651,0.190442306020206)
    104.445478092597 = slip(6.84967828238651,0.095221153010103)

    Study carefully:

    2096.44010053095 = slip(135.584676517528,44.2411450188424)
    135.584676517528 = slip(44.2411450188424,1.59867106414771)

  78. Paul Vaughan says:

    supplementary
    49.480945018333 = slip(6.84967828238651,0.399667766036927)

    anomalistic
    835.56382473927 = 1/(-14/0.615197860179071+22/1.0000262476142+9/11.8627021700857)

    nodal
    835.731063859634 = 4/(-1387/0.615194395759546+2333/1.00001071395229-930/11.862499899747)

    derive from:

    anomalistic

    1.59867106414771 = beat(1.0000262476142,0.615197860179071)
    0.799335532073854 = 1.59867106414771 / 2
    0.399667766036927 = 1.59867106414771 / 4
    0.761769224080824 = harmean(1.0000262476142,0.615197860179071)
    0.380884612040412 = axial(1.0000262476142,0.615197860179071)
    0.190442306020206 = 0.380884612040412 / 2
    0.095221153010103 = 0.380884612040412 / 4

    0.814043420635227 = beat(11.8627021700857,0.761769224080824)
    0.407021710317613 = 0.814043420635227 / 2
    0.203510855158807 = 0.814043420635227 / 4
    1.43160709790728 = harmean(11.8627021700857,0.761769224080824)
    0.715803548953639 = axial(11.8627021700857,0.761769224080824)
    0.35790177447682 = 0.715803548953639 / 2
    0.17895088723841 = 0.715803548953639 / 4

    44.2411450188424 = slip(1.59867106414771,0.814043420635227)
    22.1205725094212 = slip(1.59867106414771,0.407021710317613)
    11.0602862547106 = slip(1.59867106414771,0.203510855158807)

    6.84967828238651 = slip(1.59867106414771,0.715803548953639)
    3.42483914119326 = slip(1.59867106414771,0.35790177447682)
    24.0670906604157 = slip(1.59867106414771,0.17895088723841)

    nodal

    1.59868736807262 = beat(1.00001071395229,0.615194395759546)
    0.799343684036311 = 1.59868736807262 / 2
    0.399671842018155 = 1.59868736807262 / 4
    0.761762061330659 = harmean(1.00001071395229,0.615194395759546)
    0.38088103066533 = axial(1.00001071395229,0.615194395759546)
    0.190440515332665 = 0.38088103066533 / 2
    0.0952202576663324 = 0.38088103066533 / 4

    0.81403619360271 = beat(11.862499899747,0.761762061330659)
    0.407018096801355 = 0.81403619360271 / 2
    0.203509048400678 = 0.81403619360271 / 4
    1.43159297613223 = harmean(11.862499899747,0.761762061330659)
    0.715796488066113 = axial(11.862499899747,0.761762061330659)
    0.357898244033057 = 0.715796488066113 / 2
    0.178949122016528 = 0.715796488066113 / 4

    44.2875117414503 = slip(1.59868736807262,0.81403619360271)
    22.1437558707251 = slip(1.59868736807262,0.407018096801355)
    11.0718779353626 = slip(1.59868736807262,0.203509048400678)

    6.84843333335951 = slip(1.59868736807262,0.715796488066113)
    3.42421666667975 = slip(1.59868736807262,0.357898244033057)
    24.1324516929722 = slip(1.59868736807262,0.178949122016528)

  79. Paul Vaughan says:

    sh: most replayed time index? 209
    99476.8446931352 = beat(130901699.437495,99401.3061146969)
    “…wwwo˚k T˚he lline like Can. edge? yep…”

  80. Paul Vaughan says:

    Call off ice$UNhhowe???0vertThe

    Horizon$1929.72222222222 (in sidereal hindsight NA!SA)
    99401.3061147312 = 2/(1/11.8619848807702+1/29.4571542179636-1/11.8627021700857-1/29.4701958106261)

    Sidellmann (1992) tropical wwwithconvert$yen
    25761.5669315114 = beat(1.00001743371442,0.999978616353183) give$ :
    99438.5571830642 = 2/(1/11.8619843895747+1/29.4571637875065-1/11.8627021700857-1/29.4701958106261)

    11.8619855226385 = beat(25746.362539063,11.8565229295003)
    29.4571824642891 = beat(25746.362539063,29.4235181382615)
    223486.379079769 = beat(11.8626151546089,11.8619855226385)
    89574.6192358693 = beat(29.4571824642891,29.4474984673838)
    63945.0695493213 = axial(223486.379079769,89574.6192358693)
    99438.3330129135 = harmean(111743.189539885,89574.6192358693)

    llunisolr biasXbias
    11.8619844613515 = 1/(-1/0.999978616353183+1/1.00001741273101-1/27500000+1/11.8565229295003)
    29.4571759194281 = 1/(-1/0.999978616353183+1/1.00001741273101-1/27500000+1/29.4235181382615)
    223110.291681529 = beat(11.8626151546089,11.8619844613515)
    89635.1786377776 = beat(29.457175919428,29.4474984673838)
    63945.0695493126 = axial(223110.291681529,89635.1786377776)
    99401.0566990933 = harmean(111555.145840765,89635.1786377776)

    catchSSTand!sh(1992)fi$[hh]calefidereal
    99476.8155050703 = 2/(1/11.861990807677+1/29.4571309198874-1/11.8627021700857-1/29.4701958106261)

    1.00001743390371 = (1-(1/σ(209))^1)^(0/1)/(1-(1/σ(209))^2)^(2/2)/(1-(1/σ(209))^3)^(3/3)/(1-(1/σ(209))^4)^(2/4)/(1-(1/σ(209))^5)^(5/5)/(1-(1/σ(209))^6)^(1/6)
    25761.4413131157 = beat(1.00001743390371,0.999978616353183)
    11.8619833699953 = beat(25761.4413131157,11.8565239747501)
    29.4571633444961 = beat(25761.4413131157,29.4235187441307)
    16.9132428669975 = harmean(29.4571633444961,11.8619833699953)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    99400.2228512541 = beat(16.9161211952138,16.9132428669975)

    $0 few Can˚T[hh]ellweather MI = Thor math f(ll0˚CDownTh(y/n)c$yen)$T˚ache99.9999per(inno$)centCOllatorwell

  81. Paul Vaughan says:

    ˚T˚win blew
    ˚T˚won Vert “on”
    “ain’t no cause
    5256.07266843706 = 4/(1/11.8627021700857-6/84.0331316671926-2/164.793624044745) — anomalistic
    “just abattery 4/hire” – easyDC “load up Eur. Can. on”

  82. Paul Vaughan says:

    ain’t Talkin’ (Down!)

    “more than N’folk song”
    2432.13579731858 = 2/(-1/29.4701958106261+7/84.0331316671926-8/164.793624044745)
    “$hh!keep” sayin’: Eur. IT
    1216.06789865929 = 1/(-1/29.4701958106261+7/84.0331316671926-8/164.793624044745) — anomalistic
    “give allITll˚K˚!˚C˚K˚ withh eur. fine”
    608.033949329645 = 1/(-2/29.4701958106261+14/84.0331316671926-16/164.793624044745)
    “V!$hh!UShypnoCsis”

    “$hh!achhe earn IT BRI˚C wall”

  83. Paul Vaughan says:

    ‘factsheet’ 41k ace:
    “Wrong_2Won˚T!mowwR˚Than a foe 11˚K s_on_g” – metR˚Ck
    41000.1976938051 = harmean(101554.605384649,25684.9315068493)
    406218.421537452 = 16/(-7/11.8619822039699+5/29.4571389459274+7/11.8627021700857-5/29.4701958106261)
    101554.605384363 = 4/(-7/11.8619822039699+5/29.4571389459274+7/11.8627021700857-5/29.4701958106261)
    buy us anomalistic sam pull period

    NA!SAhherdin=mowwR0˚CO2revverse
    68760.6246393283 = beat(41000.1976938051,25684.9315068493)
    C˚0llDCllear
    25684.888118931 = beat(25808.1319319395,12873.1812695315)
    25808.1319319395 = 1/(-1/83.7474058863792+2/163.723203285421+1/84.0331316671926-2/164.793624044745)
    412930.110911032 = 16/(-1/83.7474058863792+2/163.723203285421+1/84.0331316671926-2/164.793624044745)

  84. Paul Vaughan says:

    JEV nodal setup listed here

    That also leads to:
    702.588969079652 = slip(6.84843333335951,0.761762061330659)
    351.294484539826 = slip(6.84843333335951,0.38088103066533)
    175.647242269913 = slip(6.84843333335951,0.190440515332665)
    87.8236211349565 = slip(6.84843333335951,0.0952202576663324)
    alternately:
    13.696866666719 = slip(1.59868736807262,1.43159297613223)
    702.588969080693 = slip(13.696866666719,0.761762061330659)
    351.294484540347 = slip(13.696866666719,0.38088103066533)
    175.647242270173 = slip(13.696866666719,0.190440515332665)
    87.8236211350867 = slip(13.696866666719,0.0952202576663324)

    Why the systematically-structured sampling bias in the short-duration models was not acknowledged early in “climate discussion” is the crUShin’mystery.

  85. Paul Vaughan says:

    using this
    345.720408200766 = slip(130.937221254586,4.96374104873503)
    345.720408200718 = 1/(26/11.862499899747+52/29.4511026866654-61/11.8627021700857+35/29.4701958106261)
    345 review

  86. Paul Vaughan says:

    figure 7hherd dog in0˚Kin awe “what ruff?”
    691.440816401436 = 1/(13/11.862499899747+26/29.4511026866654-30.5/11.8627021700857+17.5/29.4701958106261)
    691.440816401531 = slip(130.937221254586,9.92748209747005)
    using this
    2998.18929479478 = 2/(24/11.862499899747+48/29.4511026866654-55/11.8627021700857+29/29.4701958106261)
    1499.09464739739 = 1/(24/11.862499899747+48/29.4511026866654-55/11.8627021700857+29/29.4701958106261)

  87. Paul Vaughan says:

    supplementary notes

    345.720408200718 = 1/(26/11.862499899747+52/29.4511026866654-61/11.8627021700857+35/29.4701958106261)
    691.440816401436 = 2/(26/11.862499899747+52/29.4511026866654-61/11.8627021700857+35/29.4701958106261)

    691.440816401528 = slip(323.049985568367,130.937221254586)
    691.440816401528 = slip(220.179370109776,130.937221254586)

    691.44081640155 = slip(130.937221254586,110.089685054889)
    345.720408200775 = slip(110.089685054889,65.4686106272931)

    345.720408200766 = slip(130.937221254586,4.96374104873503)
    691.440816401531 = slip(130.937221254586,9.92748209747005)

    323.049985568367 = slip(130.937221254586,19.8549641949401)

    504.343543575558 = slip(130.937221254586,16.9161211952138)
    504.343543575558 = slip(65.4686106272931,16.9161211952138)

    220.179370109776 = slip(65.4686106272931,19.8549641949401)
    220.128528176077 = slip(91.6213954766362,16.9161211952138)

    323 = 196883 – 196560
    504 = 220 + s(220) ; 220 = s(s(220))

    504.343543575571 = 2/(-16/11.862499899747-32/29.4511026866654+39/11.8627021700857-25/29.4701958106261)

    220.179370109781 = 1/(6/11.862499899747+12/29.4511026866654-14/11.8627021700857+8/29.4701958106261)

    323.049985568356 = 2/(-14/11.862499899747-28/29.4511026866654+33/11.8627021700857-19/29.4701958106261)

  88. Paul Vaughan says:

    distinction (from 220.17937010978)
    220.128528176035 = 2/(40/11.862499899747+80/29.4511026866654-149/11.8627021700857+191/29.4701958106261)

  89. Paul Vaughan says:

    AmereR0˚CO$pereRllX!

    “Perihelion (10^6 km)”
    58.0330166444241 = 1/(1/11.0133345026533-2/27.1859355652811)

    152.138222152323 = 1/(1/78.4417377748537-1/161.934271741915)
    /
    6.08396577991655 = 1/(1/11.0133345026533+2/27.1859355652811)
    ~25 = 5^2 = 317-163-67-43-19

    bias SAM+pu+ll+in’SSTop! goes withhout sayUN:JS˚T chll ouT˚

  90. Paul Vaughan says:

    Table 8 Vote$$well

    “BR˚tmI Pence$$sol.
    ‘Gov.ME$$sum˚Thhun˚T00‘wwrite Don”

    e+v=47+71=2*59
    e-v=58*2^3
    e=58*2^3+v
    “$wwheat $wwheat Wall Dough”
    58*2^3+v+v=47+71
    2v=47+71-58*2^3
    e-173=47+71
    e=47+71+173
    VA’n’hale unhhothh˚TeaChR “NOAA bout this $Ch˚˚ll”
    160.17399348638 = 1/(-173/0.615194395759546+291/1.00001071395229-116/11.862499899747)
    287.899623750069 = 1/(-173/0.615197860179071+291/1.0000262476142-116/11.8627021700857)
    D!monde Ave.: “$IT Down Wall Dough”

  91. Paul Vaughan says:

    M!55 UNllink (well flower)

    lofty weather dog’n’dig note orrery:
    132891.64579987 = 2/(1/29.4571309198874-1/29.4701958106261)

    2-PEace˚TypoV How$ensoch!ustech

    25808 = Σs(5090) – 7*55
    25808 = Σs(5090) – 7*(28^2-27^2)
    25808 = Σs(5090) – 7*(1^2+2^2+3^2+4^2+5^2)

    25808 = Σs(5090) – 7*(378-323)
    25808 = Σs(5090) – 7*(378+196560-196883)
    25808 = Σs(5090) – 7*11*√(317-163-67-43-19)

    5 = √(317-163-67-43-19)
    55 = 11*√(317-163-67-43-19)
    378 = 323+11*√(317-163-67-43-19)

    notion ignore˚Folk˚CUS$

  92. Paul Vaughan says:

    4266 (or 4267) years

    curiosity noted before

    NASA Horizons 1929.72222222222 sidereal
    60.9469869005405 = slip(29.4571542179636,11.8619848807702) ; * 70 =
    4266.28908303783

    standish sidereal long-duration
    4267.83999767789 = slip(164.786005834669,84.01495797691)
    standish sidereal SHORT-duration
    4266.98089346112 = slip(164.790305314929,84.0175261973943)

    4266.96721287228 = 1/(-10/11.8627021700857-10/29.4701958106261+10/11.862499899747+10/29.4511026866654)
    Standish anomalistic with nodal — insight from CAREFUL study of “Chandler Diversity”

    same line of inquiry clarifies:
    173927.260141636 = 1/(-4/11.8627021700857+4/11.862499899747)
    173901.37537739 = 1 / g_2 —– La(2004a,2010a)average

  93. Paul Vaughan says:

    automating discovery of 173 (or 174) ka (see preceding comment)

    anomalistic-nodal combos arising with generalized Bollinger

    4.73181582715489 = axial(16.912768715208,6.56993811757712)
    4.73207820698391 = axial(16.9161211952138,6.56993811757712)
    4.73283332391597 = axial(16.912768715208,6.57189983390616)
    4.73309581660092 = axial(16.9161211952138,6.57189983390616)
    quarter-beats
    4374.2817110478 = beat(1.18327395415023,1.18295395678872)
    5502.44887404215 = beat(1.18320833097899,1.18295395678872)
    5502.44887404534 = beat(1.18327395415023,1.18301955174598)
    7414.79116630378 = beat(1.18320833097899,1.18301955174598)
    21334.8360643294 = beat(1.18301955174598,1.18295395678872)
    21334.8360643774 = beat(1.18327395415023,1.18320833097899)

    21334.8360643294 = slip(5502.44887404534,4374.2817110478)
    21334.8360643774 = slip(7414.79116630378,5502.44887404534)
    24317.7838264046 = slip(7414.79116630378,4374.2817110478)

    173927.260144754 = slip(21334.8360643774,4374.2817110478)
    173927.260141567 = slip(21334.8360643774,5502.44887404534)
    173927.260141568 = slip(21334.8360643774,7414.79116630378)
    173927.260138382 = slip(24317.7838264046,21334.8360643774)

    alternately – via other side
    10.7419198566295 = beat(16.9161211952138,6.56993811757712)
    10.7432721445968 = beat(16.912768715208,6.56993811757712)
    10.7471650296459 = beat(16.9161211952138,6.57189983390616)
    10.7485186386365 = beat(16.912768715208,6.57189983390616)
    1/4 beats
    4374.28171104601 = beat(2.68712965965912,2.68547996415737)
    5502.44887404212 = beat(2.68679125741149,2.68547996415737)
    5502.44887404128 = beat(2.68712965965912,2.68581803614919)
    7414.79116630151 = beat(2.68679125741149,2.68581803614919)
    21334.8360643478 = beat(2.68581803614919,2.68547996415737)
    21334.8360643352 = beat(2.68712965965912,2.68679125741149)

    another way to look at it – building blocks
    pure
    1.18295395678872 = 1/(+6/11.862499899747+10/29.4511026866654)
    1.18327395415023 = 1/(6/11.8627021700857+10/29.4701958106261)
    crosses
    1.18301955174598 = 1/(2/11.8627021700857+2/29.4701958106261+4/11.862499899747+8/29.4511026866654)
    1.18320833097899 = 1/(4/11.8627021700857+8/29.4701958106261+2/11.862499899747+2/29.4511026866654)

    simple subtraction
    4374.28171104854 = 1/(-6/11.8627021700857-10/29.4701958106261+6/11.862499899747+10/29.4511026866654)
    5502.44887404102 = 1/(-4/11.8627021700857-8/29.4701958106261+4/11.862499899747+8/29.4511026866654)
    7414.7911662994 = 1/(-2/11.8627021700857-6/29.4701958106261+2/11.862499899747+6/29.4511026866654)
    21334.8360643551 = 1/(-2/11.8627021700857-2/29.4701958106261+2/11.862499899747+2/29.4511026866654)

    note with care:
    4266.96721287228 = 1/(-10/11.8627021700857-10/29.4701958106261+10/11.862499899747+10/29.4511026866654)
    arises in
    173927.260144754 = slip(21334.8360643774,4374.2817110478)
    173927.260141636 = 1/(-4/11.8627021700857+4/11.862499899747)

    so that clarifies another source of systematic bias in the short-duration models that attracted attention long ago but remained mysterious at the time (no longer so now)

  94. Paul Vaughan says:

    “Does A NY buddy no. how the SSTory really goes?”

    1.59868960462765 = beat(1.0000174152119,0.615197263077614)
    0.799344802313826 = 1.59868960462765 / 2

    0.814043321555892 = beat(11.861990807677,0.761766203759125)
    0.407021660777946 = 0.814043321555892 / 2
    0.203510830388973 = 0.814043321555892 / 4

    44.2698538014441 = slip(1.59868960462765,0.814043321555892)
    “0=range(C+R)-USh!…”
    22.1349269007221 = slip(1.59868960462765,0.407021660777946)
    “…˚Take IT away buoys”
    11.067463450361 = slip(1.59868960462765,0.203510830388973)

    143.424905399078 = slip(44.2698538014441,1.59868960462765)
    143.424905399078 = slip(22.1349269007221,1.59868960462765)
    143.424905399078 = slip(11.067463450361,1.59868960462765)
    71.7124526995389 = slip(22.1349269007221,0.799344802313826)
    71.7124526995389 = slip(11.067463450361,0.799344802313826)

    0.71+071 = 1*71*(101/100)
    1.42+142 = 2*71*(101/100)

  95. Paul Vaughan says:

    hhear dog Un?

    19.8589050137632 = beat(29.4571309198874,11.861990807677)
    6.57038184300286 = axial(14.7285654599437,11.861990807677)
    4.7320832829358 = axial(16.9132450828034,6.57038184300286)
    100.985205171956 = slip(19.8589050137632,4.7320832829358)
    1186.18276371796 = slip(100.985205171956,19.8589050137632) — ADJ0! (“well…B a dog”)
    3455.60863545854 = slip(100.985205171956,16.9132450828034) ~= 3456
    1727.80431772927 = slip(100.985205171956,8.4566225414017)

  96. Paul Vaughan says:

    Some (naively enough) believe expression reflects belief.

    casually exploring (just a little bit at a time) where some of the narratives came from

    Jupiter-Neptune Standish (1992) sidereal

    12.7821002221242 = beat(164.786005834669,11.861990807677)
    6.39105011106212 = 12.7821002221242 / 2
    3.19552505553106 = 12.7821002221242 / 4
    22.1309057968232 = harmean(164.786005834669,11.861990807677)
    11.0654528984116 = axial(164.786005834669,11.861990807677)
    5.53272644920579 = 11.0654528984116 / 2
    2.7663632246029 = 11.0654528984116 / 4

    22.2270434142659 = beat(5116.67146563023,22.1309057968232)
    11.113521707133 = 22.2270434142659 / 2
    5.55676085356648 = 22.2270434142659 / 4
    44.0711924742534 = harmean(5116.67146563023,22.1309057968232)
    22.0355962371267 = axial(5116.67146563023,22.1309057968232)
    11.0177981185634 = 22.0355962371267 / 2
    5.50889905928168 = 22.0355962371267 / 4

    85.1348300394896 = slip(22.2270434142659,12.7821002221242)
    85.1348300394896 = slip(12.7821002221242,11.113521707133)
    42.5674150197448 = slip(12.7821002221242,5.55676085356648)

    79.8222705138594 = slip(22.0355962371267,12.7821002221242)
    79.8222705138594 = slip(12.7821002221242,11.0177981185634)
    39.9111352569297 = slip(12.7821002221242,5.50889905928168)

    note well:
    555.982859223482 = slip(85.1348300394896,22.1309057968232)
    556 = 378 + 178

    378.378118042886 = slip(85.1348300394896,2.7663632246029)
    378.378 = 0.378+378 = 378*(1001/1000)

    326.007703249488 = slip(79.8222705138594,12.7821002221242)
    163.003851624744 = slip(79.8222705138594,6.39105011106212)

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s