The Metonic cycle ‘error’ – or not?

Posted: April 3, 2020 by oldbrew in Cycles, Maths, moon, solar system dynamics

Encylopaedia Britannica on the Metonic cycle:

Metonic cycle, in chronology, a period of 19 years in which there are 235 lunations, or synodic months, after which the Moon’s phases recur on the same days of the solar year, or year of the seasons. The cycle was discovered by Meton (fl. 432 bc), an Athenian astronomer.

Calendar Wiki’s opening paragraphs on the Metonic cycle say:

The Metonic cycle or Enneadecaeteris in astronomy and calendar studies is a particular approximate common multiple of the year (specifically, the seasonal i.e. tropical year) and the synodic month. Nineteen tropical years differ from 235 synodic months by about 2 hours. The Metonic cycle’s error is one full day every 219 years, or 12.4 parts per million.

19 tropical years = 6939.602 days
235 synodic months = 6939.688 days

It is helpful to recognize that this is an approximation of reality.

The period of the Moon’s orbit around the Earth and the Earth’s orbit around the Sun (ignoring also exact definition of the year) are independent and have no known physical resonance. Examples of a real harmonic lock would be Mercury, with its 3:2 spin-orbit resonance or other orbital resonance.
– – –

Wiki: no known physical resonance

But is there really an error? If we look hard enough, such a ‘physical resonance’ is found at 6441 tropical years, which is 19*339 as the chart on the right taken from this Talkshop post shows.

Wiki: Nineteen tropical years differ from 235 synodic months by about 2 hours.

In 339 occurrences of 19 years, we have ‘339*235, minus 1’ synodic months i.e. 79664 SM. If we break down the one lost month:

1 SM = 29.530589 days = 708.73413 hours
708.73413 / 339 = 2.0906611 hours — which is ‘about 2 hours’, so our chart is accurate enough in that regard.
(Update: using Earth’s rotation period returns 2.085 hours, see my comment here.)

6441 tropical years(TY) is about a quarter of the current axial precession period of the Earth:
6441*4 = 25764 TY = 25763 sidereal years
Precession is believed to be 25770 years approx. and declining.
– – –
Fun with numbers: 79664 = 339*235,-1 or 766*104.
235+104 = 339

  1. Chaeremon says:

    Re: no known physical resonance (Earth ↔ Moon)

    From 1597 to 2099 all the Lunestices occur either ~90° or ~270° longitude in the ecliptic, that is during equinox (data points from JPL DE421).

    There are 3 Lunestice cycles which run parallel through the years: at 167½, 242 (4 Inex + 7 Saros) and 409½ years. These cycles sum up Lunestices (multiples of ½Saros±½m and ½Metonic±½m) to int years.

    Academic statistics can apparently not find periodic repetitions that are braided.

  2. oldbrew says:

    Time to ask why there are 8 moon phases perhaps 😎

  3. P.A.Semi says:

    If you take any disharmonic interval and multiply it by sufficiently large number, you get the beat again…
    Repeating something in 6441 years is not any harmony or resonance, but mere math…

  4. oldbrew says:

    P.A. – there’s more to it than that, see here…

    That post pre-dates the evections posts which start here:

    — synodic months plus tropical years = tropical months
    — synodic months minus tropical years = evections in longitude

    Also: 766 synodic months = 7 lunar apsidal cycles (see chart – 766*104 = 79664 and 7*104 = 728). Obviously 7 is not a large number.

    Another related post may appear soon.

    A comment re. the 6441 years:

  5. Chaeremon says:

    @oldbrew, Re: 766 lunations and factor 7

    From JPL DE421 data points, the temporal distance at the apsides is 766.06 lunations.

    A better one with 7 as divisor is 1750 (opposite semesters, opposite apsides, opposite declination), equates 7½ LNC and 16.0 LAC …

    There is also cycle with another “root” beat running in parallel, can perhaps be nailed sometimes (looking for ideas).

  6. oldbrew says:

    Chaeremon – the chart is explained here:

    The numbers are self-consistent, I’d say. They represent long term means of course, as single occurrences will vary +/- to some degree due to the Moon’s orbital behaviour.

  7. Chaswarnertoo says:

    Chaeremon: Given the complexity and age of the solar system would you not expect any lunar resonance to be braided? Just asking, not an expert.

  8. oldbrew says:

    The main points I was going to put in another post were mostly covered by Ian Wilson’s initial comments (‘astroclimateconnection’) here:

    Basically, 8 evections-in-longitude = 8 + (1/Phi) lunations. Given that the moon has 8 phases, the number of phases per evection would also be 8 + (1/Phi).

    That was in response to the post, which showed that the full moon cycle = 8 evections-in-longitude * Phi:
    411.78443 days (FMC) / 29.530589 = 13.944335
    Subtract 1 for evections = 12.944335
    12.944335 / 8 = 1.6180418 (Phi = ~1.618034)
    – – –
    Calendar Wiki goes for 767 synodic months, versus our 766:

    However the logic of all the periods together points to 766 as it leads to exact numbers of tropical years and of Metonic cycles, amongst other things.

  9. oldbrew says:

    Chaeremon said: ‘There is also cycle with another “root” beat running in parallel, can perhaps be nailed sometimes (looking for ideas).’
    – – –
    I suggest the parallels are the anomalistic (as in the post above) and draconic, as here:

    Interestingly they are only 2 Metonic cycles apart, i.e. 339 to 337.

  10. Chaeremon says:

    @oldbrew, Re: self-consistent
    I take probes for at least half a century, and report the average (in astronomy that is possible IFF startdate and enddate are not hidden from the observer, they belong to the premises of the statement like the guts belong to the body). If you have better than that, and in the present case better than 766.06, then please show.

    @Chaswarnertoo, sure. But the specific kind of intertwingularity is always a mystery until “we” can find cause, see e.g. the idea of Newton and the absence of generator (i.e. energy does not raise / lower gravity).

  11. oldbrew says:

    Chaeremon – yes, if we multiply the numbers in the chart to get days they all come out within a day of the same total. That’s in 6441 tropical years using standard numbers for synodic, tropical and anomalistic months, tropical year and full moon cycle that can be found easily online.

    6441 TY * 365.24219 = 2352524.9 days
    79664 SM * 29.530589 = 2352524.8
    86105 TM * 27.321582 = 2352524.8
    85377 AM * 27.55455 = 2352524.8
    5713 FMC * 411.78443 = 2352524.4

    I see no reason to doubt the figures. Self-consistent here means (for example) the difference between the large numbers gives the smaller numbers:
    86105 – 79664 = 6441
    85377 – 79664 = 5713
    86105 – 85377 = 728

    All the numbers are either a multiple of 104 (SM and LAC) or 7 less than a multiple of 104, as described in the linked comment above.

    The chart doesn’t include the evections-in-longitude but the number is 73951 (79664 -5713), which is 7 *more* than a multiple of 104, because it’s the reverse of the anomalistic month, so to speak.
    73951 * 31.811941 = 2352524.8 days

  12. Chaeremon says:

    @oldbrew, Re: standard numbers

    “Standard” are only valid for the explicitly named epoch, e.g.


    Example from

    We (you and me) can hardly compare output unless you use astronomical standard (yes, this includes method and polynomials).

    For me this is easy, Solex does this all with numerical integration.

  13. Paul Vaughan says:

    The polynomials are total BS.

  14. Paul Vaughan says:

    Statistics and data differ fundamentally.

    Communication: FAILURE (like always).

  15. oldbrew says:

    Going back to the blog post:
    Nineteen tropical years differ from 235 synodic months by about 2 hours.

    339 (Metonic) * 2 hours = 678/24 days = 28.25 days
    That lies between the tropical and synodic month. The numbers in the chart show a ‘loss’ of 1 tropical and 1 synodic month, compared to multiples of 235 SM and 254 TM. So even though the chart is only a model, it does stay close to reality IMO.

  16. Chaeremon says:

    @oldbrew, Re: April 4, 2020 at 8:21 pm

    If you know the interval-end cannot occur on syzygy (line-up of Sun & Moon), then find a max sub-interval which does (observable syzygy occurs this earlier). In my work that is 1 year before expected interval-end. So the whole interval is expected to end about ~12.25 lunations later; this measurement norm can tell the earliness or lateliness of this 1 data point — observed against the lunar quadrature when it occults e.g. a planet (consider well which patch in the ecliptic shall show the interval start versus the interval end; I prefer the solstice patch and the equinox patch as norm). Examples:

    Herodot and his 11340 figure: taking 11340 as lunations, then 1 year later planet Mercury completes 3811 sidereal (incidental here equals 2893 synodic) orbits.

    Form Hipparchos 345 years: taking that as 4272 lunations (using k [Inex -2], k = 12), then 1 year later is completion of 216 (mod 8 = 0) synodic Venus orbits.

    From K. Lepsius cattle count, the “mountain sheep” section: taking that as 3152½ lunations, then 1 year later is completion of 184 sidereal Mars orbits.

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  18. Paul Vaughan says:

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    Observe “force drift” of draconic month with tropical year (polar motion, QBO, & Chandler wobble).

    ⌊ 0.999978614647502 / 0.0745030006844627 ⌉ = ⌊13.421991134057⌉ = 13
    0.999978614647502 / 13 = 0.0769214318959617
    i.e. harmonic of 0.999978614647502 nearest 0.0745030006844627 is 0.999978614647502 / 13 = 0.0769214318959617
    2.36966735541038 = (0.0769214318959617)*(0.0745030006844627) / (0.0769214318959617 – 0.0745030006844627)

    ⌊ 0.499989307323751 / 0.0372515003422313 ⌉ = ⌊13.421991134057⌉ = 13
    0.499989307323751 / 13 = 0.0384607159479808
    i.e. harmonic of 0.499989307323751 nearest 0.0372515003422313 is 0.499989307323751 / 13 = 0.0384607159479808
    1.18483367770519 = (0.0384607159479808)*(0.0372515003422313) / (0.0384607159479808 – 0.0372515003422313)

    ⌊ 0.999978614647502 / 0.0372515003422313 ⌉ = ⌊26.843982268114⌉ = 27
    0.999978614647502 / 27 = 0.0370362449869445
    i.e. harmonic of 0.999978614647502 nearest 0.0372515003422313 is 0.999978614647502 / 27 = 0.0370362449869445
    6.40939079526111 = (0.0372515003422313)*(0.0370362449869445) / (0.0372515003422313 – 0.0370362449869445)

    1.18483367770519 = (6.40939079526111)*(0.999978614647502) / (6.40939079526111 – 0.999978614647502)

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    ⌊ 5.99685290323073 / 0.999978614647502 ⌉ = ⌊5.99698115078656⌉ = 6
    5.99685290323073 / 6 = 0.999475483871788
    i.e. harmonic of 5.99685290323073 nearest 0.999978614647502 is 5.99685290323073 / 6 = 0.999475483871788
    1986.46983643257 = (0.999978614647502)*(0.999475483871788) / (0.999978614647502 – 0.999475483871788)

    ⌊ 1986.46983643257 / 19.8650360864628 ⌉ = ⌊99.9982999168204⌉ = 100

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  19. oldbrew says:

    (766 synodic months / tropical year) – (766 SM / full moon cycle) = 7 = no. of apsidal cycles in the period

    766 Saros = 223 * 7 apsidal cycles

  20. oldbrew says:

    Post: 1 SM = 29.530589 days = 708.73413 hours
    708.73413 / 339 = 2.0906611 hours — which is ‘about 2 hours’

    – – –
    Miles Mathis worked it out from the speeds of the Earth and Moon at 2.082 hours, so very close to our figure.

    That 2-hour gap is found at the end of a period of precession, and is the distance from precise conjunction. Therefore the gap is not defined by precession, but by normal orbital motion. The exact gap is 2.082 hours.
    – – –
    In fact if we use 23.9345 hours (rotation period) instead of 24 hours (day) as above:

    29.530589 days = 706.79988 hours
    706.79988 / 339 = 2.0849553 hours

    So that almost eliminates the difference from MM’s figure (2.085 – 2.082 hours = < 11 seconds).