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Originally Posted by Robert Tulip
But why cannot you have a decimal in resonance? The fact is that every 179 years, Jupiter Saturn and Neptune are in the same relative positions, with error around 0.1%, at a point one twelfth of the way further around the sun. Because this is a function of the four biggest objects of our solar system (sun + 3 gas giants), it produces a readily visible pattern in the plot of the solar system barycentre. I don't understand why this is not classed as resonance.
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Because you have not shown any convincing evidence that it meets the gravitational dynamic criterion for a resonance. Please go back and read the opening paragraph.
I don't know how you got this link to read "wiki table of orbital resonances" in the final display. They call it "Coincidental 'near' ratios of mean motion", and that is how the URL reads in the raw editing window. They say that even the closest ones listed are dynamically insignificant and you have given no gravitationally based reason to think otherwise.
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putting in the numbers Tony gave of J:S:N = 15.094: 6.036 : 1.086 produces, as I calculate it, a mismatch after one cycle of only about 0.3° of arc for Jupiter's position relative to both Saturn and Neptune. This is much less than any of the listed binary mismatches, and has a very long randomisation time of around 100,000 years. Adding in Saturn-Neptune, the mismatch is about 1° of arc per cycle with randomisation period for the whole JSN group of about 30,000 years.
Jupiter-Saturn: 9 cycles = 178.65 years
Jupiter-Neptune: 14 cycles = 178.90 years
Saturn-Neptune: 5 cycles = 179.35 years
JS : JN : SN = 9 : 14 : 5. This relationship between the three largest planets is among the closest to an exact integer of all the resonances in the solar system.
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Close, but still no demonstration of the gravitational dynamics of a true resonance. You appear to be focused on numbers and pretty pictures, rather than on gravitational consequences, if any, of the positions that yield those numbers.