The details of interest here are:
Jupiter’s orbit period (J): 11.862615 years
Jupiter-Saturn conjunction period (J-S): 19.865036 years (mean value)
Solar Hale cycle (HC): ~22.14 years (estimated mean value)
Looking for a solar-planetary beat frequency (BF):
28 J = 332.15322 years
15 HC = 332.1 years
28 – 15 = 13 = number of beats in the period
Since Jupiter’s orbit period is a known value:
BF (approx.) = 332.15322 / 13 = 25.550247 y
Turning to J-S, consider the ‘Jose cycle‘ of 9 J-S:
9 J-S = 178.78532 y
7 BF = 178.85172 y
The value of BF using J-S known value:
178.78532 / 7 = 25.54076 y
Percent match of the two BF values = > 99.96%
Result – on the basis of the selected Hale cycle period:
Jupiter’s orbit matches the calculated solar-planetary beat frequency in the ratio 28:13.
For the Jose cycle (9 J-S conjunctions) the equivalent ratio is 9:7.
The Hale cycle ratio is 13 BF:15 HC by the above definitions.
Conclusion: this beat frequency connects the three items of interest as described.
(Note: 9 J-S figure updated 14/02/17 due to a typo).