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).*