Since long-period tidal forcing is essentially a multiplicative expansion of 3 fundamental lunar cycles and the annual cycle, I set up a combinatorial expansion of the 3 sin waves synced to an annual impulse.

![](https://imagizer.imageshack.com/img924/7915/l33Rsl.png)

The fit was a result of 126 binomially expanded terms, with some more important than others. The best way to rank their impact is from the power spectra:

![](https://imagizer.imageshack.com/img923/2899/zxFLZ2.png)

These follow essentially the same amplitude ranking as expected ... Mf is always strongest, Mm is next, with the first cross-term Mf' following. The 27.09 day ostensibly [evective](https://en.wikipedia.org/wiki/Evection) term is fascinating in that it is also a complementary satellite sideband of the Mm to Mf peak via the 8.85 year perigee cycle. So when the tropical sinusoid of 2*Mf=27.32 days is multiplied by the 8.85 year cycle, it will create both the Mm=27.55 day (prograde) cycle and the 27.09 (retrograde) cycle. And then the 2 satellite sidebands around Mm at 27.44 and 27.66 days are due to the 18.6 year nodal cycle. That's why there are 126 terms, instead of the 35 expected from just the annual + 3 lunar tidal expansion to the 4th power -- adding the longer cycles introduces the necessary satellite terms to the Mm factor

Compare against this which is obtained from the power spectrum of the Earth's length of day (LOD)

![](https://imagizer.imageshack.com/img924/7623/lo3vJB.png)

> "Similarly, the stabilized AR-z spectrum sees the two Mm tidal signals at 13.20 cpy (~27.67 days) and 13.48 cpy (~27.09 days) (see inset zoom-in) that are far from resolved in the Fourier spectrum. "
Application of Stabilized AR‐z Spectrum in Harmonic Analysis for Geophysics, Hao Ding, Benjamin F. Chao https://doi.org/10.1029/2018JB015890

Described at another level of detail in this blog post: https://geoenergymath.com/2020/08/02/combinatorial-tidal-constituents/