An intriguing discovery is that the higher-resolution aspects of the SOI time-series (as illustrated by the Australian BOM 30-day SOI moving average) may also have a tidal influence. Note the fast noisy envelope that rides on top of the deep El Nino of 2015-2016 shown below:
Yet if we retain this in the 1880-present monthly ENSO model, by simultaneously isolating  the higher frequency fine structure from 2015-2017, the fine structure also emerges in the model. This is shown in the lower panel below.
This indicates that the differential equation being used currently can possibly be modified to include faster-responding derivative terms which will simultaneously show the multi-year fluctuations as well as what was thought to be a weekly-to-monthly-scale noise envelope. In fact, I had been convinced that this term was due to localized weather but a recent post suggested that this may indeed be a deterministic signal.
Lunisolar tidal effects likely do impact the ocean behavior at every known time-scale, from the well-characterized diurnal and semi-diurnal SLH tides to the long-term deep-ocean mixing proposed by Munk and Wunsch. It's not surprising that tidal forces would have an impact on the intermediate time-scale ENSO dynamics, both at the conventional low resolution (used for El Nino predictions) and at the higher-resolution that emerges from SOI measurements (the 30-day moving average shown above). Obviously, monthly and fortnightly oscillations observed in the SOI are commensurate with the standard lunar tides of periods 13-14 days and 27-28 days. And non-linear interactions may result in the 40-60 day oscillations observed in LOD.
It's entirely possible that removing the 30-day moving average on the SOI measurements can reveal even more detail/
 Isolation is accomplished by subtracting a 24-day average about the moving average value, which suppresses the longer-term SOI variation.