Ninetnigh

Let us imagine a world where the Earth has a second moon besides the Moon, in a 2 : 3 resonance.

The sidereal period of the second moon is two thirds of the sidereal period of the Moon, the synodic period is obtained by subtracting the frequency of the annual cycle from the frequency of the lunar cycle. The exact length of the year and month varies, for the present time, the rounded values 365.2422 and 29.5306 days are usually used. We will use the value 365.2421875 (= 365 + 1/4 - 1/128) for the year length. .

period	sidereal	synodic
month	27.3216	29.5306
ninetnight	18.2144	19.1704

These values were calculated as follows:

<script>
year = 365.25 - 1/128;
month = 29.5306;
sid_month = 1 / (1 / month + 1 / year) 
sid_twentnight = sid_month * 2 / 3; 
twentnight = 1 / (1 / sid_twentnight - 1 / year) 
document.write(year
 + "<br/>\n" + sid_month
 + " " + sid_twentnight
 + "<br/>\n" + month
 + " " + twentnight
 );
</script>

which gives:
365.2421875
27.321591770525725 18.214394513683818
29.5306 19.17041064324263