First Triumvirate Triumvir or tresvir: member of a college of three members. The expression is mostly used to describe the First Triumvirate (60 BCE; Pompey the Great, Crassus, and Julius Caesar) and Second Triumvirate (43 BCE; Marc Antony, Lepidus, and Octavian).
With 8 nights, I suggest you spend 3-4 in Tokyo, 1 in Hakone for ryokan, 3-4 in Kyoto. Hakone offers Mt. Fuji views on a clear day, and it sits right off the train line between Tokyo and Kyoto so it's an easy detour. While at Kyoto, spend a day at Nara.
October is one of the best month to travel Japan. Not too hot, not too cold. Average temperature in Tokyo is 17 celcious degree. October is not a busy season but if you travel from October 6th to 8th, it's better to book hotels/trains early. This is 3-day holidays in Japan and many people go out for vacation.