The mod of an oscillator or spaceship is the smallest number of generations that it takes for it to reappear in its original form, possibly subject to some rotation or reflection.
The mod of a given oscillator may be equal to its period, but it may also be a quarter of the period (for oscillators that rotate 90 degrees every quarter period) or half the period (for other oscillators that rotate 180 degrees every half period, oscillators with 180 degree rotational symmetry that rotate 90 degrees every half period, and also for flippers).
The toad is an example of an oscillator with a mod identical to its period, as its two phases are completely distinct and one cannot be mapped to the other purely through rotation and translation (click above to open LifeViewer) RLE:herePlaintext:here
The clock is an example of an oscillator with a mod which is half its period, as both of its phases can clearly be turned into each other by a diagonal reflection (click above to open LifeViewer) RLE:herePlaintext:here
The phoenix 1 is another example of an oscillator with a mod which is half its period, as both of its phases are interchangeable through orthogonal mirroring (click above to open LifeViewer) RLE:herePlaintext:here
Sixty-nine is an example of an oscillator with a mod which is a quarter its period, as all four of its phases are identical and differ only by 90-degree rotation (click above to open LifeViewer) RLE:herePlaintext:here
On a hexagonal or triangular grid
The mod of an oscillator can be identical to its period, but it can also be half of its period (for oscillators which flip halfway through their evolution), a third of its period (for oscillators that rotate 120 degrees a third of the way through their evolution), or a sixth of its period (for oscillators that rotate 60 degrees a third of the way through their evolution).
Oscillators with a mod one sixth of their period must be centered around either a hexagonal cell or a triangular grid's vertex.
Other definitions
As the Conway's Game of Life community is a mathematical community, the term "mod" is sometimes used for its standard mathematical definition, such as "≡ 2 (mod 8)" for numbers that are 2 greater than a multiple of 8. This definition is not to be confused with the mod of a pattern.