Confused about the period of a discretised time-series
On the continuous setting, the period is the minimum $T\ne 0$ such that $\theta(t+T)=\theta(t)$. Once you decide to have $N$ regularly sampled points inside a period, you introduce a sampling period $T_N$ that is directly proportional to the period: $T_N=T/N$. Hence, you can describe your period as a quantity with a time unit (second) or as a unitless number of samples, equivalently. 

Now, if for some reason you normalize the time axis with an affine function, the mean will only shift the whole axis (no change for the period), and the scaling only will dilate or shrink the interval between samples.

So here, the effect of  $s(1)$ will turn $T_N$ into $T_N/s(1)$, and also  $T$ into $T/s(1)$. 

In your case, you can directly compute $s(1)=T_N\sqrt{\frac{N(N+1)}{12}}$. In this formula, the scale factor depends on $N$ only, and not on the start and end indices.
For $N=100$, the new period should be now a unitless quantity of 
$$T/s(1) = \sqrt{\frac{12N}{N+1}}\,.$$