What is the mathematical term for the frequency as stops per cycle (similar to roots of unity)?

Question

Background:

For context, in discrete Fourier transforms $N$ samples in temporal space are transformed into $N$ frequency domain coefficients, each one corresponding to the dot product between the sampled signal and each $\exp\left(-i \frac{2\pi}{N}kn\right)$ with $n=0,\dots,N-1$ corresponding to the sampled points. These dot products are carried out independently for each bin number $k=\{0,\dots,N-1\}.$

Each one of these $\exp\left(-i \frac{2\pi}{N}\right)$ values being a root of unity, such that for every increasing $k$ index the angular frequency is increased by an equal amount so as to return to zero at $k=N-1.$ Each $k$ value contributes a harmonic to the fundamental frequency $\exp\left(-i \frac{2\pi}{N}\right),$ for $N=1.$

Each primitive root of unity $\exp\left(-i \frac{2\pi}{N}\right)$ determines $N$ points around the circle before returning to $0.$ There is no time involved.

Question:

What is the name of the frequency expressed as the number of stops or points equally spaced around the circle independent of time? So if an object moved around an axis and flashed $5$ times before completing one revolution, what would be the name of this frequency ($5 \text{ dots/cycle})$ or $\frac{2\pi}{5}\text{ rad/sample}$, which is independent of time? It is not angular frequency, which does depend on time, and it is a concept similar to the root of unity $\exp\left(-i \frac{2\pi}{5}\right).$

In other words, the "frequency term" I am looking for would give the exact same result ($5$) for both spiraling discrete dots around an axis in the picture below, despite one set of points being more elongated along the axis of rotation than the other

Answer 1

The digital frequency:

The digital frequency is cyclic with period $2\pi.$ The range for the digital frequency can be chosen to be any interval of length $2\pi:$

$$\begin{align} (-\pi,\pi] &\text{ or} [-\pi,\pi)\\ &[0,2\pi) \end{align}$$

the units are $\text{rad/sample}:$ $2\pi \frac{f_o}{F_s}\to \frac{\text{rad}}{\text{cycle}}\frac{\frac{\text{cycles}}{\text{sec}}}{\frac{\text{samples}}{\text{sec}}}$

I have also found it as radian frequency:

... we will omit mention of an explicit sampling interval $ T=1/f_s$, as this is most typical in the digital signal processing literature. It is often said that the sampling frequency is $ f_s=1$. In this case, a radian frequency $ \omega_k := 2\pi k/N$ is in units of $\text{radians / sample}.$ Elsewhere in this book, $ \omega_k$ usually means $\text{radians / second}.$ (Of course, there's no difference when the sampling rate is really $ 1$.) Another term we use in connection with the $ f_s=1$ convention is normalized frequency: All normalized radian frequencies lie in the range $ [-\pi,\pi)$, and all normalized frequencies in Hz lie in the range $ [-0.5,0.5).$ Note that physical units of seconds and Hertz can be reintroduced by the substitution $$\displaystyle e^{i 2\pi nk/N} = e^{i 2\pi k (f_s/N) nT} := e^{i \omega_k t_n}. $$

The related term normalized frequency is described in Wikipedia as:

Normalized frequency is a unit of measurement of frequency equivalent to $\text{cycles/sample}.$ In digital signal processing (DSP), the continuous time variable, $t,$ with units of seconds, is replaced by the discrete integer variable, $n,$ with units of samples. More precisely, the time variable, in seconds, has been normalized (divided) by the sampling interval, $T\text{ (seconds/sample),}$ which causes time to have convenient integer values at the moments of sampling. This practice is analogous to the concept of natural units, meaning that the natural unit of time in a DSP system is samples.

...

Sometimes, the unnormalized frequency is represented in units of $\text{radians/second}$ (angular frequency), and denoted by ${\displaystyle \textstyle \omega }$. When ${\displaystyle \textstyle \omega }$ is normalized by the sample-rate $(\text{samples/sec}),$ the resulting units are $\text{radians/sample}.$ The normalized Nyquist frequency is $\pi \text{ radians/sample},$ and the normalized sample-rate is $2\pi \text{ radians/sample}.$